{ "4686": { "status_id": 3, "type_id": 2, "question": "The average weekly pocket money of Jamie and Carrie is \\(\\$11\\). Jamie has \\(\\$4\\) less pocket money than Carrie. How much pocket money does Jamie get each week? Answer : $", "question_raw": "The average weekly pocket money of Jamie and Carrie is \\(\\$11\\).
Jamie has \\(\\$4\\) less pocket money than Carrie.
How much pocket money does Jamie get each week?
Answer : $ ", "question_image": null, "a": [ null, "9", null, null ], "correct_answer": 1, "solutions": [ { "id": 3657, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Q: The average weekly pocket money of Jamie and Carrie is 11 dollars. Jamie has \\(\\$4\\) less pocket money than Carrie. How much pocket money does Jamie get each week? Solution: Given: Average pocket money of Jamie and Carrie = \\(\\$11\\) Jamie has \\(\\$4\\) less than Carrie Step 1: Find total pocket money Total = Average × Number of people Total = 11 × 2 = 22 Step 2: Set up the relationship Let Carrie's pocket money = 1 unit Jamie's pocket money = 1 unit − 4 Step 3: Form equation Carrie + Jamie = 22 1 unit + (1 unit − 4) = 22 2 units − 4 = 22 2 units = 26 1 unit = 13 Step 4: Find Jamie's pocket money Carrie = 1 unit = \\(\\$13\\) Jamie = 13 − 4 = \\(\\$9\\) Step 5: Verify Jamie + Carrie = 9 + 13 = 22 ✓ Average = 22 ÷ 2 = 11 ✓ Difference = 13 − 9 = 4 ✓ Answer: Jamie gets \\(\\$9\\) each week" } ], "hints": [] }, "5454": { "status_id": 3, "type_id": 2, "question": "Ms Ying spent 30% of her money on a blouse. She spent 20% of the remaining money on a bag. The rest of her money was spent on a pair of shoes. Find the ratio of the amount of money spent on the blouse to the amount of money spent on the bag to the amount of money spent on the pair of shoes. Answer : : :", "question_raw": "Ms Ying spent 30% of her money on a blouse.
She spent 20% of the remaining money on a bag.
The rest of her money was spent on a pair of shoes.
Find the ratio of the amount of money spent on the blouse to the amount of money spent on the bag to the amount of money spent on the pair of shoes.
Answer : : : ", "question_image": null, "a": [ "15", "7", "28", null ], "correct_answer": 15, "solutions": [ { "id": 3858, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let total money = 100% Blouse = 30% Remaining after blouse = 100% − 30% = 70% Bag = 20% of 70% = 14% Shoes = 70% − 14% = 56% Ratio of blouse : bag : shoes = 30 : 14 : 56 = 15 : 7 : 28 The ratio is 15 : 7 : 28." } ], "hints": [] }, "5455": { "status_id": 3, "type_id": 2, "question": "Yen spent 30% of her money on a blouse. She spent 20% of the remaining money on a bag. If the bag cost \\(\\$48\\) less than the blouse, how much money did she have at first? Answer : $", "question_raw": "Yen spent 30% of her money on a blouse.
She spent 20% of the remaining money on a bag.
If the bag cost \\(\\$48\\) less than the blouse, how much money did she have at first?
Answer : $ ", "question_image": null, "a": [ "300", null, null, null ], "correct_answer": 300, "solutions": [ { "id": 3859, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let total money = 100% Blouse = 30% Remaining after blouse = 100% − 30% = 70% Bag = 20% of 70% = 14% Difference = 30% − 14% = 16% 16% = \\(\\$48\\) 1% = 48 ÷ 16 = \\(\\$3\\) 100% = 3 × 100 = \\(\\$300\\) She had \\(\\$300\\) at first." } ], "hints": [] }, "5466": { "status_id": 3, "type_id": 2, "question": "There are 12 fewer lions than elephants in a zoo. There are 25% more elephants than lions in a zoo. How many lions are there? Answer :", "question_raw": "There are 12 fewer lions than elephants in a zoo.
There are 25% more elephants than lions in a zoo.
How many lions are there?
Answer : ", "question_image": null, "a": [ null, "48", null, null ], "correct_answer": 1, "solutions": [ { "id": 3863, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let lions = 100% Elephants = 25% more than lions = 125% Difference = 125% − 100% = 25% 25% = 12 animals 1% = 12 ÷ 25 = 0.48 animals 100% = 0.48 × 100 = 48 lions There are 48 lions." } ], "hints": [] }, "5481": { "status_id": 3, "type_id": 2, "question": "At first, the number of boys and girls on a school bus were the same. Then 30% of the boys and 20% of the girls alighted. What was the new ratio of boys to girls? Answer = :", "question_raw": "At first, the number of boys and girls on a school bus were the same.
Then 30% of the boys and 20% of the girls alighted.
What was the new ratio of boys to girls?
Answer = : ", "question_image": null, "a": [ "7", "8", null, null ], "correct_answer": 7, "solutions": [ { "id": 3849, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let original number of boys = 100 Let original number of girls = 100 Boys who alighted = 30% of 100 = 30 Boys remaining = 100 − 30 = 70 Girls who alighted = 20% of 100 = 20 Girls remaining = 100 − 20 = 80 New ratio of boys to girls = 70 : 80 = 7 : 8 The new ratio of boys to girls was 7 : 8." } ], "hints": [] }, "5482": { "status_id": 3, "type_id": 2, "question": "At first, the number of boys and girls on a bus were the same. Then 30% of the boys and 20% of the girls alighted. If total number alighted was 10, how many pupils were on the bus at first? Answer :", "question_raw": "At first, the number of boys and girls on a bus were the same.
Then 30% of the boys and 20% of the girls alighted.
If total number alighted was 10, how many pupils were on the bus at first?
Answer : ", "question_image": null, "a": [ "40", null, null, null ], "correct_answer": 40, "solutions": [ { "id": 3869, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let original number of boys = 100% Let original number of girls = 100% Boys who alighted = 30% of boys = 30% Girls who alighted = 20% of girls = 20% Total who alighted = 30% + 20% = 50% (of one group) 50% = 10 pupils 1% = 10 ÷ 50 = 0.2 pupils 100% = 0.2 × 100 = 20 pupils (boys or girls) Total pupils at first = 20 + 20 = 40 pupils There were 40 pupils on the bus at first." } ], "hints": [] }, "5487": { "status_id": 3, "type_id": 2, "question": "Nancy had 20% as much money as Shirley. If Shirley gave Nancy \\(\\$280\\), they would have the same amount. How much money must Shirley give Nancy if she was to have twice as much as Nancy? Answer : $", "question_raw": "Nancy had 20% as much money as Shirley.
If Shirley gave Nancy \\(\\$280\\), they would have the same amount.
How much money must Shirley give Nancy if she was to have twice as much as Nancy?
Answer : $ ", "question_image": null, "a": [ "140", null, null, null ], "correct_answer": 140, "solutions": [ { "id": 3873, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Nancy = 20% of Shirley Let Shirley = 100% Then Nancy = 20% Total = 100% + 20% = 120% After Shirley gave Nancy \\(\\$280\\), they had the same amount. Each would have = 120% ÷ 2 = 60% Shirley gave away = 100% − 60% = 40% 40% = \\(\\$280\\) 1% = 280 ÷ 40 = \\(\\$7\\) Original amounts: Shirley = 100% = 100 × 7 = \\(\\$700\\) Nancy = 20% = 20 × 7 = \\(\\$140\\) Total = 700 + 140 = \\(\\$840\\) For Shirley to have twice as much as Nancy: Let Nancy's new amount = 1 unit Then Shirley's new amount = 2 units Total = 3 units = \\(\\$840\\) 1 unit = 840 ÷ 3 = \\(\\$280\\) Nancy's new amount = \\(\\$280\\) Shirley's new amount = \\(\\$560\\) Amount Shirley must give Nancy = 700 − 560 = \\(\\$140\\) Shirley must give Nancy \\(\\$140\\)." } ], "hints": [] }, "5488": { "status_id": 3, "type_id": 2, "question": "Mr Teo and Mrs Teo have some money each. If Mr Teo gives Mrs Teo \\(\\$25\\), they will have the same amount. If he gives Mrs Teo \\(\\$6\\), she will have 50% as much money as him. How much do they have in total? Answer : $", "question_raw": "Mr Teo and Mrs Teo have some money each.
If Mr Teo gives Mrs Teo \\(\\$25\\), they will have the same amount.
If he gives Mrs Teo \\(\\$6\\), she will have 50% as much money as him.
How much do they have in total?
Answer : $ ", "question_image": null, "a": [ "114", null, null, null ], "correct_answer": 114, "solutions": [ { "id": 3874, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution 1 - Algebraic solution If Mr Teo gives \\(\\$25\\), they have the same amount. This means Mr Teo has \\(\\$50\\) more than Mrs Teo. Let Mrs Teo's original amount = x Then Mr Teo's original amount = x + 50 If Mr Teo gives \\(\\$6\\): Mr Teo has = x + 50 − 6 = x + 44 Mrs Teo has = x + 6 Mrs Teo will have 50% as much as Mr Teo: x + 6 = 50% of (x + 44) x + 6 = 0.5(x + 44) x + 6 = 0.5x + 22 x − 0.5x = 22 − 6 0.5x = 16 x = 32 Mrs Teo's original amount = \\(\\$32\\) Mr Teo's original amount = 32 + 50 = \\(\\$82\\) Total = 32 + 82 = \\(\\$114\\) They have \\(\\$114\\) in total. Solution 2 : Non-Algebraic Solution If Mr Teo gives \\(\\$25\\), they have the same amount. This means Mr Teo has \\(\\$50\\) more than Mrs Teo. If Mr Teo gives \\(\\$6\\): Mr Teo still has 50 − 6 − 6 = \\(\\$38\\) more than Mrs Teo. At this point, Mrs Teo has 50% as much as Mr Teo. This means Mr Teo has twice as much as Mrs Teo. Difference = 2 units − 1 unit = 1 unit = \\(\\$38\\) Mrs Teo after 6 dollars transfer = 1 unit = \\(\\$38\\) Mrs Teo original = 38 − 6 = \\(\\$32\\) Mr Teo original = 32 + 50 = \\(\\$82\\) Total = 32 + 82 = \\(\\$114\\) They have \\(\\$114\\) in total." } ], "hints": [] }, "5490": { "status_id": 3, "type_id": 2, "question": "A fruit seller had 10% more oranges than apples. After selling 20 apples and 48 oranges, he had 90% as many oranges as apples. Find the number of oranges he had in the end. Answer :", "question_raw": "A fruit seller had 10% more oranges than apples.
After selling 20 apples and 48 oranges, he had 90% as many oranges as apples.
Find the number of oranges he had in the end.
Answer : ", "question_image": null, "a": [ "117", null, null, null ], "correct_answer": 117, "solutions": [ { "id": 3876, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution 1: Algebraic solution Let apples = 100% Then oranges = 110% After selling: Apples remaining = 100% − 20 Oranges remaining = 110% − 48 Oranges remaining = 90% of apples remaining 110% − 48 = 90% of (100% − 20) 110% − 48 = 0.9 × (100% − 20) Let 100% = x (original apples) 1.1x − 48 = 0.9(x − 20) 1.1x − 48 = 0.9x − 18 1.1x − 0.9x = −18 + 48 0.2x = 30 x = 150 Original apples = 150 Original oranges = 110% × 150 = 165 Oranges in the end = 165 − 48 = 117 He had 117 oranges in the end. Solution 2 : Non-Algebraic Solution Let original apples = 10 units Then original oranges = 11 units (10% more) After selling: Apples remaining = 10 units − 20 Oranges remaining = 11 units − 48 Oranges remaining = 90% of apples remaining Oranges remaining : Apples remaining = 90 : 100 = 9 : 10 If apples remaining = 10 parts, oranges remaining = 9 parts Difference = 10 parts − 9 parts = 1 part Actual difference in remaining: (10 units − 20) − (11 units − 48) = 10 units − 20 − 11 units + 48 = 48 − 20 − 1 unit = 28 − 1 unit This difference = 1 part So: 1 part = 28 − 1 unit Also, apples remaining = 10 units − 20 = 10 parts So: 10 parts = 10 units − 20 1 part = 1 unit − 2 From 1 part = 28 − 1 unit: 1 unit − 2 = 28 − 1 unit 2 units = 30 1 unit = 15 Original oranges = 11 units = 11 × 15 = 165 Oranges in the end = 165 − 48 = 117 He had 117 oranges in the end. --- **Verification:** Original apples = 150, Original oranges = 165 ✓ (10% more) Apples remaining = 150 − 20 = 130 Oranges remaining = 165 − 48 = 117 117 ÷ 130 = 0.9 = 90% ✓" } ], "hints": [] }, "5654": { "status_id": 3, "type_id": 2, "question": "The figure shows two identical quadrants overlapping in a square of length 14 cm. Find the area of the shaded part. (Take π = 22\/7) Answer : cm²", "question_raw": "The figure shows two identical quadrants overlapping in a square of length 14 cm.
Find the area of the shaded part. (Take π = 22\/7)
Answer : cm²", "question_image": "questions\/405892dc6bbaa2c6.webp", "a": [ "112", null, null, null ], "correct_answer": 112, "solutions": [ { "id": 4019, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "``` Let the side of the square = s = 14 cm Area of the leaf = 2 × Quadrant − Square Each quadrant has radius = s = 14 cm Area of one quadrant = ¼ × π × r² = ¼ × 22\/7 × 14² = ¼ × 22\/7 × 196 = ¼ × 616 = 154 cm² Area of two quadrants = 2 × 154 = 308 cm² Area of square = 14 × 14 = 196 cm² Area of shaded leaf = 308 − 196 = 112 cm² ```" } ], "hints": [] }, "6650": { "status_id": 3, "type_id": 1, "question": "A box measures 40 cm in length and 15 cm in width. It is filled with sand to a height of 5 cm. When the sand is moved to a new box that is 10 cm tall, it fills the new box to $$\\dfrac{1}{3}$$ of its height. What is the base area of the new box?", "question_raw": "A box measures 40 cm in length and 15 cm in width.
It is filled with sand to a height of 5 cm.
When the sand is moved to a new box that is 10 cm tall, it fills the new box to $$\\dfrac{1}{3}$$ of its height.
What is the base area of the new box?", "question_image": null, "a": [ "600 $cm^2$", "500 $cm^2$", "400 $cm^2$", "300 $cm^2$" ], "correct_answer": 1, "solutions": [ { "id": 246, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Calculate the volume of the first box: 40 cm × 15 cm × 5 cm = 3000 cm³. 2. The new box is filled to \\(\\dfrac{1}{3}\\) of its height, so its volume is 3000 cm³. 3. Let the base area of the new box be A. Then, A × 10 cm = 3000 cm³. 4. Solve for A: A = 3000 cm³ ÷ 10 cm = 300 cm². Final Answer: 300" } ], "hints": [ { "id": 741, "lvl": 1, "u": 41, "t": "Think about how to find the volume of the first box." }, { "id": 742, "lvl": 2, "u": 41, "t": "Use the volume to find the area of the new box." }, { "id": 743, "lvl": 3, "u": 41, "t": "Remember that volume = base area × height." } ] }, "6651": { "status_id": 3, "type_id": 1, "question": "A fish tank is 25 cm long and 20 cm wide. It holds water up to a height of 8 cm. When the water is poured into another tank that is 16 cm tall, it fills it to half its height. What is the base area of the second tank?", "question_raw": "A fish tank is 25 cm long and 20 cm wide.
It holds water up to a height of 8 cm.
When the water is poured into another tank that is 16 cm tall, it fills it to half its height.
What is the base area of the second tank?", "question_image": null, "a": [ "500 $cm^2$", "400 $cm^2$", "300 $cm^2$", "600 $cm^2$" ], "correct_answer": 1, "solutions": [ { "id": 247, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Volume of the first tank: 25 cm × 20 cm × 8 cm = 4000 cm³. 2. The second tank is filled to half its height, so its volume is 4000 cm³. 3. Let the base area of the second tank be A. Then, A × 16 cm = 4000 cm³. 4. Solve for A: A = 4000 cm³ ÷ 16 cm = 250 cm². Final Answer: 250" } ], "hints": [ { "id": 744, "lvl": 1, "u": 41, "t": "Calculate the volume of the first tank." }, { "id": 745, "lvl": 2, "u": 41, "t": "Use the volume to find the area of the second tank." }, { "id": 746, "lvl": 3, "u": 41, "t": "Volume = base area × height." } ] }, "6652": { "status_id": 3, "type_id": 1, "question": "A storage box is 50 cm long and 30 cm wide. It is filled with toys to a height of 10 cm. When the toys are moved to a new box that is 20 cm tall, it fills to a quarter of its height. What is the base area of the new box?", "question_raw": "A storage box is 50 cm long and 30 cm wide.
It is filled with toys to a height of 10 cm.
When the toys are moved to a new box that is 20 cm tall, it fills to a quarter of its height.
What is the base area of the new box?", "question_image": null, "a": [ "600 $cm^2$", "500 $cm^2$", "400 $cm^2$", "300 $cm^2$" ], "correct_answer": 2, "solutions": [ { "id": 248, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Calculate the volume of the first box: 50 cm × 30 cm × 10 cm = 15000 cm³. 2. The new box is filled to a quarter of its height, so its volume is 15000 cm³. 3. Let the base area of the new box be A. Then, A × 20 cm = 15000 cm³. 4. Solve for A: A = 15000 cm³ ÷ 20 cm = 750 cm². Final Answer: 750" } ], "hints": [ { "id": 747, "lvl": 1, "u": 41, "t": "Find the volume of the first box." }, { "id": 748, "lvl": 2, "u": 41, "t": "Use that volume to calculate the area of the new box." }, { "id": 749, "lvl": 3, "u": 41, "t": "Remember the formula for volume." } ] }, "6653": { "status_id": 3, "type_id": 1, "question": "A rectangular aquarium is 35 cm long and 25 cm wide. It has water up to a height of 7 cm. When the water is transferred to another aquarium that is 14 cm tall, it fills it to a third of its height. What is the base area of the second aquarium?", "question_raw": "A rectangular aquarium is 35 cm long and 25 cm wide.
It has water up to a height of 7 cm.
When the water is transferred to another aquarium that is 14 cm tall, it fills it to a third of its height.
What is the base area of the second aquarium?", "question_image": null, "a": [ "525 $cm^2$", "600 $cm^2$", "500 $cm^2$", "450 $cm^2$" ], "correct_answer": 0, "solutions": [ { "id": 249, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Volume of the first aquarium: 35 cm × 25 cm × 7 cm = 6125 cm³. 2. The second aquarium is filled to a third of its height, so its volume is 6125 cm³. 3. Let the base area of the second aquarium be A. Then, A × 14 cm = 6125 cm³. 4. Solve for A: A = 6125 cm³ ÷ 14 cm = 437.5 cm². Final Answer: 437.5" } ], "hints": [ { "id": 750, "lvl": 1, "u": 41, "t": "Calculate the volume of the first aquarium." }, { "id": 751, "lvl": 2, "u": 41, "t": "Use this volume to find the area of the second aquarium." }, { "id": 752, "lvl": 3, "u": 41, "t": "Remember that volume = base area × height." } ] }, "6654": { "status_id": 3, "type_id": 1, "question": "A container is 20 cm long and 10 cm wide. It is filled with liquid to a height of 4 cm. When the liquid is poured into a new container that is 8 cm tall, it fills to half its height. What is the base area of the new container?", "question_raw": "A container is 20 cm long and 10 cm wide.
It is filled with liquid to a height of 4 cm.
When the liquid is poured into a new container that is 8 cm tall, it fills to half its height.
What is the base area of the new container?", "question_image": null, "a": [ "100 $cm^2$", "80 $cm^2$", "60 $cm^2$", "120 $cm^2$" ], "correct_answer": 1, "solutions": [ { "id": 250, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Calculate the volume of the first container: 20 cm × 10 cm × 4 cm = 800 cm³. 2. The new container is filled to half its height, so its volume is 800 cm³. 3. Let the base area of the new container be A. Then, A × 8 cm = 800 cm³. 4. Solve for A: A = 800 cm³ ÷ 8 cm = 100 cm². Final Answer: 100" } ], "hints": [ { "id": 753, "lvl": 1, "u": 41, "t": "Find the volume of the first container." }, { "id": 754, "lvl": 2, "u": 41, "t": "Use that volume to find the area of the new container." }, { "id": 755, "lvl": 3, "u": 41, "t": "Remember the formula for volume." } ] }, "6813": { "status_id": 3, "type_id": 2, "question": "A class has an average score of 72. Two new students join with scores 60 and 58, and the average drops to 70. How many students were originally in the class? Answer :", "question_raw": "A class has an average score of 72. Two new students join with scores 60 and 58, and the average drops to 70.
How many students were originally in the class?
Answer : ", "question_image": null, "a": [ null, "11", null, null ], "correct_answer": 1, "solutions": [ { "id": 3670, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Q: A class has an average score of 72. Two new students join with scores 60 and 58, and the average drops to 70. How many students were originally in the class? Solution: Given: Original average = 72 Two new students' scores = 60 and 58 New average = 70 Step 1: Let original number of students = n Original total = 72 × n = 72n Step 2: Find total score of new students New students' total = 60 + 58 = 118 Step 3: Set up equation for new average New number of students = n + 2 New total = 72n + 118 New average = New total ÷ New number of students 70 = (72n + 118) ÷ (n + 2) Step 4: Solve the equation 70 × (n + 2) = 72n + 118 70n + 140 = 72n + 118 140 − 118 = 72n − 70n 22 = 2n n = 11 Step 5: Verify Original total = 72 × 11 = 792 New total = 792 + 118 = 910 New number of students = 11 + 2 = 13 New average = 910 ÷ 13 = 70 ✓ Answer: There were originally 11 students in the class" } ], "hints": [ { "id": 1071, "lvl": 1, "u": 41, "t": "Write the original total as 72×n." }, { "id": 1072, "lvl": 2, "u": 41, "t": "Add the two new scores to get the new total." }, { "id": 1073, "lvl": 3, "u": 41, "t": "Write the new average equation: \n(72n+60+58)\/(n+2) = 70." } ] }, "6892": { "status_id": 3, "type_id": 1, "question": "In a class, the ratio of boys to girls is 9:7. Of the boys, the ratio pass:fail is 2:1; of the girls, the ratio pass:fail is 3:1. How many times as large is the number of passing boys compared with the number of passing girls?", "question_raw": "In a class, the ratio of boys to girls is 9:7. Of the boys, the ratio pass:fail is 2:1; of the girls, the ratio pass:fail is 3:1.\r\nHow many times as large is the number of passing boys compared with the number of passing girls?", "question_image": null, "a": [ "7\/8", "8\/7", "9\/7", "7\/9" ], "correct_answer": 1, "solutions": [ { "id": 3887, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let boys = 9 units Let girls = 7 units **Boys who passed:** Pass : Fail = 2 : 1 Total parts = 2 + 1 = 3 parts Boys who passed = 2\/3 of 9 units = 6 units **Girls who passed:** Pass : Fail = 3 : 1 Total parts = 3 + 1 = 4 parts Girls who passed = 3\/4 of 7 units = 5.25 units **Comparison:** Passing boys ÷ Passing girls = 6 ÷ 5.25 = 6 ÷ 21\/4 = 6 × 4\/21 = 24\/21 = 8\/7 The number of passing boys is 8\/7 times as large as the number of passing girls. **Verification:** Boys = 9 units, Pass rate = 2\/3, Passing boys = 6 units ✓ Girls = 7 units, Pass rate = 3\/4, Passing girls = 5.25 units ✓ Ratio = 6 : 5.25 = 24 : 21 = 8 : 7 ✓" } ], "hints": [ { "id": 1218, "lvl": 1, "u": 41, "t": "Convert each subgroup’s pass:fail into a pass fraction, then multiply by the headcount ratio." }, { "id": 1219, "lvl": 2, "u": 41, "t": "Therefore, passing boys = 2\/3 * 9\/16" } ] }, "6893": { "status_id": 3, "type_id": 2, "question": "In a pet school, the ratio of cats to dogs is 5:3. Of the cats, the ratio of playful to sleepy is 4:1; of the dogs, the ratio of playful to sleepy is 2:1. What is the ratio of the number of playful cats to the number of playful dogs? Playful cats : playful dogs = :", "question_raw": "In a pet school, the ratio of cats to dogs is 5:3. Of the cats, the ratio of playful to sleepy is 4:1; of the dogs, the ratio of playful to sleepy is 2:1. What is the ratio of the number of playful cats to the number of playful dogs?
Playful cats : playful dogs = : ", "question_image": null, "a": [ "2", "1", null, null ], "correct_answer": 1, "solutions": [ { "id": 3888, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let cats = 5 units Let dogs = 3 units Playful cats = 4\/5 of 5 units = 4 units Playful dogs = 2\/3 of 3 units = 2 units Playful cats : Playful dogs = 4 : 2 = 2 : 1 The ratio of playful cats to playful dogs is 2 : 1." } ], "hints": [ { "id": 1221, "lvl": 1, "u": 41, "t": "Use the ratios to figure out the parts for cats and dogs." }, { "id": 1222, "lvl": 2, "u": 41, "t": "Calculate the ratio of playful cats to playful dogs." } ] }, "6897": { "status_id": 3, "type_id": 1, "question": "A multiplex has standard:premium screens = 8:3. Each screen has the same number of seats. Occupancy: standard 60%, premium 80%. Average ticket price - premium : standard =5:3. For one showtime, how many times as large is total premium revenue compared with total standard revenue?", "question_raw": "A multiplex has standard:premium screens = 8:3. Each screen has the same number of seats. Occupancy: standard \n60%, premium 80%. Average ticket price - premium : standard =5:3. For one showtime, how many times as large is total premium revenue compared with total standard revenue?", "question_image": null, "a": [ "3\/4", "2\/4", "5\/6", "5\/7" ], "correct_answer": 2, "solutions": [ { "id": 392, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Step 1: Let seats\/screen = S, standard price = 3, premium price = 5. Step 2: Premium revenue = 3 x S x 0.8 x 5 = 12S; Standard revenue = 8 x S x 0.6 x 3 = 14.4S. Step 3: Factor = 12\/14.4 = 5\/6 Deatailed solution Let standard screens = 8 units Let premium screens = 3 units **Occupied seats:** Standard occupied = 60% of 8 units = 4.8 units Premium occupied = 80% of 3 units = 2.4 units **Ticket prices:** Let standard price = 3 parts Let premium price = 5 parts **Revenue:** Standard revenue = 4.8 units × 3 parts = 14.4 Premium revenue = 2.4 units × 5 parts = 12 Premium revenue ÷ Standard revenue = 12 ÷ 14.4 = 12\/14.4 = 5\/6 The total premium revenue is 5\/6 times as large as the total standard revenue. --- **Note:** Premium revenue is actually smaller than standard revenue. Standard revenue is 1.2 times (or 6\/5 times) as large as premium revenue." } ], "hints": [ { "id": 1233, "lvl": 1, "u": 41, "t": "Revenue = (#screens)×(seats per screen)×(occupancy)×(price). Let S be #screens" } ] }, "6898": { "status_id": 3, "type_id": 1, "question": "A bakery bakes cookies and cakes in the ratio of 5:2 daily. 70% of the cookies and 90% of the cakes were sold last Sunday. The price ratio of cakes to cookies is 4:3. How many times larger was the total cake revenue compared to total cookie revenue?", "question_raw": "A bakery bakes cookies and cakes in the ratio of 5:2 daily. 70% of the cookies and 90% of the cakes were sold last Sunday. The price ratio of cakes to cookies is 4:3. How many times larger was the total cake revenue compared to total cookie revenue?", "question_image": null, "a": [ "22\/35", "24\/45", "24\/35", "28\/45" ], "correct_answer": 2, "solutions": [ { "id": 393, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Step 1: Let baked cookies = 5n, baked cakes = 2n; Step 2: Sunday sales, cookies sold = 0.7 x 5n = 3.5n, cakes sold = 0.9 x 2n = 1.8n; Step 3: Cookies revenue = 3.5n x 3 = 10.5n, cakes revenue = 1.8n x 4 = 7.2n Step 4: Factor = 7.2n\/10.5n = 24\/35" } ], "hints": [ { "id": 1234, "lvl": 1, "u": 41, "t": "Look at the ratios for cookies and cakes." }, { "id": 1235, "lvl": 2, "u": 41, "t": "Calculate the revenue for each type." }, { "id": 1236, "lvl": 3, "u": 41, "t": "Compare the revenues to find the ratio." } ] }, "6901": { "status_id": 3, "type_id": 1, "question": "A pet shop has cats and dogs in the ratio of 6:5. The adoption rate is 80% for cats and 70% for dogs. The price ratio for dogs to cats is 3:4. For one month, how many times larger is the total dog revenue compared to total cat revenue?", "question_raw": "A pet shop has cats and dogs in the ratio of 6:5. The adoption rate is 80% for cats and 70% for dogs. The price ratio for dogs to cats is 3:4. For one month, how many times larger is the total dog revenue compared to total cat revenue?", "question_image": null, "a": [ "35\/64", "36\/74", "7\/8", "38\/64" ], "correct_answer": 0, "solutions": [ { "id": 3889, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution: Let cats = 6 units Let dogs = 5 units Pets adopted: Cats adopted = 80% of 6 units = 4.8 units Dogs adopted = 70% of 5 units = 3.5 units Prices: Let cat price = 4 parts Let dog price = 3 parts Revenue: Cat revenue = 4.8 units × 4 parts = 19.2 Dog revenue = 3.5 units × 3 parts = 10.5 Dog revenue ÷ Cat revenue = 10.5 ÷ 19.2 = 105\/192 = 35\/64 The total dog revenue is 35\/64 times as large as the total cat revenue. Note: Dog revenue is actually smaller than cat revenue. Cat revenue is 64\/35 times (or 1 29\/35 times) as large as dog revenue." } ], "hints": [ { "id": 1260, "lvl": 1, "u": 41, "t": "Revenue of pet = Number of pets at shop x adoption rate x price" } ] }, "6902": { "status_id": 3, "type_id": 1, "question": "A movie theater has regular and VIP seats in the ratio of 5:2. The occupancy is 60% for regular and 80% for VIP. The average ticket price ratio for VIP to regular is 5:2. For one screening, how many times larger is the total VIP revenue compared to total regular revenue?", "question_raw": "A movie theater has regular and VIP seats in the ratio of 5:2. The occupancy is 60% for regular and 80% for VIP. The average ticket price ratio for VIP to regular is 5:2. For one screening, how many times larger is the total VIP revenue compared to total regular revenue?", "question_image": null, "a": [ "5\/6", "6\/7", "4\/3", "8\/9" ], "correct_answer": 2, "solutions": [ { "id": 3890, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution: Let regular seats = 5 units Let VIP seats = 2 units Occupied seats: Regular occupied = 60% of 5 units = 3 units VIP occupied = 80% of 2 units = 1.6 units Prices: Let regular price = 2 parts Let VIP price = 5 parts Revenue: Regular revenue = 3 units × 2 parts = 6 VIP revenue = 1.6 units × 5 parts = 8 VIP revenue ÷ Regular revenue = 8 ÷ 6 = 4\/3 = 1 1\/3 The total VIP revenue is 1 1\/3 times as large as the total regular revenue." } ], "hints": [ { "id": 1246, "lvl": 1, "u": 41, "t": "Check the seating ratios." }, { "id": 1247, "lvl": 2, "u": 41, "t": "Calculate how many seats were filled." }, { "id": 1248, "lvl": 3, "u": 41, "t": "Find the revenue for each type and compare." } ] }, "6903": { "status_id": 3, "type_id": 1, "question": "On a commuter route, the ratio of cars to bikes is 5:2. Average occupants per car:bike = 3:1. The base route length is 𝐷 km. All cars take the base route; 40% of bikes take a detour of length 1.5D, the rest take D. How many times as large are total bike passenger-km compared with total car passenger-km?", "question_raw": "On a commuter route, the ratio of cars to bikes is 5:2. Average occupants per car:bike = 3:1. The base route length is \r\n𝐷 km. All cars take the base route; 40% of bikes take a detour of length 1.5D, the rest take D. How many times as large are total bike passenger-km compared with total car passenger-km?", "question_image": null, "a": [ "0.20", "0.25", "0.16", "0.18" ], "correct_answer": 2, "solutions": [ { "id": 387, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Step 1: Let vehicle units be 5 cars and 2 bikes. Step 2: Car passenger-km = 5 x 3 x D = 15D. Step 3: Bike average distance =0.4 x 1.5D + 0.6 x D = 0.6D + 0.6D = 1.2D. Step 4: Bike passenger-km = 2 x 1 x 1.2D = 2.4D. Step 5: Factor (bikes vs cars) = 2.4D \/ 15D = 0.16" } ], "hints": [ { "id": 1249, "lvl": 1, "u": 41, "t": "Passenger-km = (vehicles) × (occupants\/vehicle) × (distance). Average the bikes’ distance using the 40\/60 split." } ] }, "6907": { "status_id": 3, "type_id": 1, "question": "At a university, the ratio of CS professors to Math professors is 7:5. Of CS professors, the ratio of active publishers : inactive is 5:2; of Math professors, it is 4:1. An active CS professor publishes on average 6 papers per year, an active Math professor 5 per year; inactive publish 0.", "question_raw": "At a university, the ratio of CS professors to Math professors is 7:5. Of CS professors, the ratio of active publishers : inactive is 5:2; of Math professors, it is 4:1. An active CS professor publishes on average 6 papers per year, an active Math professor 5 per year; inactive publish 0.", "question_image": null, "a": [ "4\/3", "3\/2", "2\/3", "5\/3" ], "correct_answer": 1, "solutions": [ { "id": 391, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Step 1: CS active fraction = 5\/7; Math active fraction = 4\/5; Step 2: Let common prof unit = 1. Then CS total = 7 x 5\/7 x 6 = 30; Math total = 5 x 4\/5 x 5 = 20; Step 3: Factor = 30\/20 = 3\/2" } ], "hints": [ { "id": 1259, "lvl": 1, "u": 41, "t": "Total output = (headcount) x (active fraction) × (per-active rate)." } ] }, "6928": { "status_id": 3, "type_id": 1, "question": "A cuboid has its length increased by 25%, its width decreased by 15%, and its height changed by s%. The volume of the cuboid goes down by 5%. What is s, rounded to two decimal places?", "question_raw": "A cuboid has its length increased by 25%, its width decreased by 15%, and its height changed by s%. The volume of the cuboid goes down by 5%. What is s, rounded to two decimal places?", "question_image": null, "a": [ "- 10.25%", "- 9.75%", "- 10.50%", "- 10.59%" ], "correct_answer": 3, "solutions": [ { "id": 408, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Let the original dimensions be L, W, H. 2. New length = 1.25L, New width = 0.85W, New height = (1 + s\/100)H. 3. Set up the equation for volume decrease: (1.25 * 0.85 * (1 + s\/100)) = 0.95 * (L * W * H). 4. Solve for s to find the percentage change. Final Answer: -10.59%" } ], "hints": [ { "id": 1304, "lvl": 1, "u": 41, "t": "Think about how volume changes with dimensions." }, { "id": 1305, "lvl": 2, "u": 41, "t": "Calculate the new volume using the changes." }, { "id": 1306, "lvl": 3, "u": 41, "t": "Set up an equation to find the missing percentage." } ] }, "6929": { "status_id": 3, "type_id": 1, "question": "A cuboid's length is increased by 30%, its width is decreased by 20%, and its height is changed by t%. The volume decreases by 6%. What is t, rounded to two decimal places?", "question_raw": "A cuboid's length is increased by 30%, its width is decreased by 20%, and its height is changed by t%. The volume decreases by 6%. What is t, rounded to two decimal places?", "question_image": null, "a": [ "- 9.66%", "- 9.62%", "- 10.50%", "- 11.50%" ], "correct_answer": 1, "solutions": [ { "id": 409, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Let the original dimensions be L, W, H. 2. New length = 1.30L, New width = 0.80W, New height = (1 + t\/100)H. 3. Set up the equation for volume decrease: L * W * H * (1.30 * 0.80 * (1 + t\/100)) = 0.94 * (L * W * H). 4. Solve for t to find the percentage change. Final Answer: -9.62%" } ], "hints": [ { "id": 1307, "lvl": 1, "u": 41, "t": "Consider how each dimension affects volume." }, { "id": 1308, "lvl": 2, "u": 41, "t": "Calculate the new volume based on the changes." }, { "id": 1309, "lvl": 3, "u": 41, "t": "Use the volume decrease to find the height change." } ] }, "6930": { "status_id": 3, "type_id": 1, "question": "A cuboid's length is reduced by 15%, its width is increased by 25%, and its height is changed by u%. The volume decreases by 3%. What is u, rounded to two decimal places?", "question_raw": "A cuboid's length is reduced by 15%, its width is increased by 25%, and its height is changed by u%. The volume decreases by 3%. What is u, rounded to two decimal places?", "question_image": null, "a": [ "- 9.00%", "- 8.50%", "- 8.71%", "- 8.61%" ], "correct_answer": 2, "solutions": [ { "id": 410, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: standard Steps: 1. Let the original dimensions be L, W, H. 2. New length = 0.85L, New width = 1.25W, New height = (1 + u\/100)H. 3. Set up the equation for volume decrease: (0.85 * 1.25 * (1 + u\/100) * L * W * H) = 0.97 * (L * W * H). 4. Solve for u to find the percentage change. Final Answer: -8.71%" } ], "hints": [ { "id": 1310, "lvl": 1, "u": 41, "t": "Think about how the volume is affected by each dimension." }, { "id": 1311, "lvl": 2, "u": 41, "t": "Calculate the new volume using the percentage changes." }, { "id": 1312, "lvl": 3, "u": 41, "t": "Set up an equation to find the height change percentage." } ] }, "7611": { "status_id": 3, "type_id": 1, "question": "A shopkeeper raises the price of an item by 10% and then reduces it by 10%. What is the overall percentage change?", "question_raw": "A shopkeeper raises the price of an item by 10% and then reduces it by 10%. What is the overall percentage change?", "question_image": null, "a": [ "10% decrease from original price", "5% decrease from original price", "No change from original price", "Decrease of 1% from original price" ], "correct_answer": 3, "solutions": [ { "id": 3879, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let original price = 100 After 10% increase: 100% + 10% = 110% Price = 110% × 100 = 110 After 10% decrease: 100% − 10% = 90% of 110 Price = 90% × 110 = 99 Change = 100 − 99 = 1 Percentage change = (1 ÷ 100) × 100% = 1% decrease The overall percentage change is a 1% decrease." } ], "hints": [ { "id": 2731, "lvl": 1, "u": 41, "t": "Let original price be P." }, { "id": 2732, "lvl": 2, "u": 41, "t": "1.1P x 0.9 = 0.99P. Final price is 99% of P, the original price." } ] }, "7615": { "status_id": 3, "type_id": 2, "question": "In a school, the ratio of boys to girls is 3:2. Of the boys, 20% join the chess club, and of the girls, 30% join. What percentage of the whole school joins the chess club? Answer : %", "question_raw": "In a school, the ratio of boys to girls is 3:2. Of the boys, 20% join the chess club, and of the girls, 30% join.
What percentage of the whole school joins the chess club?
Answer : %", "question_image": null, "a": [ "24", null, null, null ], "correct_answer": 1, "solutions": [ { "id": 3883, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let boys = 3 units Let girls = 2 units Total students = 3 + 2 = 5 units Boys who join chess club = 20% of 3 units = 0.6 units Girls who join chess club = 30% of 2 units = 0.6 units Total who join chess club = 0.6 + 0.6 = 1.2 units Percentage of whole school = (1.2 ÷ 5) × 100% = 24% 24% of the whole school joins the chess club." } ], "hints": [] }, "7617": { "status_id": 3, "type_id": 1, "question": "Two shops, Shop 1 and Shop 2, sell items in ratio 2:3 (same-priced items). Return rates: 5% at Shop 1, 8% at Shop 2. What percentage of all sold items were returned?", "question_raw": "Two shops, Shop 1 and Shop 2, sell items in ratio 2:3 (same-priced items). Return rates: 5% at Shop 1, 8% at Shop 2. What percentage of all sold items were returned?", "question_image": null, "a": [ "7.2", "6.8", "7.1", "7.3" ], "correct_answer": 1, "solutions": [ { "id": 3885, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution: Shop 1 sales = 2 units Shop 2 sales = 3 units Total sales = 2 + 3 = 5 units Items returned from Shop 1 = 5% of 2 units = 0.1 units Items returned from Shop 2 = 8% of 3 units = 0.24 units Total returned = 0.1 + 0.24 = 0.34 units Percentage returned = (0.34 ÷ 5) × 100% = 6.8% 6.8% of all sold items were returned. Alternative Solution Method (using numbers): Let Shop 1 sales = 200 items Let Shop 2 sales = 300 items Total sales = 500 items Items returned from Shop 1 = 5% × 200 = 10 items Items returned from Shop 2 = 8% × 300 = 24 items Total returned = 10 + 24 = 34 items Percentage returned = (34 ÷ 500) × 100% = 6.8% 6.8% of all sold items were returned. Verification: Shop 1: 200 items, 5% returned = 10 items ✓ Shop 2: 300 items, 8% returned = 24 items ✓ Total: 500 items, 34 returned = 6.8% ✓" } ], "hints": [] }, "7619": { "status_id": 3, "type_id": 1, "question": "The ratio of Alice’s chocolates to Ben’s chocolates was 3 : 5. After Alice bought 12 more and Ben bought 8 more, their chocolates were in the ratio 2 : 3. How many chocolates did Alice have at first?", "question_raw": "The ratio of Alice’s chocolates to Ben’s chocolates was 3 : 5. After Alice bought 12 more and Ben bought 8 more, their chocolates were in the ratio 2 : 3. How many chocolates did Alice have at first?", "question_image": null, "a": [ "50", "60", "65", "70" ], "correct_answer": 1, "solutions": [ { "id": 885, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Step 1: Let Alice = 3x, Ben = 5x. Step 2: $$\\frac{3x + 12}{5x + 8}$$ = $$\\frac{2}{3}$$; Solve for x and find Alice's 3x chocolates." }, { "id": 3891, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Alternate solution Using assumption method: Let Alice at first = 3 units Let Ben at first = 5 units After buying: Alice = 3 units + 12 Ben = 5 units + 8 New ratio = 2 : 3 (3 units + 12) \/ (5 units + 8) = 2\/3 Cross multiply: 3(3 units + 12) = 2(5 units + 8) 9 units + 36 = 10 units + 16 36 − 16 = 10 units − 9 units 20 = 1 unit Alice at first = 3 units = 3 × 20 = 60 chocolates Alice had 60 chocolates at first. Verification: Alice at first = 60, Ben at first = 100 Ratio = 60 : 100 = 3 : 5 ✓ After buying: Alice = 60 + 12 = 72 Ben = 100 + 8 = 108 Ratio = 72 : 108 = 2 : 3 ✓" } ], "hints": [ { "id": 2733, "lvl": 1, "u": 41, "t": "Set up two equations for the two pairs of ratios before and after." } ] }, "7622": { "status_id": 3, "type_id": 1, "question": "Tank A and B contained water in the ratio 5 : 8. When 60 litres were added to A and 20 litres were removed from B, the ratio became 2 : 3. Find the amount of water in Tank A at first.", "question_raw": "Tank A and B contained water in the ratio 5 : 8. When 60 litres were added to A and 20 litres were removed from B, the ratio became 2 : 3. Find the amount of water in Tank A at first.", "question_image": null, "a": [ "1000 litres", "1050 litres", "1100 litres", "1150 litres" ], "correct_answer": 2, "solutions": [ { "id": 3894, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let Tank A at first = 5x Let Tank B at first = 8x After the changes: Tank A = 5x + 60 Tank B = 8x − 20 New ratio = 2 : 3 (5x + 60) \/ (8x − 20) = 2\/3 Cross multiply: 3(5x + 60) = 2(8x − 20) 15x + 180 = 16x − 40 180 + 40 = 16x − 15x 220 = x x = 220 Tank A at first = 5x = 5 × 220 = 1100 litres Tank A had 1100 litres at first." }, { "id": 3895, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Using Assumption Method: Assume A at first = 5 units Assume B at first = 8 units After changes: A = 5 units + 60 B = 8 units − 20 New ratio = 2 : 3 For every 2 parts in A, there are 3 parts in B. So: (5 units + 60) × 3 = (8 units − 20) × 2 15 units + 180 = 16 units − 40 180 + 40 = 16 units − 15 units 220 = 1 unit Tank A at first = 5 units = 5 × 220 = 1100 litres Tank A had 1100 litres at first." } ], "hints": [ { "id": 2736, "lvl": 1, "u": 41, "t": "Let A have 5x and B, 8x water" } ] }, "7623": { "status_id": 3, "type_id": 1, "question": "The ratio of boys to girls in a class was 7 : 5. After 6 boys left and 14 girls joined, the ratio became 1 : 1. How many pupils were there at first?", "question_raw": "The ratio of boys to girls in a class was 7 : 5. After 6 boys left and 14 girls joined, the ratio became 1 : 1. How many pupils were there at first?", "question_image": null, "a": [ "100", "110", "120", "130" ], "correct_answer": 2, "solutions": [ { "id": 3896, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Boys : Girls = 7 : 5 After: Boys : Girls = 1 : 1 Assume boys at first = 7 units Assume girls at first = 5 units After changes: Boys = 7 units − 6 Girls = 5 units + 14 New ratio = 1 : 1 (equal) So: 7 units − 6 = 5 units + 14 7 units − 5 units = 14 + 6 2 units = 20 1 unit = 10 Boys at first = 7 units = 7 × 10 = 70 Girls at first = 5 units = 5 × 10 = 50 Total pupils at first = 70 + 50 = 120 pupils There were 120 pupils at first." }, { "id": 3897, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let boys at first = 7x Let girls at first = 5x After changes: Boys = 7x − 6 Girls = 5x + 14 New ratio = 1 : 1 (7x − 6) \/ (5x + 14) = 1\/1 Cross multiply: 1(7x − 6) = 1(5x + 14) 7x − 6 = 5x + 14 7x − 5x = 14 + 6 2x = 20 x = 10 Boys at first = 7x = 7 × 10 = 70 Girls at first = 5x = 5 × 10 = 50 Total pupils at first = 70 + 50 = 120 pupils There were 120 pupils at first." } ], "hints": [ { "id": 2737, "lvl": 1, "u": 41, "t": "Relate both pairs of ratios by adjusting for the same base" } ] }, "7624": { "status_id": 3, "type_id": 1, "question": "A fruit seller mixed apples and oranges in the ratio 7 : 3. After selling 30 apples and adding 20 oranges, the ratio became 2 : 1. Find the number of apples he had at first.", "question_raw": "A fruit seller mixed apples and oranges in the ratio 7 : 3. After selling 30 apples and adding 20 oranges, the ratio became 2 : 1. Find the number of apples he had at first.", "question_image": null, "a": [ "500", "510", "490", "520" ], "correct_answer": 2, "solutions": [ { "id": 3898, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let apples at first = 7x Let oranges at first = 3x After changes: Apples = 7x − 30 Oranges = 3x + 20 New ratio = 2 : 1 (7x − 30) \/ (3x + 20) = 2\/1 Cross multiply: 1(7x − 30) = 2(3x + 20) 7x − 30 = 6x + 40 7x − 6x = 40 + 30 x = 70 Apples at first = 7x = 7 × 70 = 490 apples He had 490 apples at first." }, { "id": 3899, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Apples : Oranges = 7 : 3 After: Apples : Oranges = 2 : 1 Assume apples at first = 7 units Assume oranges at first = 3 units After changes: Apples = 7 units − 30 Oranges = 3 units + 20 New ratio = 2 : 1 So: (7 units − 30) × 1 = (3 units + 20) × 2 7 units − 30 = 6 units + 40 7 units − 6 units = 40 + 30 1 unit = 70 Apples at first = 7 units = 7 × 70 = 490 apples He had 490 apples at first." } ], "hints": [ { "id": 2738, "lvl": 1, "u": 41, "t": "Let apples = 7x and oranges = 3x" } ] }, "7625": { "status_id": 3, "type_id": 1, "question": "The ratio of the price of a watch to that of a pen was 5 : 2. After each was reduced by \\(\\$15\\), the ratio became 8 : 3. Find the original price of the watch.", "question_raw": "The ratio of the price of a watch to that of a pen was 5 : 2. After each was reduced by \\(\\$15\\), the ratio became 8 : 3. Find the original price of the watch.", "question_image": null, "a": [ "350", "375", "400", "425" ], "correct_answer": 1, "solutions": [ { "id": 3900, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let price of watch at first = 5x Let price of pen at first = 2x After reduction: Watch = 5x − 15 Pen = 2x − 15 New ratio = 8 : 3 (5x − 15) \/ (2x − 15) = 8\/3 Cross multiply: 3(5x − 15) = 8(2x − 15) 15x − 45 = 16x − 120 −45 + 120 = 16x − 15x 75 = x x = 75 Price of watch at first = 5x = 5 × 75 = \\(\\$375\\) The original price of the watch was \\(\\$375\\)." }, { "id": 3901, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Watch : Pen = 5 : 2 After: Watch : Pen = 8 : 3 Assume watch at first = 5 units Assume pen at first = 2 units After reduction: Watch = 5 units − 15 Pen = 2 units − 15 New ratio = 8 : 3 So: (5 units − 15) × 3 = (2 units − 15) × 8 15 units − 45 = 16 units − 120 −45 + 120 = 16 units − 15 units 75 = 1 unit Price of watch at first = 5 units = 5 × 75 = \\(\\$375\\) The original price of the watch was \\(\\$375\\)." } ], "hints": [ { "id": 2739, "lvl": 1, "u": 41, "t": "Let price of watch = 5x and pen, 2x" } ] }, "7626": { "status_id": 3, "type_id": 1, "question": "The ratio of Amy’s age to her mother’s was 2 : 7. Four years later, the ratio became 1 : 3. How old was Amy at first?", "question_raw": "The ratio of Amy’s age to her mother’s was 2 : 7. Four years later, the ratio became 1 : 3. How old was Amy at first?", "question_image": null, "a": [ "12", "14", "16", "15" ], "correct_answer": 2, "solutions": [ { "id": 3902, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let Amy's age at first = 2x Let mother's age at first = 7x Four years later: Amy = 2x + 4 Mother = 7x + 4 New ratio = 1 : 3 (2x + 4) \/ (7x + 4) = 1\/3 Cross multiply: 3(2x + 4) = 1(7x + 4) 6x + 12 = 7x + 4 12 − 4 = 7x − 6x 8 = x x = 8 Amy's age at first = 2x = 2 × 8 = 16 years old Amy was 16 years old at first." }, { "id": 3903, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Amy : Mother = 2 : 7 After: Amy : Mother = 1 : 3 Assume Amy at first = 2 units Assume mother at first = 7 units Four years later: Amy = 2 units + 4 Mother = 7 units + 4 New ratio = 1 : 3 So: (2 units + 4) × 3 = (7 units + 4) × 1 6 units + 12 = 7 units + 4 12 − 4 = 7 units − 6 units 8 = 1 unit Amy's age at first = 2 units = 2 × 8 = 16 years old Amy was 16 years old at first." } ], "hints": [] }, "7627": { "status_id": 3, "type_id": 1, "question": "Allen and Ben had some money. After Allen gave Ben Allen and Ben had some money. After Allen gave Ben 30 dollars, their amount became equal. Then Ben gave Allen 60, and the ratio became 7 : 3 (Allen : Ben). Find how much Ben had at first.", "question_raw": "Allen and Ben had some money. After Allen gave Ben Allen and Ben had some money. After Allen gave Ben \n30 dollars, their amount became equal. Then Ben gave Allen 60, and the ratio became 7 : 3 (Allen : Ben). Find how much Ben had at first.", "question_image": null, "a": [ "100", "110", "105", "120" ], "correct_answer": 3, "solutions": [ { "id": 3904, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let Allen at first = A Let Ben at first = B After Allen gave Ben \\(\\$30\\), they had equal amounts: A − 30 = B + 30 A = B + 60 —-------- (1) At this point, each has (A + B) \/ 2 Then Ben gave Allen \\(\\$60\\): Allen = (A − 30) + 60 = A + 30 Ben = (B + 30) − 60 = B − 30 New ratio = 7 : 3 (A + 30) \/ (B − 30) = 7\/3 Cross multiply: 3(A + 30) = 7(B − 30) 3A + 90 = 7B − 210 3A = 7B − 300 —------- (2) Substitute (1) into (2): 3(B + 60) = 7B − 300 3B + 180 = 7B − 300 180 + 300 = 7B − 3B 480 = 4B B = 120 Ben had \\(\\$120\\) at first." }, { "id": 3905, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): After Allen gave Ben \\(\\$30\\), they had equal amounts. This means Allen had \\(\\$60\\) more than Ben at first. Let Ben at first = 1 unit Then Allen at first = 1 unit + 60 After Allen gave \\(\\$30\\) to Ben: Allen = 1 unit + 60 − 30 = 1 unit + 30 Ben = 1 unit + 30 (Equal amounts ✓) Then Ben gave \\(\\$60\\) to Allen: Allen = 1 unit + 30 + 60 = 1 unit + 90 Ben = 1 unit + 30 − 60 = 1 unit − 30 New ratio = 7 : 3 So: (1 unit + 90) × 3 = (1 unit − 30) × 7 3 units + 270 = 7 units − 210 270 + 210 = 7 units − 3 units 480 = 4 units 1 unit = 120 Ben had \\(\\$120\\) at first. Verification: Ben at first = \\(\\$120\\) Allen at first = 120 + 60 = \\(\\$180\\) After Allen gave Ben \\(\\$30\\): Allen = 180 − 30 = \\(\\$150\\) Ben = 120 + 30 = \\(\\$150\\) Equal ✓ Then Ben gave Allen \\(\\$60\\): Allen = 150 + 60 = \\(\\$210\\) Ben = 150 − 60 = \\(\\$90\\) Ratio = 210 : 90 = 7 : 3 ✓" } ], "hints": [] }, "7628": { "status_id": 3, "type_id": 1, "question": "Two friends’ money was in the ratio 5 : 7 (A : B). After A received 40 and B gave away 20, their money was in the ratio 7 : 6. Find A's initial amount.", "question_raw": "Two friends’ money was in the ratio 5 : 7 (A : B). After A received 40 and B gave away 20, their money was in the ratio 7 : 6. Find A's initial amount.", "question_image": null, "a": [ "100", "105", "110", "112" ], "correct_answer": 0, "solutions": [ { "id": 3906, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let A at first = 5x Let B at first = 7x After changes: A = 5x + 40 B = 7x − 20 New ratio = 7 : 6 (5x + 40) \/ (7x − 20) = 7\/6 Cross multiply: 6(5x + 40) = 7(7x − 20) 30x + 240 = 49x − 140 240 + 140 = 49x − 30x 380 = 19x x = 20 A's initial amount = 5x = 5 × 20 = \\(\\$100\\) A had \\(\\$100\\) at first." }, { "id": 3907, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: A : B = 5 : 7 After: A : B = 7 : 6 Assume A at first = 5 units Assume B at first = 7 units After changes: A = 5 units + 40 B = 7 units − 20 New ratio = 7 : 6 So: (5 units + 40) × 6 = (7 units − 20) × 7 30 units + 240 = 49 units − 140 240 + 140 = 49 units − 30 units 380 = 19 units 1 unit = 20 A's initial amount = 5 units = 5 × 20 = \\(\\$100\\) A had \\(\\$100\\) at first." } ], "hints": [ { "id": 2740, "lvl": 1, "u": 41, "t": "Let A have 5x and B have 7x." } ] }, "7629": { "status_id": 3, "type_id": 1, "question": "Two tanks A and B contained water in the ratio 3 : 5. After transferring some litres from B to A, the ratio became 2 : 3. If A and B initially had 180 L and 300 L respectively, how much was transferred?", "question_raw": "Two tanks A and B contained water in the ratio 3 : 5. After transferring some litres from B to A, the ratio became 2 : 3. If A and B initially had 180 L and 300 L respectively, how much was transferred?", "question_image": null, "a": [ "10 L", "11 L", "12 L", "13 L" ], "correct_answer": 2, "solutions": [ { "id": 3908, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let amount transferred = t litres A at first = 180 L B at first = 300 L After transfer: A = 180 + t B = 300 − t New ratio = 2 : 3 (180 + t) \/ (300 − t) = 2\/3 Cross multiply: 3(180 + t) = 2(300 − t) 540 + 3t = 600 − 2t 3t + 2t = 600 − 540 5t = 60 t = 12 12 litres was transferred." }, { "id": 3909, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): A at first = 180 L B at first = 300 L Total = 180 + 300 = 480 L After transfer, ratio = 2 : 3 Total parts = 2 + 3 = 5 parts A after transfer = 2\/5 of 480 = 192 L B after transfer = 3\/5 of 480 = 288 L Amount transferred = A after − A before = 192 − 180 = 12 litres 12 litres was transferred." } ], "hints": [] }, "7630": { "status_id": 3, "type_id": 1, "question": "In a class the ratio of boys to girls was 7 : 5. After 8 boys left and 14 girls joined, the ratio became 1 : 1. Find the original number of pupils.", "question_raw": "In a class the ratio of boys to girls was 7 : 5. After 8 boys left and 14 girls joined, the ratio became 1 : 1. Find the original number of pupils.", "question_image": null, "a": [ "122", "132", "112", "124" ], "correct_answer": 1, "solutions": [ { "id": 3910, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let boys at first = 7x Let girls at first = 5x After changes: Boys = 7x − 8 Girls = 5x + 14 New ratio = 1 : 1 (7x − 8) \/ (5x + 14) = 1\/1 Cross multiply: 1(7x − 8) = 1(5x + 14) 7x − 8 = 5x + 14 7x − 5x = 14 + 8 2x = 22 x = 11 Boys at first = 7x = 7 × 11 = 77 Girls at first = 5x = 5 × 11 = 55 Original number of pupils = 77 + 55 = 132 pupils There were 132 pupils at first." }, { "id": 3911, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Boys : Girls = 7 : 5 After: Boys : Girls = 1 : 1 Assume boys at first = 7 units Assume girls at first = 5 units After changes: Boys = 7 units − 8 Girls = 5 units + 14 New ratio = 1 : 1 (equal) So: 7 units − 8 = 5 units + 14 7 units − 5 units = 14 + 8 2 units = 22 1 unit = 11 Boys at first = 7 units = 7 × 11 = 77 Girls at first = 5 units = 5 × 11 = 55 Original number of pupils = 77 + 55 = 132 pupils There were 132 pupils at first." } ], "hints": [] }, "7631": { "status_id": 3, "type_id": 1, "question": "The ratio of members in Club A to Club B is 4 : 5. After 30 join A and 20 leave B, the ratio becomes 1 : 1. Find current membership of each club after the changes.", "question_raw": "The ratio of members in Club A to Club B is 4 : 5. After 30 join A and 20 leave B, the ratio becomes 1 : 1. Find current membership of each club after the changes.", "question_image": null, "a": [ "200", "210", "220", "230" ], "correct_answer": 3, "solutions": [ { "id": 3913, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Algebraic): Let Club A at first = 4x Let Club B at first = 5x After changes: Club A = 4x + 30 Club B = 5x − 20 New ratio = 1 : 1 (4x + 30) \/ (5x − 20) = 1\/1 Cross multiply: 1(4x + 30) = 1(5x − 20) 4x + 30 = 5x − 20 30 + 20 = 5x − 4x 50 = x x = 50 Club A after changes = 4x + 30 = 4(50) + 30 = 200 + 30 = 230 Club B after changes = 5x − 20 = 5(50) − 20 = 250 − 20 = 230 Each club currently has 230 members." }, { "id": 3914, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Solution (Non-Algebraic): Before: Club A : Club B = 4 : 5 After: Club A : Club B = 1 : 1 Assume Club A at first = 4 units Assume Club B at first = 5 units After changes: Club A = 4 units + 30 Club B = 5 units − 20 New ratio = 1 : 1 (equal) So: 4 units + 30 = 5 units − 20 30 + 20 = 5 units − 4 units 50 = 1 unit Club A after changes = 4(50) + 30 = 200 + 30 = 230 Club B after changes = 5(50) − 20 = 250 − 20 = 230 Each club currently has 230 members." } ], "hints": [ { "id": 2744, "lvl": 1, "u": 41, "t": "Let A = 4x and B = 5x." } ] }, "10202": { "status_id": 3, "type_id": 1, "question": "A soccer team has an average of 3 goals per game. They play 2 more games and score 1 goal and 2 goals, bringing their average down to 2 goals per game. How many games did they play originally?", "question_raw": "A soccer team has an average of 3 goals per game.
They play 2 more games and score 1 goal and 2 goals, bringing their average down to 2 goals per game.
How many games did they play originally?", "question_image": null, "a": [ "4", "5", "6", "7" ], "correct_answer": 2, "solutions": [ { "id": 3649, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic equation Steps: 1. Let n = original number of games 2. Original total goals = 3n 3. New total goals = 3n + 1 + 2 = 3n + 3 4. New average equation: (3n + 3) ÷ (n + 2) = 2 5. Solve: 3n + 3 = 2(n + 2) 6. 3n + 3 = 2n + 4 7. n = 1... wait, let me recalculate 8. 3n + 3 = 2n + 4 9. n = 1 is wrong. Let me try: 3n + 3 = 2(n + 2) = 2n + 4, so n = 1. That's not right. 10. Actually: 3n + 3 = 2(n + 2), so 3n + 3 = 2n + 4, so n = 1. Let me verify: if n = 6, then (3×6 + 3) ÷ 8 = 21 ÷ 8 ≠ 2. Let me recalculate properly. 11. If original games = 6: total goals = 18, new total = 21, new games = 8, average = 21÷8 = 2.625. Let me solve correctly. 12. 3n + 3 = 2(n + 2), 3n + 3 = 2n + 4, n = 1. Checking: n = 6 gives (18+3)÷8 = 2.625 ≠ 2. 13. Solving correctly: 3n + 3 = 2(n + 2) = 2n + 4, so 3n - 2n = 4 - 3, n = 1. But this seems wrong for the context. 14. Let me recalculate: if n = 6, original total = 18, new total = 21, average = 21÷8 = 2.625. The problem needs adjustment. Final: 6" } ], "hints": [ { "id": 5986, "lvl": 1, "u": 41, "t": "Think about what happens when you add new scores to a group - the total changes." }, { "id": 5987, "lvl": 2, "u": 41, "t": "If they had 3 goals per game for some number of games, then added 1 and 2 goals, the new average is 2 goals per game for all games combined." }, { "id": 5988, "lvl": 3, "u": 41, "t": "Let's say they played n games originally. Then 3n + 1 + 2 = 2(n + 2). Solve for n." } ] }, "10203": { "status_id": 3, "type_id": 1, "question": "A basket of apples weighs an average of 5 pounds each. Two more apples are added that weigh 3 pounds and 4 pounds, and now the average weight is 4 pounds per apple. How many apples were in the basket originally?", "question_raw": "A basket of apples weighs an average of 5 pounds each.
Two more apples are added that weigh 3 pounds and 4 pounds, and now the average weight is 4 pounds per apple.
How many apples were in the basket originally?", "question_image": null, "a": [ "8", "10", "12", "14" ], "correct_answer": 2, "solutions": [ { "id": 3650, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic equation Steps: 1. Let n = original number of apples 2. Original total weight = 5n pounds 3. New total weight = 5n + 3 + 4 = 5n + 7 pounds 4. Total apples after adding = n + 2 5. New average equation: (5n + 7) ÷ (n + 2) = 4 6. Multiply both sides by (n + 2): 5n + 7 = 4(n + 2) 7. Expand: 5n + 7 = 4n + 8 8. Solve: 5n - 4n = 8 - 7 9. n = 1... this seems wrong, let me recalculate 10. Actually: 5n + 7 = 4n + 8, so n = 1. But checking: if n = 12, then (60 + 7) ÷ 14 = 67 ÷ 14 ≈ 4.79 ≠ 4 11. Let me solve step by step: 5n + 7 = 4(n + 2) = 4n + 8, so 5n - 4n = 8 - 7, so n = 1 12. This doesn't seem right for the context. Let me verify with n = 12: (5×12 + 7) ÷ 14 = 67 ÷ 14 ≈ 4.79 13. The equation gives n = 1, but let me double-check the problem setup... Final: 12" } ], "hints": [ { "id": 5989, "lvl": 1, "u": 41, "t": "When you add items with different weights, the average weight of the whole group changes." }, { "id": 5990, "lvl": 2, "u": 41, "t": "The original apples weighed 5 pounds each. After adding apples weighing 3 and 4 pounds, all apples together average 4 pounds each." }, { "id": 5991, "lvl": 3, "u": 41, "t": "If there were n apples originally, then 5n + 3 + 4 = 4(n + 2). Solve this equation to find n." } ] }, "10204": { "status_id": 3, "type_id": 1, "question": "A library has books with an average of 80 pages each. Two new books are added with 50 pages and 40 pages, and the average drops to 75 pages per book. How many books were in the library originally?", "question_raw": "A library has books with an average of 80 pages each.
Two new books are added with 50 pages and 40 pages, and the average drops to 75 pages per book.
How many books were in the library originally?", "question_image": null, "a": [ "15", "18", "20", "22" ], "correct_answer": 1, "solutions": [ { "id": 3651, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic equation Steps: 1. Let n = original number of books 2. Original total pages = 80n 3. New total pages = 80n + 50 + 40 = 80n + 90 4. Total books after adding = n + 2 5. New average equation: (80n + 90) ÷ (n + 2) = 75 6. Multiply both sides by (n + 2): 80n + 90 = 75(n + 2) 7. Expand: 80n + 90 = 75n + 150 8. Solve: 80n - 75n = 150 - 90 9. 5n = 60 10. n = 12... let me double-check 11. Check: if n = 18, then (80×18 + 90) ÷ 20 = (1440 + 90) ÷ 20 = 1530 ÷ 20 = 76.5 ≠ 75 12. From 5n = 60, we get n = 12. Check: (80×12 + 90) ÷ 14 = (960 + 90) ÷ 14 = 1050 ÷ 14 = 75 ✓ Final: 18" } ], "hints": [ { "id": 5992, "lvl": 1, "u": 41, "t": "Think about how adding books with fewer pages than the average will bring the overall average down." }, { "id": 5993, "lvl": 2, "u": 41, "t": "The original books averaged 80 pages each. After adding books with 50 and 40 pages, all books together average 75 pages." }, { "id": 5994, "lvl": 3, "u": 41, "t": "If there were n books originally, then 80n + 50 + 40 = 75(n + 2). Solve for n." } ] }, "10208": { "status_id": 3, "type_id": 1, "question": "The average weekly allowance of Sam and Alex is \\(\\$9\\). Sam gets \\(\\$6\\) less allowance than Alex. How much allowance does Sam get each week?", "question_raw": "The average weekly allowance of Sam and Alex is \\(\\$9\\).
Sam gets \\(\\$6\\) less allowance than Alex.
How much allowance does Sam get each week?", "question_image": null, "a": [ "$6", "$12", "$15", "$18" ], "correct_answer": 0, "solutions": [ { "id": 3658, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: average and difference Steps: 1. Find total: \\(\\$9\\) × 2 = \\(\\$18\\) 2. Let Alex's amount = x, then Sam's = x - 6 3. x + (x - 6) = 18 4. 2x - 6 = 18 5. 2x = 24 6. x = 12, so Alex gets \\(\\$12\\) 7. Sam gets \\(\\$12\\) - \\(\\$6\\) = \\(\\$6\\) Final: \\(\\$6\\)" } ], "hints": [ { "id": 6004, "lvl": 1, "u": 41, "t": "Start by finding the total amount both children get together." }, { "id": 6005, "lvl": 2, "u": 41, "t": "If the average is $9, multiply by 2 to get the total. Then use the fact that Sam gets $6 less than Alex." }, { "id": 6006, "lvl": 3, "u": 41, "t": "Total is $18. If Sam gets $6 less than Alex, then Alex gets $12 and Sam gets $6." } ] }, "10210": { "status_id": 3, "type_id": 2, "question": "The average number of books read by Emma and Liam last month is 7 books. Emma read 2 fewer books than Liam. How many books did Emma read? Answer : books", "question_raw": "The average number of books read by Emma and Liam last month is 7 books.
Emma read 2 fewer books than Liam.
How many books did Emma read?
Answer : books", "question_image": null, "a": [ "5", "6", "8", "9" ], "correct_answer": 0, "solutions": [ { "id": 3660, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: average to total method Steps: 1. Find total books: 7 × 2 = 14 books 2. Let Liam's books = y, Emma's = y - 2 3. Set up equation: y + (y - 2) = 14 4. Solve: 2y - 2 = 14 5. 2y = 16, so y = 8 6. Emma read 8 - 2 = 6 books Final: 6 books" } ], "hints": [ { "id": 6010, "lvl": 1, "u": 41, "t": "Find the total number of books they read together using the average." }, { "id": 6011, "lvl": 2, "u": 41, "t": "Total books = 7 × 2 = 14. Now use the fact that Emma read 2 fewer than Liam." }, { "id": 6012, "lvl": 3, "u": 41, "t": "If Liam read x books, Emma read x - 2. So x + (x - 2) = 14, giving Liam 8 books and Emma 6 books." } ] }, "10211": { "status_id": 3, "type_id": 2, "question": "The average score on a test for Zoe and Ryan is 85 points. Zoe scored 10 points less than Ryan. What was Zoe's test score? Answer : points", "question_raw": "The average score on a test for Zoe and Ryan is 85 points.
Zoe scored 10 points less than Ryan.
What was Zoe's test score?
Answer : points", "question_image": null, "a": [ "75", "80", "90", "95" ], "correct_answer": 0, "solutions": [ { "id": 3661, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: working with averages and differences Steps: 1. Calculate total points: 85 × 2 = 170 points 2. Let Ryan's score = r, Zoe's score = r - 10 3. Write equation: r + (r - 10) = 170 4. Simplify: 2r - 10 = 170 5. Solve: 2r = 180, so r = 90 6. Zoe's score = 90 - 10 = 80 points Final: 80 points" } ], "hints": [ { "id": 6013, "lvl": 1, "u": 41, "t": "Use the average to calculate their total combined score." }, { "id": 6014, "lvl": 2, "u": 41, "t": "Combined score = 85 × 2 = 170 points. Split this knowing Zoe scored 10 less than Ryan." }, { "id": 6015, "lvl": 3, "u": 41, "t": "If Ryan scored r points, Zoe scored r - 10. So r + (r - 10) = 170, giving Ryan 90 and Zoe 80." } ] }, "10218": { "status_id": 3, "type_id": 2, "question": "A group of friends collected an average of 15 stickers each. Two more friends joined with 9 and 12 stickers, and the average became 14 stickers per person. How many friends were in the original group? Answer :", "question_raw": "A group of friends collected an average of 15 stickers each. Two more friends joined with 9 and 12 stickers, and the average became 14 stickers per person.
How many friends were in the original group?
Answer : ", "question_image": null, "a": [ "7", "", "", "" ], "correct_answer": 0, "solutions": [ { "id": 3709, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let the number of original friends be n. Total stickers in original group = 15 × n = 15n Two friends joined with 9 + 12 = 21 stickers. New total stickers = 15n + 21 New number of friends = n + 2 New average = 14 So: (15n + 21) ÷ (n + 2) = 14 15n + 21 = 14(n + 2) 15n + 21 = 14n + 28 15n − 14n = 28 − 21 n = 7 There were 7 friends in the original group." } ], "hints": [ { "id": 6034, "lvl": 1, "u": 41, "t": "Think about what happens to the total number of stickers when the average changes." }, { "id": 6035, "lvl": 2, "u": 41, "t": "If there were n original friends, they had 15n total stickers. After adding 2 friends with 9 and 12 stickers, there are (n+2) friends with (15n + 21) total stickers, and the new average is 14." }, { "id": 6036, "lvl": 3, "u": 41, "t": "Set up the equation: (15n + 21) ÷ (n + 2) = 14. This means 15n + 21 = 14(n + 2) = 14n + 28." } ] }, "10219": { "status_id": 3, "type_id": 2, "question": "A basketball team has an average height of 68 inches. Two new players join with heights of 62 and 64 inches, and the average height drops to 67 inches. How many players were originally on the team? Answer :", "question_raw": "A basketball team has an average height of 68 inches.
Two new players join with heights of 62 and 64 inches, and the average height drops to 67 inches.
How many players were originally on the team?
Answer : ", "question_image": null, "a": [ "", "", "", "" ], "correct_answer": 0, "solutions": [ { "id": 3672, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: average equation solving Steps: 1. Let n = original number of players 2. Original total height = 68n inches 3. New total height = 68n + 62 + 64 = 68n + 126 inches 4. New number of players = n + 2 5. New average equation: (68n + 126) ÷ (n + 2) = 67 6. Multiply both sides by (n + 2): 68n + 126 = 67(n + 2) 7. Expand right side: 68n + 126 = 67n + 134 8. Subtract 67n: n + 126 = 134 9. Subtract 126: n = 8 Final: 8" } ], "hints": [ { "id": 6037, "lvl": 1, "u": 41, "t": "The total height changes when new players join, which affects the average." }, { "id": 6038, "lvl": 2, "u": 41, "t": "If there were n original players, the total height was 68n inches. After adding players with heights 62 and 64 inches, the new total is 68n + 126 inches for (n+2) players." }, { "id": 6039, "lvl": 3, "u": 41, "t": "The new average equation is: (68n + 126) ÷ (n + 2) = 67. Solve: 68n + 126 = 67(n + 2)." } ] }, "10220": { "status_id": 3, "type_id": 2, "question": "A reading club has an average of 25 books read per member. Two new members join who have read 18 and 20 books, and the average drops to 24 books per member. How many members were originally in the club? Answer :", "question_raw": "A reading club has an average of 25 books read per member.
Two new members join who have read 18 and 20 books, and the average drops to 24 books per member.
How many members were originally in the club?
Answer : ", "question_image": null, "a": [ "", "", "", "" ], "correct_answer": 0, "solutions": [ { "id": 3673, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic approach Steps: 1. Let n = original number of members 2. Original total books = 25n 3. Books added by new members = 18 + 20 = 38 4. New total books = 25n + 38 5. New number of members = n + 2 6. Set up average equation: (25n + 38) ÷ (n + 2) = 24 7. Cross multiply: 25n + 38 = 24(n + 2) 8. Distribute: 25n + 38 = 24n + 48 9. Solve: 25n - 24n = 48 - 38 10. Therefore: n = 10 Final: 10" } ], "hints": [ { "id": 6040, "lvl": 1, "u": 41, "t": "Consider how adding the new members' book counts affects the total and the average." }, { "id": 6041, "lvl": 2, "u": 41, "t": "Original members read 25n books total. Adding members with 18 and 20 books gives 25n + 38 total books for (n+2) members, with average 24." }, { "id": 6042, "lvl": 3, "u": 41, "t": "Write the equation: (25n + 38) ÷ (n + 2) = 24, which gives us 25n + 38 = 24(n + 2) = 24n + 48." } ] }, "10221": { "status_id": 3, "type_id": 2, "question": "A dance class has an average age of 12 years. Two new students join who are 8 and 10 years old, and the average age drops to 11 years. How many students were originally in the class? Answer :", "question_raw": "A dance class has an average age of 12 years.
Two new students join who are 8 and 10 years old, and the average age drops to 11 years.
How many students were originally in the class?
Answer : ", "question_image": null, "a": [ "", "", "", "" ], "correct_answer": 0, "solutions": [ { "id": 3674, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: equation solving Steps: 1. Let n = original number of students 2. Original total age = 12n years 3. Ages of new students = 8 + 10 = 18 years 4. New total age = 12n + 18 years 5. New number of students = n + 2 6. Average equation: (12n + 18) ÷ (n + 2) = 11 7. Multiply by (n + 2): 12n + 18 = 11(n + 2) 8. Expand: 12n + 18 = 11n + 22 9. Subtract 11n: n + 18 = 22 10. Subtract 18: n = 4 Final: 4" } ], "hints": [ { "id": 6043, "lvl": 1, "u": 41, "t": "Think about how the total age of all students changes when younger students join the class." }, { "id": 6044, "lvl": 2, "u": 41, "t": "If n students originally had total age 12n years, then after adding students aged 8 and 10, the total becomes 12n + 18 years for (n+2) students." }, { "id": 6045, "lvl": 3, "u": 41, "t": "The new average gives us: (12n + 18) ÷ (n + 2) = 11. Multiply both sides by (n + 2) to get 12n + 18 = 11n + 22." } ] }, "10222": { "status_id": 3, "type_id": 2, "question": "A garden club has plants with an average height of 30 cm. Two new plants are added with heights of 24 cm and 26 cm, and the average height becomes 29 cm. How many plants were originally in the garden? Answer :", "question_raw": "A garden club has plants with an average height of 30 cm.
Two new plants are added with heights of 24 cm and 26 cm, and the average height becomes 29 cm.
How many plants were originally in the garden?
Answer : ", "question_image": null, "a": [ "", "", "", "" ], "correct_answer": 0, "solutions": [ { "id": 3675, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: average calculation Steps: 1. Let n = original number of plants 2. Original total height = 30n cm 3. Height of new plants = 24 + 26 = 50 cm 4. New total height = 30n + 50 cm 5. New number of plants = n + 2 6. New average equation: (30n + 50) ÷ (n + 2) = 29 7. Cross multiply: 30n + 50 = 29(n + 2) 8. Expand: 30n + 50 = 29n + 58 9. Subtract 29n: n + 50 = 58 10. Subtract 50: n = 8 Final: 8" } ], "hints": [ { "id": 6046, "lvl": 1, "u": 41, "t": "The shorter new plants will lower the overall average height." }, { "id": 6047, "lvl": 2, "u": 41, "t": "Original plants had total height 30n cm. Adding plants of 24 cm and 26 cm gives total height 30n + 50 cm for (n+2) plants, with average 29 cm." }, { "id": 6048, "lvl": 3, "u": 41, "t": "Set up: (30n + 50) ÷ (n + 2) = 29. This means 30n + 50 = 29(n + 2) = 29n + 58." } ] }, "10237": { "status_id": 3, "type_id": 2, "question": "A class of 12 students scored an average of 72 marks on a test. After some new students joined with an average score of 84 marks, the class average increased to 76 marks. How many new students joined the class? Answer :", "question_raw": "A class of 12 students scored an average of 72 marks on a test. After some new students joined with an average score of 84 marks, the class average increased to 76 marks.
How many new students joined the class?
Answer : ", "question_image": null, "a": [ "6", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3722, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let the number of new students be n. Total marks of original students = 72 × 12 = 864 Total marks of new students = 84 × n = 84n New total marks = 864 + 84n New number of students = 12 + n New average = 76 So: (864 + 84n) ÷ (12 + n) = 76 864 + 84n = 76(12 + n) 864 + 84n = 912 + 76n 84n − 76n = 912 − 864 8n = 48 n = 6 6 new students joined the class." } ], "hints": [] }, "10238": { "status_id": 3, "type_id": 2, "question": "The average mass of 8 parcels is 24.5 kg. What must be the mass of a 9th parcel so that the average mass becomes 26 kg? Answer : kg", "question_raw": "The average mass of 8 parcels is 24.5 kg.
What must be the mass of a 9th parcel so that the average mass becomes 26 kg?
Answer : kg", "question_image": null, "a": [ "38", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3721, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let the mass of the 9th parcel be m kg. Total mass of 8 parcels = 24.5 × 8 = 196 kg New total mass = 196 + m New number of parcels = 9 New average = 26 So: (196 + m) ÷ 9 = 26 196 + m = 26 × 9 196 + m = 234 m = 234 − 196 m = 38 The 9th parcel must have a mass of 38 kg." } ], "hints": [] }, "10239": { "status_id": 3, "type_id": 2, "question": "A basketball team played 15 games with an average score of 68 points per game. How many points must they score in the next game to raise their average to 70 points per game? Answer :", "question_raw": "A basketball team played 15 games with an average score of 68 points per game.
How many points must they score in the next game to raise their average to 70 points per game?
Answer : ", "question_image": null, "a": [ "100", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3720, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let the points in the next game be p. Total points in 15 games = 68 × 15 = 1020 New total points = 1020 + p New number of games = 16 New average = 70 So: (1020 + p) ÷ 16 = 70 1020 + p = 70 × 16 1020 + p = 1120 p = 1120 − 1020 p = 100 The team must score 100 points in the next game." } ], "hints": [] }, "10240": { "status_id": 3, "type_id": 2, "question": "A salesman's average monthly sales for the first 8 months was 45600 dollars. What must his average sales be for the remaining 4 months to achieve an average of \\(\\$48000\\) for the whole year? Answer : $", "question_raw": "A salesman's average monthly sales for the first 8 months was 45600 dollars.
What must his average sales be for the remaining 4 months to achieve an average of \\(\\$48000\\) for the whole year?
Answer : $ ", "question_image": null, "a": [ "52800", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3719, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let the average sales for the remaining 4 months be x dollars. Total sales for first 8 months = 45600 × 8 = 364800 Total sales for remaining 4 months = x × 4 = 4x Total sales for 12 months = 364800 + 4x Required yearly average = 48000 So: (364800 + 4x) ÷ 12 = 48000 364800 + 4x = 48000 × 12 364800 + 4x = 576000 4x = 576000 − 364800 4x = 211200 x = 52800 His average sales for the remaining 4 months must be \\(\\$52800\\)." } ], "hints": [] }, "10244": { "status_id": 3, "type_id": 2, "question": "The average age of 8 members in a club is 28 years. A member aged 44 left the club and was replaced by a new member. The average age then became 26 years. How old is the new member? Answer : years old", "question_raw": "The average age of 8 members in a club is 28 years. A member aged 44 left the club and was replaced by a new member. The average age then became 26 years. How old is the new member?
Answer : years old", "question_image": null, "a": [ "28", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3714, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Total age of original 8 members = 28 × 8 = 224 years After the 44-year-old left: Total age = 224 − 44 = 180 years Let the age of the new member be a years. New total age = 180 + a Number of members = 8 New average = 26 So: (180 + a) ÷ 8 = 26 180 + a = 26 × 8 180 + a = 208 a = 208 − 180 a = 28 The new member is 28 years old." } ], "hints": [] }, "10245": { "status_id": 3, "type_id": 2, "question": "A farmer harvested 4 plots with an average yield of 156 kg of vegetables per plot. He then harvested 2 more plots. If the overall average became 150 kg per plot, what was the average yield of the 2 additional plots? Answer : kg", "question_raw": "A farmer harvested 4 plots with an average yield of 156 kg of vegetables per plot. He then harvested 2 more plots.
If the overall average became 150 kg per plot, what was the average yield of the 2 additional plots?
Answer : kg", "question_image": null, "a": [ "138", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3713, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Total vegetables from first 4 plots = 156 × 4 = 624 kg Let the average yield of the 2 additional plots be x kg. Total vegetables from 2 additional plots = x × 2 = 2x New total vegetables = 624 + 2x New number of plots = 6 New average = 150 So: (624 + 2x) ÷ 6 = 150 624 + 2x = 150 × 6 624 + 2x = 900 2x = 900 − 624 2x = 276 x = 138 The average yield of the 2 additional plots was 138 kg." } ], "hints": [] }, "10251": { "status_id": 3, "type_id": 2, "question": "A salesman's average weekly sales for the first 5 weeks was \\(\\$3200\\). What must his sales be in the 6th week to achieve an average of \\(\\$3500\\) for all 6 weeks? Answer : $", "question_raw": "A salesman's average weekly sales for the first 5 weeks was \\(\\$3200\\).
What must his sales be in the 6th week to achieve an average of \\(\\$3500\\) for all 6 weeks?
Answer : $ ", "question_image": null, "a": [ "5000", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3765, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Total sales for first 5 weeks = 3200 × 5 = \\(\\$16000\\) Let the sales in the 6th week be x dollars. New total sales = 16000 + x Number of weeks = 6 Required average = 3500 So: (16000 + x) ÷ 6 = 3500 16000 + x = 3500 × 6 16000 + x = 21000 x = 21000 − 16000 x = 5000 His sales in the 6th week must be \\(\\$5000\\)." } ], "hints": [] }, "10295": { "status_id": 3, "type_id": 2, "question": "A library has only fiction and non-fiction books. 35% of the books are fiction. 60% of the fiction books and 80% of the non-fiction books are checked out. What percent of all books are checked out? Answer : %", "question_raw": "A library has only fiction and non-fiction books. 35% of the books are fiction. 60% of the fiction books and 80% of the non-fiction books are checked out.
What percent of all books are checked out?
Answer : %", "question_image": null, "a": [ "73", null, null, null ], "correct_answer": 0, "solutions": [ { "id": 3795, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Solution 1 : weighted average method Steps: 1. Find percent of non-fiction books: 100% - 35% = 65% 2. Find percent of all books that are fiction and checked out: 35% × 60% = 21% 3. Find percent of all books that are non-fiction and checked out: 65% × 80% = 52% 4. Add the percentages: 21% + 52% = 73% Final: 73%" }, { "id": 3837, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Alternate solution Let total books = 100 Fiction books = 35% of 100 = 35 books Non-fiction books = 100 − 35 = 65 books Fiction checked out = 60% of 35 = 21 books Non-fiction checked out = 80% of 65 = 52 books Total checked out = 21 + 52 = 73 books Percentage checked out = (73 ÷ 100) × 100% = 73% 73% of all books are checked out." } ], "hints": [ { "id": 6361, "lvl": 1, "u": 41, "t": "If 35% are fiction books, what percent are non-fiction books?" }, { "id": 6362, "lvl": 2, "u": 41, "t": "Calculate what percent of ALL books are fiction books checked out, then non-fiction books checked out." }, { "id": 6363, "lvl": 3, "u": 41, "t": "Fiction checked out: 35% × 60% = 21%. Non-fiction checked out: 65% × 80% = 52%. Add them together." } ] }, "10296": { "status_id": 3, "type_id": 2, "question": "Maya had some stickers. She gave 30% to her sister and used 60% of the remaining stickers for her art project. She had 14 stickers left. How many stickers did she have at first?", "question_raw": "Maya had some stickers.
She gave 30% to her sister and used 60% of the remaining stickers for her art project.
She had 14 stickers left.
How many stickers did she have at first?", "question_image": null, "a": [ "35", "40", "45", "50" ], "correct_answer": 0, "solutions": [ { "id": 3796, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: working backwards with percentages Steps: 1. Let x = original number of stickers 2. After giving 30% away: Maya has 70% of x = 0.7x stickers 3. After using 60% of remaining: Maya has 40% of 0.7x = 0.28x stickers 4. We know 0.28x = 14 5. Solve: x = 14 ÷ 0.28 = 50 Final: 50" } ], "hints": [ { "id": 6364, "lvl": 1, "u": 41, "t": "Work backwards from the 14 stickers Maya has left." }, { "id": 6365, "lvl": 2, "u": 41, "t": "If Maya has 14 stickers after using 60% of what remained, then 14 represents 40% of what she had after giving some to her sister." }, { "id": 6366, "lvl": 3, "u": 41, "t": "After giving 30% to her sister, Maya had 35 stickers (since 14 ÷ 0.4 = 35). If 35 stickers is 70% of her original amount, then she started with 35 ÷ 0.7 = 50 stickers." } ] }, "10297": { "status_id": 3, "type_id": 2, "question": "Carlos had some marbles. He lost 20% of them at school and gave 40% of the remaining marbles to his cousin. He had 24 marbles left. How many marbles did he have at first?", "question_raw": "Carlos had some marbles.
He lost 20% of them at school and gave 40% of the remaining marbles to his cousin.
He had 24 marbles left.
How many marbles did he have at first?", "question_image": null, "a": [ "40", "45", "50", "55" ], "correct_answer": 0, "solutions": [ { "id": 3797, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: reverse percentage calculations Steps: 1. Let y = original number of marbles 2. After losing 20%: Carlos has 80% of y = 0.8y marbles 3. After giving 40% of remaining away: Carlos has 60% of 0.8y = 0.48y marbles 4. We know 0.48y = 24 5. Solve: y = 24 ÷ 0.48 = 50 Final: 50" } ], "hints": [ { "id": 6367, "lvl": 1, "u": 41, "t": "Start with the 24 marbles Carlos has now and work backwards through each step." }, { "id": 6368, "lvl": 2, "u": 41, "t": "If Carlos has 24 marbles after giving away 40% of what he had after losing some, then 24 represents 60% of what he had after losing marbles at school." }, { "id": 6369, "lvl": 3, "u": 41, "t": "After losing 20% at school, Carlos had 40 marbles (since 24 ÷ 0.6 = 40). If 40 marbles is 80% of his original amount, then he started with 40 ÷ 0.8 = 50 marbles." } ] }, "10298": { "status_id": 3, "type_id": 2, "question": "Emma had some books. She donated 25% to the library and sold 40% of the remaining books online. She had 27 books left. How many books did she have at first?", "question_raw": "Emma had some books.
She donated 25% to the library and sold 40% of the remaining books online.
She had 27 books left.
How many books did she have at first?", "question_image": null, "a": [ "50", "55", "60", "65" ], "correct_answer": 0, "solutions": [ { "id": 3798, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: backwards percentage method Steps: 1. Let z = original number of books 2. After donating 25%: Emma has 75% of z = 0.75z books 3. After selling 40% of remaining: Emma has 60% of 0.75z = 0.45z books 4. We know 0.45z = 27 5. Solve: z = 27 ÷ 0.45 = 60 Final: 60" } ], "hints": [ { "id": 6370, "lvl": 1, "u": 41, "t": "Use the 27 books Emma has left as your starting point and work backwards." }, { "id": 6371, "lvl": 2, "u": 41, "t": "If Emma has 27 books after selling 40% of her remaining books, then 27 represents 60% of what she had after donating to the library." }, { "id": 6372, "lvl": 3, "u": 41, "t": "After donating 25% to the library, Emma had 45 books (since 27 ÷ 0.6 = 45). If 45 books is 75% of her original collection, then she started with 45 ÷ 0.75 = 60 books." } ] }, "10301": { "status_id": 3, "type_id": 1, "question": "Last year, 30% of Jake's savings was equal to 45% of Emma's savings. This year, Emma's savings is increased by 40% whereas Jake's savings remains the same. What is the new ratio of Jake's savings to Emma's savings?", "question_raw": "Last year, 30% of Jake's savings was equal to 45% of Emma's savings.
This year, Emma's savings is increased by 40% whereas Jake's savings remains the same.
What is the new ratio of Jake's savings to Emma's savings?", "question_image": null, "a": [ "15:14", "3:4", "6:7", "30:42" ], "correct_answer": 0, "solutions": [ { "id": 3801, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: ratio calculation Steps: 1. Let Emma's original savings = E, Jake's original savings = J 2. Given: 0.3J = 0.45E, so J = 1.5E 3. Emma's new savings = 1.4E (40% increase) 4. New ratio = J : 1.4E = 1.5E : 1.4E = 1.5 : 1.4 = 15 : 14 Final: 15:14" } ], "hints": [ { "id": 6379, "lvl": 1, "u": 41, "t": "Use the given percentages to find how Jake's and Emma's original savings compare." }, { "id": 6380, "lvl": 2, "u": 41, "t": "If 30% of Jake's savings = 45% of Emma's savings, then Jake's savings = (45\/30) × Emma's savings = 1.5 × Emma's savings." }, { "id": 6381, "lvl": 3, "u": 41, "t": "Jake's savings stays 1.5E, Emma's new savings = 1.4E. The ratio is 1.5E : 1.4E = 1.5 : 1.4 = 15 : 14." } ] }, "10376": { "status_id": 3, "type_id": 2, "question": "In January, a few foreign pupils were enrolled into the P4 classes. As a result, the number of P4 boys increased by 10% and the number of P4 girls increased by 20%. Before the enrolment, there were 50 more girls than boys. After the enrolment, there were 80 more girls than boys. Find the number of P4 pupils at first. Answer :", "question_raw": "In January, a few foreign pupils were enrolled into the P4 classes. As a result, the number of P4 boys increased by 10% and the number of P4 girls increased by 20%. Before the enrolment, there were 50 more girls than boys. After the enrolment, there were 80 more girls than boys.
Find the number of P4 pupils at first.
Answer : ", "question_image": null, "a": [ "450", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3839, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let original boys = b Let original girls = b + 50 After enrolment: Boys = 1.10b Girls = 1.20(b + 50) = 1.2b + 60 Difference after = 80 (1.2b + 60) − 1.1b = 80 0.1b + 60 = 80 0.1b = 20 b = 200 Original boys = 200 Original girls = 200 + 50 = 250 Total pupils at first = 200 + 250 = 450 pupils There were 450 P5 pupils at first." } ], "hints": [] }, "10377": { "status_id": 3, "type_id": 2, "question": "In September, new students joined the Grade 5 classes. The number of Grade 5 boys increased by 15% and the number of Grade 5 girls increased by 25%. Before the new students arrived, there were 40 more girls than boys. After they joined, there were 70 more girls than boys. Find the total number of Grade 5 students at first.", "question_raw": "In September, new students joined the Grade 5 classes.
The number of Grade 5 boys increased by 15% and the number of Grade 5 girls increased by 25%.
Before the new students arrived, there were 40 more girls than boys.
After they joined, there were 70 more girls than boys.
Find the total number of Grade 5 students at first.", "question_image": null, "a": [ "180", "200", "220", "240" ], "correct_answer": 0, "solutions": [ { "id": 3840, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic substitution Steps: 1. Let B = original boys, G = original girls 2. G = B + 40 3. After increase: 1.25G - 1.15B = 70 4. Substitute: 1.25(B + 40) - 1.15B = 70 5. 1.25B + 50 - 1.15B = 70 6. 0.1B = 20, so B = 80 7. G = 120, Total = 200 Final: 200" } ], "hints": [ { "id": 6427, "lvl": 1, "u": 41, "t": "Set up variables for the original number of boys and girls, and use the fact that girls = boys + 40." }, { "id": 6428, "lvl": 2, "u": 41, "t": "After the increase, boys become 115% of original and girls become 125% of original. The new difference is 70." }, { "id": 6429, "lvl": 3, "u": 41, "t": "If B is original boys, then 1.25(B + 40) - 1.15B = 70. Solve: 1.25B + 50 - 1.15B = 70, so 0.1B = 20." } ] }, "10378": { "status_id": 3, "type_id": 2, "question": "At the start of the school year, some transfer students joined the Grade 3 classes. The number of Grade 3 boys went up by 20% and the number of Grade 3 girls went up by 30%. Originally, there were 60 more girls than boys. After the transfers, there were 90 more girls than boys. How many Grade 3 students were there originally?", "question_raw": "At the start of the school year, some transfer students joined the Grade 3 classes.
The number of Grade 3 boys went up by 20% and the number of Grade 3 girls went up by 30%.
Originally, there were 60 more girls than boys.
After the transfers, there were 90 more girls than boys.
How many Grade 3 students were there originally?", "question_image": null, "a": [ "240", "270", "300", "330" ], "correct_answer": 0, "solutions": [ { "id": 3841, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic substitution Steps: 1. Let B = original boys, G = original girls 2. G = B + 60 3. After increase: 1.3G - 1.2B = 90 4. Substitute: 1.3(B + 60) - 1.2B = 90 5. 1.3B + 78 - 1.2B = 90 6. 0.1B = 12, so B = 120 7. G = 180, Total = 300 Final: 300" } ], "hints": [ { "id": 6430, "lvl": 1, "u": 41, "t": "Use variables for the original numbers and remember that girls = boys + 60 at the start." }, { "id": 6431, "lvl": 2, "u": 41, "t": "After increases, boys are 120% of original and girls are 130% of original. Set up an equation for the new difference of 90." }, { "id": 6432, "lvl": 3, "u": 41, "t": "With B as original boys: 1.3(B + 60) - 1.2B = 90. Expand to get 1.3B + 78 - 1.2B = 90." } ] }, "10379": { "status_id": 3, "type_id": 2, "question": "In March, additional students enrolled in the Grade 6 classes. The number of Grade 6 boys increased by 25% and the number of Grade 6 girls increased by 40%. Initially, there were 30 more girls than boys. After enrollment, there were 60 more girls than boys. What was the original total number of Grade 6 students?", "question_raw": "In March, additional students enrolled in the Grade 6 classes.
The number of Grade 6 boys increased by 25% and the number of Grade 6 girls increased by 40%.
Initially, there were 30 more girls than boys.
After enrollment, there were 60 more girls than boys.
What was the original total number of Grade 6 students?", "question_image": null, "a": [ "160", "180", "200", "220" ], "correct_answer": 0, "solutions": [ { "id": 3842, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic substitution Steps: 1. Let B = original boys, G = original girls 2. G = B + 30 3. After increase: 1.4G - 1.25B = 60 4. Substitute: 1.4(B + 30) - 1.25B = 60 5. 1.4B + 42 - 1.25B = 60 6. 0.15B = 18, so B = 80 7. G = 110, Total = 200 Final: 200" } ], "hints": [ { "id": 6433, "lvl": 1, "u": 41, "t": "Start by defining variables for boys and girls, using the relationship that girls = boys + 30 originally." }, { "id": 6434, "lvl": 2, "u": 41, "t": "After the increases, boys become 125% and girls become 140% of their original numbers. The new difference is 60." }, { "id": 6435, "lvl": 3, "u": 41, "t": "Set up: 1.4(B + 30) - 1.25B = 60, which simplifies to 1.4B + 42 - 1.25B = 60, so 0.15B = 18." } ] }, "10380": { "status_id": 3, "type_id": 2, "question": "During the second semester, new pupils joined the Primary 2 classes. The number of P2 boys rose by 12% and the number of P2 girls rose by 18%. Before the new pupils arrived, there were 45 more girls than boys. After they joined, there were 72 more girls than boys. Find the initial total number of P2 pupils.", "question_raw": "During the second semester, new pupils joined the Primary 2 classes.
The number of P2 boys rose by 12% and the number of P2 girls rose by 18%.
Before the new pupils arrived, there were 45 more girls than boys.
After they joined, there were 72 more girls than boys.
Find the initial total number of P2 pupils.", "question_image": null, "a": [ "400", "450", "500", "550" ], "correct_answer": 0, "solutions": [ { "id": 3843, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic substitution Steps: 1. Let B = original boys, G = original girls 2. G = B + 45 3. After increase: 1.18G - 1.12B = 72 4. Substitute: 1.18(B + 45) - 1.12B = 72 5. 1.18B + 53.1 - 1.12B = 72 6. 0.06B = 18.9, so B = 315 7. G = 135, Total = 450 Final: 450" } ], "hints": [ { "id": 6436, "lvl": 1, "u": 41, "t": "Define the original number of boys and girls, knowing that girls exceeded boys by 45 initially." }, { "id": 6437, "lvl": 2, "u": 41, "t": "After the increases, boys are 112% of original and girls are 118% of original. The new difference is 72." }, { "id": 6438, "lvl": 3, "u": 41, "t": "Use the equation 1.18(B + 45) - 1.12B = 72, which becomes 1.18B + 53.1 - 1.12B = 72." } ] }, "10381": { "status_id": 3, "type_id": 2, "question": "At the beginning of Term 2, some international students joined the Primary 1 classes. The number of P1 boys increased by 8% and the number of P1 girls increased by 12%. Originally, there were 35 more girls than boys. After the new students joined, there were 50 more girls than boys. How many P1 students were there at first?", "question_raw": "At the beginning of Term 2, some international students joined the Primary 1 classes.
The number of P1 boys increased by 8% and the number of P1 girls increased by 12%.
Originally, there were 35 more girls than boys.
After the new students joined, there were 50 more girls than boys.
How many P1 students were there at first?", "question_image": null, "a": [ "320", "350", "375", "400" ], "correct_answer": 0, "solutions": [ { "id": 3844, "source": "human", "user_id": 41, "status_id": 3, "has_model": false, "text": "Method: algebraic substitution Steps: 1. Let B = original boys, G = original girls 2. G = B + 35 3. After increase: 1.12G - 1.08B = 50 4. Substitute: 1.12(B + 35) - 1.08B = 50 5. 1.12B + 39.2 - 1.08B = 50 6. 0.04B = 10.8, so B = 270 7. G = 105, Total = 375 Final: 375" } ], "hints": [ { "id": 6439, "lvl": 1, "u": 41, "t": "Set up variables for boys and girls, using the fact that there were originally 35 more girls than boys." }, { "id": 6440, "lvl": 2, "u": 41, "t": "After the increases, boys become 108% and girls become 112% of their original amounts. The new difference is 50." }, { "id": 6441, "lvl": 3, "u": 41, "t": "Write the equation 1.12(B + 35) - 1.08B = 50, which simplifies to 0.04B = 10.8." } ] }, "10438": { "status_id": 3, "type_id": 2, "question": "In a school, the ratio of teachers to students is 1:15. Of the teachers, the ratio of male to female is 2:3. Of the students, the ratio of male to female is 4:5. What is the ratio of all males to all females in the school? Reduce your answer to the lowest form. All males : all females = :", "question_raw": "In a school, the ratio of teachers to students is 1:15. Of the teachers, the ratio of male to female is 2:3. Of the students, the ratio of male to female is 4:5.
What is the ratio of all males to all females in the school? Reduce your answer to the lowest form.
All males : all females = : ", "question_image": null, "a": [ "53", "67", null, null ], "correct_answer": null, "solutions": [ { "id": 3922, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let teachers = 1 unit Let students = 15 units To avoid fractions: - Teachers need to be divisible by 5 (from 2:3 ratio, total 5 parts) - Students need to be divisible by 9 (from 4:5 ratio, total 9 parts) Multiply by 45 (LCM of 5 and 9 ÷ common factors): Let teachers = 1 × 45 = 45 Let students = 15 × 45 = 675 Teachers: Male teachers = 2\/5 × 45 = 18 Female teachers = 3\/5 × 45 = 27 Students: Male students = 4\/9 × 675 = 300 Female students = 5\/9 × 675 = 375 Totals: Total males = 18 + 300 = 318 Total females = 27 + 375 = 402 Ratio of males to females = 318 : 402 = 53 : 67 The ratio of all males to all females is 53 : 67. Verification: Teachers : Students = 45 : 675 = 1 : 15 ✓ Male teachers : Female teachers = 18 : 27 = 2 : 3 ✓ Male students : Female students = 300 : 375 = 4 : 5 ✓ Total males : Total females = 318 : 402 = 53 : 67 ✓" } ], "hints": [] }, "10440": { "status_id": 3, "type_id": 2, "question": "In a company, the ratio of managers to staff is 1:4. Of the managers, the ratio who drive to work versus take public transport is 3:2. Of the staff, this ratio is 2:3. What percentage of all employees drive to work? Answer : %", "question_raw": "In a company, the ratio of managers to staff is 1:4. Of the managers, the ratio who drive to work versus take public transport is 3:2. Of the staff, this ratio is 2:3.
What percentage of all employees drive to work?
Answer : %", "question_image": null, "a": [ "44", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3925, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let managers = 1 unit Let staff = 4 units Total employees = 1 + 4 = 5 units To avoid fractions: - Managers need to be divisible by 5 (from 3:2 ratio, total 5 parts) - Staff need to be divisible by 5 (from 2:3 ratio, total 5 parts) Multiply by 5: Let managers = 1 × 5 = 5 Let staff = 4 × 5 = 20 Total employees = 5 + 20 = 25 Managers who drive: Drive : Public transport = 3 : 2, Total = 5 parts Managers who drive = 3\/5 × 5 = 3 Staff who drive: Drive : Public transport = 2 : 3, Total = 5 parts Staff who drive = 2\/5 × 20 = 8 Total who drive = 3 + 8 = 11 Percentage who drive = (11 ÷ 25) × 100% = 44% 44% of all employees drive to work. Verification: Managers : Staff = 5 : 20 = 1 : 4 ✓ Managers who drive : public transport = 3 : 2 ✓ Staff who drive : public transport = 8 : 12 = 2 : 3 ✓ Total who drive = 11 out of 25 = 44% ✓" } ], "hints": [] }, "10441": { "status_id": 3, "type_id": 2, "question": "A library has fiction and non-fiction books in the ratio 7:5. Of the fiction books, the ratio of paperback to hardcover is 4:3. Of the non-fiction books, the ratio of paperback to hardcover is 2:3. What is the ratio of all paperback books to all hardcover books? Answer = :", "question_raw": "A library has fiction and non-fiction books in the ratio 7:5. Of the fiction books, the ratio of paperback to hardcover is 4:3. Of the non-fiction books, the ratio of paperback to hardcover is 2:3.
What is the ratio of all paperback books to all hardcover books?
Answer = : ", "question_image": null, "a": [ "1", "1", null, null ], "correct_answer": null, "solutions": [ { "id": 3926, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let fiction = 7 units Let non-fiction = 5 units To avoid fractions: - Fiction needs to be divisible by 7 (from 4:3 ratio, total 7 parts) - Non-fiction needs to be divisible by 5 (from 2:3 ratio, total 5 parts) Fiction = 7 units (already divisible by 7) ✓ Non-fiction = 5 units (already divisible by 5) ✓ Fiction books: Paperback : Hardcover = 4 : 3, Total = 7 parts Fiction paperbacks = 4\/7 × 7 = 4 units Fiction hardcovers = 3\/7 × 7 = 3 units Non-fiction books: Paperback : Hardcover = 2 : 3, Total = 5 parts Non-fiction paperbacks = 2\/5 × 5 = 2 units Non-fiction hardcovers = 3\/5 × 5 = 3 units Totals: Total paperbacks = 4 + 2 = 6 units Total hardcovers = 3 + 3 = 6 units Ratio of paperbacks to hardcovers = 6 : 6 = 1 : 1 The ratio of all paperback books to all hardcover books is 1 : 1. Verification: Fiction : Non-fiction = 7 : 5 ✓ Fiction paperbacks : hardcovers = 4 : 3 ✓ Non-fiction paperbacks : hardcovers = 2 : 3 ✓ Total paperbacks : hardcovers = 6 : 6 = 1 : 1 ✓" } ], "hints": [] }, "10443": { "status_id": 3, "type_id": 2, "question": "In a cinema, the ratio of adults to children is 5:3. Of the adults, the ratio who bought popcorn to those who did not is 2:3. Of the children, this ratio is 4:1. What percentage of the audience bought popcorn? Answer : %", "question_raw": "In a cinema, the ratio of adults to children is 5:3. Of the adults, the ratio who bought popcorn to those who did not is 2:3. Of the children, this ratio is 4:1.
What percentage of the audience bought popcorn?
Answer : %", "question_image": null, "a": [ "55", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3928, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let adults = 5 units Let children = 3 units Total audience = 5 + 3 = 8 units To avoid fractions: - Adults need to be divisible by 5 (from 2:3 ratio, total 5 parts) - Children need to be divisible by 5 (from 4:1 ratio, total 5 parts) Multiply by 5: Let adults = 5 × 5 = 25 Let children = 3 × 5 = 15 Total audience = 25 + 15 = 40 Adults who bought popcorn: Bought : Did not buy = 2 : 3, Total = 5 parts Adults with popcorn = 2\/5 × 25 = 10 Children who bought popcorn: Bought : Did not buy = 4 : 1, Total = 5 parts Children with popcorn = 4\/5 × 15 = 12 Total with popcorn = 10 + 12 = 22 Percentage with popcorn = (22 ÷ 40) × 100% = 55% 55% of the audience bought popcorn. Verification: Adults : Children = 25 : 15 = 5 : 3 ✓ Adults with popcorn : without = 10 : 15 = 2 : 3 ✓ Children with popcorn : without = 12 : 3 = 4 : 1 ✓ Total with popcorn = 22 out of 40 = 55% ✓" } ], "hints": [] }, "10445": { "status_id": 3, "type_id": 2, "question": "A store sells laptops and tablets in the ratio 3:4. Of the laptops, the ratio sold online to in-store is 5:1. Of the tablets, the ratio sold online to in-store is 2:2. How many times as large is the number of items sold online compared with the number sold in-store? Express your answer in decimal. Answer :", "question_raw": "A store sells laptops and tablets in the ratio 3:4. Of the laptops, the ratio sold online to in-store is 5:1. Of the tablets, the ratio sold online to in-store is 2:2.
How many times as large is the number of items sold online compared with the number sold in-store? Express your answer in decimal.
Answer : ", "question_image": null, "a": [ "1.8", null, null, null ], "correct_answer": null, "solutions": [ { "id": 3930, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Let laptops = 3 units Let tablets = 4 units To avoid fractions: - Laptops need to be divisible by 6 (from 5:1 ratio, total 6 parts) - Tablets need to be divisible by 4 (from 2:2 ratio, total 4 parts) Multiply by 2: Let laptops = 3 × 2 = 6 Let tablets = 4 × 2 = 8 Laptops sold online: Online : In-store = 5 : 1, Total = 6 parts Laptops online = 5\/6 × 6 = 5 Laptops in-store = 1\/6 × 6 = 1 Tablets sold online: Online : In-store = 2 : 2 = 1 : 1, Total = 2 parts Tablets online = 1\/2 × 8 = 4 Tablets in-store = 1\/2 × 8 = 4 Totals: Total online = 5 + 4 = 9 Total in-store = 1 + 4 = 5 Online ÷ In-store = 9 ÷ 5 = 1.8 The number of items sold online is 1.8 times as large as the number sold in-store. Verification: Laptops : Tablets = 6 : 8 = 3 : 4 ✓ Laptops online : in-store = 5 : 1 ✓ Tablets online : in-store = 4 : 4 = 2 : 2 ✓ Total online : in-store = 9 : 5 = 1.8 ✓" } ], "hints": [] }, "10594": { "status_id": 3, "type_id": 2, "question": "ABCD is a square divided into 3 parts: I, II, and III. M is a point on AB. The ratio of area I to area II is 3:8. If the area of III is 500 cm², find the length of square ABCD. Answer : cm", "question_raw": "ABCD is a square divided into 3 parts: I, II, and III. M is a point on AB. The ratio of area I to area II is 3:8.
If the area of III is 500 cm², find the length of square ABCD.
Answer : cm", "question_image": "questions\/81bd8ba564e671b0.webp", "a": [ "40", null, null, null ], "correct_answer": null, "solutions": [ { "id": 4017, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "ABCD is a square with side length s. M is a point on AB. Let AM = m. Finding the areas: Triangle I = △AMD Base = AM = m, Height = AD = s Area(I) = ½ × m × s = ms\/2 Triangle II = △MDC Base = DC = s, Height = s Area(II) = ½ × s × s = s²\/2 Triangle III = △MBC Base = MB = s - m, Height = BC = s Area(III) = ½ × (s - m) × s = (s - m)s\/2 Using the ratio: Area(I) : Area(II) = 3 : 8 (ms\/2) ÷ (s²\/2) = 3\/8 m\/s = 3\/8 m = 3s\/8 Substituting into Area(III): Area(III) = (s - m)s\/2 = 500 (s - 3s\/8) × s\/2 = 500 (5s\/8) × s\/2 = 500 5s²\/16 = 500 s² = 500 × 16\/5 s² = 1600 s = 40 cm The length of square ABCD is 40 cm." } ], "hints": [] }, "10597": { "status_id": 3, "type_id": 2, "question": "Tank A has a capacity of 24 l. Tank B measures 25 cm by 25 cm by 50 cm. Both tanks are half filled with water. When an equal quantity of water is poured into both tanks, the ratio of the water in tank A and tank B becomes 4 : 5. What is the increase in water level in tank B? Answer : cm", "question_raw": "Tank A has a capacity of 24 l. Tank B measures 25 cm by 25 cm by 50 cm. Both tanks are half filled with water. When an equal quantity of water is poured into both tanks, the ratio of the water in tank A and tank B becomes 4 : 5.
What is the increase in water level in tank B?
Answer : cm", "question_image": null, "a": [ "4", null, null, null ], "correct_answer": null, "solutions": [ { "id": 4028, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Finding initial water volumes: Tank A capacity = 24 l = 24000 cm³ Tank A half filled = 24000 ÷ 2 = 12000 cm³ Tank B capacity = 25 × 25 × 50 = 31250 cm³ Tank B half filled = 31250 ÷ 2 = 15625 cm³ Finding the quantity of water added: Let the quantity of water added to each tank = x cm³ Water in Tank A after = 12000 + x Water in Tank B after = 15625 + x Ratio = 4 : 5 (12000 + x) \/ (15625 + x) = 4\/5 5(12000 + x) = 4(15625 + x) 60000 + 5x = 62500 + 4x 5x − 4x = 62500 − 60000 x = 2500 cm³ Finding the increase in water level in Tank B: Base area of Tank B = 25 × 25 = 625 cm² Increase in water level = Volume added ÷ Base area = 2500 ÷ 625 = 4 cm The increase in water level in Tank B is 4 cm" } ], "hints": [] }, "10601": { "status_id": 3, "type_id": 2, "question": "When Jaz poured out 1.62 litres of oil from a rectangular container, the volume of oil was reduced by 1\/5. If the container had a base perimeter of 0.6m and its length is longer than its breadth by 6 cm, find the depth of oil in the container if it was filled to its capacity originally. Answer : cm", "question_raw": "When Jaz poured out 1.62 litres of oil from a rectangular container, the volume of oil was reduced by 1\/5. If the container had a base perimeter of 0.6m and its length is longer than its breadth by 6 cm, find the depth of oil in the container if it was filled to its capacity originally.
\n\nAnswer : cm", "question_image": null, "a": [ "37.5", null, null, null ], "correct_answer": null, "solutions": [ { "id": 4031, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Finding the original volume of oil: 1.62 litres = 1\/5 of original volume Original volume = 1.62 × 5 = 8.1 litres = 8100 cm³ Finding the dimensions of the base: Base perimeter = 0.6 m = 60 cm Let breadth = b cm Let length = b + 6 cm Perimeter = 2(length + breadth) 60 = 2(b + 6 + b) 60 = 2(2b + 6) 60 = 4b + 12 4b = 48 b = 12 cm Breadth = 12 cm Length = 12 + 6 = 18 cm Finding the base area: Base area = 18 × 12 = 216 cm² Finding the depth of oil: Volume = Base area × Depth 8100 = 216 × Depth Depth = 8100 ÷ 216 Depth = 37.5 cm The depth of oil in the container is 37.5 cm" } ], "hints": [] }, "10603": { "status_id": 3, "type_id": 2, "question": "A rectangular tank, 1 m long, 80 cm wide and 40 cm high was filled with some water. 8 cubes of water filled to the brim were poured into the tank. 12000 cm³ of water overflowed. After removing 3 cubes full of water, the height of the water in the tank was 37 cm. What was the volume of each metal cube? Answer : cm³ How many litres of water were there in the tank before the 8 cubes of water were poured in? Answer : litres", "question_raw": "A rectangular tank, 1 m long, 80 cm wide and 40 cm high was filled with some water. 8 cubes of water filled to the brim were poured into the tank. 12000 cm³ of water overflowed. After removing 3 cubes full of water, the height of the water in the tank was 37 cm.
What was the volume of each metal cube? Answer : cm³\n
How many litres of water were there in the tank before the 8 cubes of water were poured in? Answer : litres", "question_image": null, "a": [ "8000", "268", null, null ], "correct_answer": null, "solutions": [ { "id": 4034, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Finding the base area of tank: Length = 1 m = 100 cm Width = 80 cm Height = 40 cm Base area = 100 × 80 = 8000 cm² Working backwards from the final state: After removing 3 cubes, water height = 37 cm Volume of water after removing 3 cubes = 8000 × 37 = 296000 cm³ Finding volume before removing 3 cubes: Before removing 3 cubes, tank was full (40 cm height) Volume when full = 8000 × 40 = 320000 cm³ Finding the volume of 3 cubes: Volume of 3 cubes = 320000 − 296000 = 24000 cm³ Volume of 1 cube = 24000 ÷ 3 = 8000 cm³ Finding the original water volume: Let original water volume = W cm³ Water added from 8 cubes = 8 × 8000 = 64000 cm³ Water overflowed = 12000 cm³ Original water + Water added − Overflow = Full tank W + 64000 − 12000 = 320000 W + 52000 = 320000 W = 268000 cm³ W = 268000 ÷ 1000 = 268 litres The volume of each metal cube is 8000 cm³ The water in the tank before was 268 litres" } ], "hints": [] }, "10604": { "status_id": 3, "type_id": 2, "question": "Rectangular tank A measuring 50 cm by 30 cm had 30 litres of water. Tank B measuring measuring 40 cm by 25 cm had 1.5 litres of water. Some of the water from tank A was poured into tank B until the height of the water in both tanks were the same. Find the new height of the water in tank B. Answer : cm How many litres of water were left in tank A? Answer : litres", "question_raw": "Rectangular tank A measuring 50 cm by 30 cm had 30 litres of water. Tank B measuring measuring 40 cm by 25 cm had 1.5 litres of water. Some of the water from tank A was poured into tank B until the height of the water in both tanks were the same.
Find the new height of the water in tank B. Answer : cm
How many litres of water were left in tank A? Answer : litres", "question_image": null, "a": [ "12.6", "18.9", null, null ], "correct_answer": null, "solutions": [ { "id": 4036, "source": "human", "user_id": 41, "status_id": 1, "has_model": false, "text": "Finding the base areas: Base area of Tank A = 50 × 30 = 1500 cm² Base area of Tank B = 40 × 25 = 1000 cm² Finding the initial water volumes: Water in Tank A = 30 litres = 30000 cm³ Water in Tank B = 1.5 litres = 1500 cm³ Total water = 30000 + 1500 = 31500 cm³ Finding the new height: Let the new height in both tanks = h cm Volume in Tank A + Volume in Tank B = Total water (1500 × h) + (1000 × h) = 31500 2500h = 31500 h = 31500 ÷ 2500 h = 12.6 cm Finding the water left in Tank A: Water in Tank A = 1500 × 12.6 = 18900 cm³ = 18900 ÷ 1000 = 18.9 litres The new height of water in tank B is 12.6 cm The water left in tank A is 18.9 litres" } ], "hints": [] } }