Find how much money remains after Day 2, then determine how many days of \(\(\frac{1}{10}\)\) spending are needed to use it up.<br><br>
&bull; Used on Day 1: \(\(\frac{1}{5} = \frac{4}{20}\)\)<br>
&bull; Used on Day 2: \(\(\frac{1}{4} = \frac{5}{20}\)\)<br>
&bull; Total used by end of Day 2: \(\(\frac{4}{20} + \frac{5}{20} = \frac{9}{20}\)\)<br>
&bull; Remaining after Day 2: \(\(1 - \frac{9}{20} = \frac{11}{20}\)\)<br>
&bull; From Day 3 onwards, he uses \(\(\frac{1}{10} = \frac{2}{20}\)\) per day<br>
&bull; Days needed: \(\(\frac{11}{20} \div \frac{2}{20} = \frac{11}{2} = 5\frac{1}{2}\)\) days<br>
&bull; Days 3, 4, 5, 6, 7 (5 full days): uses \(\(5 \times \frac{2}{20} = \frac{10}{20}\)\). Total = \(\(\frac{9}{20} + \frac{10}{20} = \frac{19}{20}\)\). Remaining = \(\(\frac{1}{20}\)\)<br>
&bull; On Day 8, he needs only \(\(\frac{1}{20}\)\) (half of his usual \(\(\frac{2}{20}\)\)), so he uses up all his money on Day 8<br>
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He uses up all his money on Day 8.<br>
Check: Total = \(\(\frac{4}{20} + \frac{5}{20} + 5 \times \frac{2}{20} + \frac{1}{20} = \frac{20}{20}\)\) = 1 whole. ✓<br><br>
Final Answer: Day 8