Work backwards from the end using the exchange information.<br><br>
[[model:questions/solutions/q5332.png]]<br><br>
After exchange, each girl has 16 fruits and \( \frac{1}{3} \) as many apples as oranges.<br>
&bull; For each girl: apples : oranges = 1 : 3.<br>
&bull; Each girl has 4 units of fruit → 4 units = 16, 1 unit = 4.<br>
&bull; So each has 4 apples and 12 oranges after exchange.<br><br>
Before Belle gave half her oranges to Angie:<br>
&bull; Belle had 12 × 2 = 24 oranges (she kept 12, gave 12 to Angie).<br>
&bull; Angie had 12 - 12 = 0 oranges originally.<br><br>
Before Angie gave half her apples to Belle:<br>
&bull; Angie had 4 × 2 = 8 apples (she kept 4, gave 4 to Belle).<br>
&bull; Belle had 4 - 4 = 0 apples originally.<br><br>
Angie bought 8 apples and Belle bought 24 oranges.<br>
Check: Exchange → Angie: 4A+12O=16; Belle: 4A+12O=16. Ratio 4:12=1:3. ✓<br><br>
Final Answer: Angie bought 8 apples, Belle bought 24 oranges.