The Summer Camp Ratio Problem
A summer camp has 84 places and receives applications in the ratio 5:3:4 for three age groups - juniors, intermediates and seniors. The camp must fill every place and keep the ratio exactly. A coordinator notices that the ratio does not divide 84 cleanly and proposes adjusting the senior intake upward by one 'share' to make the numbers work. Separately, the camp's insurance requires that no single age group exceed 40% of total places. The coordinator also wants to know how many additional juniors would be needed to reach exactly 40% juniors if the ratio were abandoned.
1. Test whether the ratio 5:3:4 can fill all 84 places exactly. If not, find the closest whole-number allocation that keeps the ratio as nearly as possible.
Read the worked answer
Total shares = 5 + 3 + 4 = 12. 84 ÷ 12 = 7 exactly - so the ratio does divide 84 cleanly.
Juniors = 5 × 7 = 35; intermediates = 3 × 7 = 21; seniors = 4 × 7 = 28. Sum = 84. ✓
The coordinator is mistaken: no adjustment is needed.
2. Check the insurance condition against the correct allocation, and identify which group (if any) breaches the 40% cap.
Read the worked answer
40% of 84 = 33.6 places.
Juniors = 35 places. 35 > 33.6, so juniors breach the 40% cap (35 ÷ 84 = 41.7%). Intermediates = 21 (25%), seniors = 28 (33.3%) - both fine.
So the ratio-compliant allocation is arithmetically correct but fails the insurance condition.
3. If the ratio were abandoned and juniors capped at exactly 40% of 84, how many juniors would that be, and how many places would remain for the other two groups?
Read the worked answer
40% of 84 = 33.6, which is not a whole number of people, so exactly 40% is impossible. The maximum whole number at or below 40% is 33 juniors (39.3%).
Remaining places = 84 − 33 = 51 for intermediates and seniors.
This is the real judgement call: you cannot hit 40% exactly with 84 places, so you must round down to 33 to stay compliant.