How many char siu rice boxes were there?

Question

An office ordered lunch boxes from a roast-meat shop: char siu rice at $37 a box, roast duck rice at $40 and two-meat rice at $50, paying $2900 in all.

There were 3 more roast duck boxes than three times the two-meat boxes.

Had the numbers of char siu and two-meat boxes been swapped, the bill would have been $3004.

Accepted Answer

Answer: 20 boxes. After the swap the roast duck is the same. Swapping char siu ($37) with two-meat ($50) adds $3004 − $2900 = $104. Where the counts match, swapping changes nothing; each extra char siu turned two-meat adds $50 − $37 = $13. So there are 104 ÷ 13 = 8 more char siu than two-meat. Bundle 1 char siu, 3 roast duck and 1 two-meat: $37 + 3 × $40 + $50 = $207 a set. One set per two-meat box leaves 8 char siu and 3 roast duck. The leftovers are worth 37 × 8 + 40 × 3 = $416, so the sets come to $2900 − $416 = $2484: 2484 ÷ 207 = 12 sets, 12 two-meat boxes. Char siu: 12 + 8 = 20 boxes. Check: roast duck 3 × 12 + 3 = 39. 37 × 20 + 40 × 39 + 50 × 12 = 2900 ✓; swapped, 50 × 20 + 40 × 39 + 37 × 12 = 3004 ✓ Common mistake: thinking every char siu and two-meat box changes the bill; only the 8 extra char siu do. And when bundling, don't forget the 3 extra roast duck.

Chickens and rabbits: olympiad question 8

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