Problem 24 · 2002 AMC 8
Stretch
Ratios, Rates & Proportions
ratioproportion
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
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Answer: B — 40%.
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Hint 1 of 2
The "dozen" is pure bait — she uses an *equal* number of each fruit, so however many that is divides out. All that matters is the juice from *one* pear versus *one* orange.
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Hint 2 of 2
Pears: 8 oz from 3, so 8/3 oz each. Oranges: 8 oz from 2, so 4 oz each. The blend's pear share is just one pear's juice over (one pear + one orange).
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Approach: compare juice per fruit
- Since she blends *equal counts* of each fruit, the common count cancels — work per fruit. One pear yields 8/3 oz; one orange yields 8/2 = 4 oz.
- Pear-to-orange juice is 8/3 : 4, and clearing the 3 gives 8 : 12 = 2 : 3.
- Pear's share = 2 ÷ (2 + 3) = 2/5 = 40%.
- *Worth keeping:* when two things are mixed in equal *counts*, the actual count is irrelevant — reduce to one of each and compare. Chasing the dozen (32 oz vs 48 oz) gives the same answer with bigger numbers.
Another way — scale up to the full dozen:
- 12 pears give 12 × 8/3 = 32 oz; 12 oranges give 12 × 4 = 48 oz.
- Pear fraction = 32 ÷ (32 + 48) = 32/80 = 2/5 = 40% — same ratio, just unscaled.
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