Problem 14 · 1986 AJHSME
Hard
Fractions, Decimals & Percents
maximize-fraction
If 200 ≤ a ≤ 400 and 600 ≤ b ≤ 1200, then the largest value of the quotient b ⁄ a is
Show answer
Answer: C — 6.
Show hints
Hint 1 of 2
A fraction grows two ways: a bigger top *or* a smaller bottom. To make b⁄a as large as it can be, push each part to its own extreme — which way for b, which way for a?
Still stuck? Show hint 2 →
Hint 2 of 2
Top as big as allowed, bottom as small as allowed: use b at its maximum and a at its minimum. The two choices don't interfere, so you can pick both extremes at once.
Show solution
Approach: push top up, bottom down
- A quotient is largest when its numerator is largest and its denominator is smallest — and those two pushes are independent, so you can do both. Pick b at its top (1200) and a at its bottom (200).
- b ⁄ a = 1200 ⁄ 200 = 6.
- Watch the trap: it's tempting to use the two biggest numbers (1200 and 400), but a big denominator *shrinks* the fraction — 1200⁄400 = 3, smaller than 6. Smallest bottom is what matters.
- Why this transfers: any 'maximize a fraction in a range' problem reduces to max-numerator-over-min-denominator (assuming everything stays positive).
Mark:
· log in to save