Problem 20 · 1986 AJHSME
Hard
Fractions, Decimals & Percents
round-then-estimate
The value of the expression (304)⁵ ⁄ ((29.7)(399)⁴) is closest to
Show answer
Answer: D — 3.
Show hints
Hint 1 of 3
Never compute a fifth power here — the word 'closest' plus those round-ish numbers (304, 399, 29.7) is begging you to round to 300, 400, 30 first.
Still stuck? Show hint 2 →
Hint 2 of 3
After rounding, you have 300⁵ on top and 400⁴ on the bottom. Don't expand them — pair up four of the 300s with the four 400s to make (300⁄400)⁴ = (3⁄4)⁴, and you're left with one spare 300 over the 30.
Still stuck? Show hint 3 →
Hint 3 of 3
(3⁄4)⁴ is a bit under ⅓, and the leftover 300⁄30 = 10, so the product is roughly 10 × (a third).
Show solution
Approach: round, then pair the powers into (3⁄4)⁴
- 'Closest' means estimate, so round to friendly numbers: 304 → 300, 399 → 400, 29.7 → 30. The expression becomes 300⁵ ⁄ (30 · 400⁴).
- Don't expand the powers — regroup instead. Match four 300s against the four 400s: (300⁄400)⁴ = (3⁄4)⁴. The fifth 300 pairs with the 30 to give 300⁄30 = 10.
- So it's (3⁄4)⁴ · 10 = (81⁄256) · 10 ≈ 0.316 · 10 ≈ 3.16, closest to 3.
- Sanity check the size: 81⁄256 is just under ⅓, and ⅓ of 10 is about 3.3 — so the answer is a small single-digit number, instantly ruling out the .003, .03, .3, and 30 options. The art is grouping powers before multiplying, not crunching huge numbers.
Mark:
· log in to save