1986 AMC 8 Problem 20

Attempt Problem 20 of the 1986 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1986 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

20.

The value of the expression
(304)5(29.7)(399)4\frac{(304)^5}{(29.7)(399)^4}
is closest to
0.0030.003
0.030.03
0.30.3
33
3030
Answer: D
Concepts:estimationexponent
Small hint:
Round the numbers: 304≈300,304 \approx 300, 29.7≈30,29.7 \approx 30, and 399≈400399 \approx 400
Big hint:
Simplify 300530⋅4004=10(34)4\dfrac{300^5}{30 \cdot 400^4} = 10 \left(\dfrac34\right)^4
Solution:
Estimating,
300530⋅4004=10(300400)4=10(34)4=10⋅81256≈3. \begin{aligned} \dfrac{300^5}{30 \cdot 400^4} &= 10 \left(\dfrac{300}{400}\right)^4 \\ &= 10 \left(\dfrac34\right)^4 \\ &= 10 \cdot \dfrac{81}{256} \approx 3. \end{aligned}
Thus, the correct answer is D .
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