1991 AMC 8 Problem 7

Below is the professionally curated solution for Problem 7 of the 1991 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1991 AMC 8 solutions, or check the answer key.

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Concepts:estimation distributive property

Difficulty rating: 890

The value of $\frac{(487{,}000)(12{,}027{,}300) + (9{,}621{,}001)(487{,}000)}{(19{,}367)(.05)}$ is closest to

$10{,}000{,}000$

$100{,}000{,}000$

$1{,}000{,}000{,}000$

$10{,}000{,}000{,}000$

$100{,}000{,}000{,}000$

Solution:

Factor the numerator: $487{,}000\,(12{,}027{,}300 + 9{,}621{,}001).$

Rounding to leading digits, this is about $500{,}000 \times (10{,}000{,}000 + 10{,}000{,}000) = 500{,}000 \times 2 \times 10^{7}.$ The denominator is about $20{,}000 \times .05 = 1000.$

So the value is roughly $\dfrac{500{,}000 \times 2 \times 10^{7}}{1000} = 10^{10} = 10{,}000{,}000{,}000.$

Thus, the correct answer is D .

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