2011 AMC 10B Problem 10

Attempt Problem 10 of the 2011 AMC 10B 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 2011 AMC 10B solutions, or check the answer key.

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10.

Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}.$ The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

$1$

$9$

$10$

$11$

$101$

Answer: B

Concepts:geometric sequence estimation

Difficulty rating: 1280

Solution:

The largest number is $10^{10}.$ The other ten numbers have sum $S=\sum_{i=0}^9 10^i=\dfrac{10^{10}-1}{9}.$ Therefore their ratio is $\dfrac{10^{10}}{S}=9\cdot\dfrac{10^{10}}{10^{10}-1},$ which is just slightly greater than $9$ and hence closest to $9.$

Thus, the correct answer is B .

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