1981 AMC 12 Problem 13

Attempt Problem 13 of the 1981 AMC 12 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 1981 AMC 12 solutions, or check the answer key.

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

13.

Suppose that at the end of any year, a unit of money has lost

10\%

of the value it had at the beginning of that year. Find the smallest integer

n

such that after

n

years the unit of money will have lost at least

90\%

of its value. (To the nearest thousandth

\log_{10}3

0.477.

)

14

16

18

20

22

After

n

years, the fraction of value remaining is

0.9^n

Use

\log_{10}0.9=2\log_{10}3-1

At least

90\%

lost means

0.9^n\le0.1.

Now

\begin{aligned} \log_{10}0.9 &=\log_{10}9-1\\ &=2(0.477)-1\\ &=-0.046. \end{aligned}

Thus

n(-0.046)\le-1,

n\ge21.739\ldots.

The least integer is

22.

Therefore, the correct answer is E.

Problem 13 in other years