1998 CEMC Gauss (Grade 7) Problems/Problem 22

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Problem

Each time a bar of soap is used, its volume decreases by 10%. What is the minimum number of times a new bar would have to be used so that less than one-half its volume remains?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Solution

After $x$ uses, the volume of the soap is $(0.9)^x.$ So, we are trying to solve \[(0.9)^x < 0.5.\] We can see that the solution is $x < 7,$ so the answer is $\text{(C)}.$

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