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?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9
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?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Solution

Number of Times Soap Used

Approximate Volume Remaining (as \%)

1

2

3

4

5

6

7
0.9 or $90 \%$

$(0.9)^{2}$ or $81 \%$

$(0.9)^{3}$ or $72.9 \%$

$(0.9)^{4}$ or $65.61 \%$

$(0.9)^{5}$ or $59.1 \%$

$(0.9)^{6}$ or $53.1 \%$

$(0.9)^{7}$ or $47.8 \%$

So if the soap is used 7 times the volume will be less than $\frac{1}{2}$ of the original volume.

NOTE: In essence, we are trying to find a positive integer $x$ so that $(0.9)^{x}<0.5$. The value of $x$ can be found by using the $y^{x}$ button on your calculator where $y=0.9$ and experimenting to find a value for $x$.

ANSWER: (C)