# SAT/ACT problem of the week, January 12, 2017 solution

Hint for process of elimination: This is considered a difficult problem to solve on the SAT directly, since the test makers don’t necessarily expect you to know logarithms. However, they do expect you to know the equation for percent growth, so you should have an equation to plug into. The equation is described below in the full solution.

Notice that the investment grows from $10,000 to$100,000, which is a tenfold increase in value of the investment. In this case, you might get a hint that the growth rate of the investment is pretty big. If the investment grows at a rate of 20% per year, then it takes less than 5 years for the value of the investment to double. That means it takes less than 15 years for the investment to double three times, which is a factor of $latex 2^3 = 8$. To grow by a factor of 10, it really should not take more than 15 years, at the roughest estimate. This leaves choices A (5 years), B (10 years), and C (12.6 years). Clearly 5 years is too short. After all, an investment that grows at 20% per year cannot grow 1000% in five years. This leaves choice B and C, giving you a 50% chance of getting the answer right. Even if you can’t decide on which answer is correct at this point, you should still give an answer.

Read on for a full solution.

Solution: You are expected to know and memorize the growth rate formula, which is:

$latex A = P(1+\frac{r}{n})^{nt}$

In this formula

• $latex A$ represents the final value of the investment
• $latex P$ is equal to the principal, i.e. the initial investment
• $latex r$ is the yearly interest rate, converted to a decimal. In this case $latex 20\% = .20$
• $latex n$ is the number of times the interest is compounded per year
• $latex t$ is the number of years that have passed

Using process of elimination, trying the numbers one at a time will give you the final answer. To demonstrate a substitution/check, we will plug in C, the correct answer.

$latex 100000 \approx 10000 (1 + \frac{.20}{1})^{1 \times 12.6} = 99468$

$99468 is very close to$100000.

Work it out! Make sure you get the same answer.