The value, in dollars, of an investment t years after it was... | SAT Nonlinear functions Question, Answer Explained

Math · Advanced MathHard

The value, in dollars, of an investment $t$ years after it was made is modeled by $V(t) = 8{,}000(1.21)^t$ . According to the model, by what percent does the value of the investment increase every $6$ months?

$10\%$

$10.5\%$

$21\%$

$42\%$

Show answer and explanation

Answer: A

Six months is $\frac{1}{2}$ year, so every $6$ months the value is multiplied by $(1.21)^{\frac{1}{2}} = \sqrt{1.21} = 1.1$ . Multiplying by $1.1$ is a $10\%$ increase. Check: two $10\%$ increases give $1.1^2 = 1.21$ , a $21\%$ increase per year.

Choice B splits the $21\%$ yearly increase into two equal parts, but percent increases compound, so the half-year increase is less than $10.5\%$ .

Why the other choices are wrong

B. This divides 21% by 2. Growth compounds, and 1.105² is about 1.221, which is more than 1.21.
C. 21% is the increase for a full year, not for 6 months.
D. This doubles the yearly rate. The increase over 6 months has to be smaller than the yearly increase, not larger.

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