Question 9·200 Super-Hard SAT Math Questions·Advanced Math

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The function $f$ is defined by

f(x)=150(0.4)^x(2.5)^{1-x}.

For any positive integer $n$ , the value of $f(n)$ is $p$ % less than the value of $f(n-1)$ . Which choice is the value of $p$ ?

In an exponential change question, compare an output with the one immediately before it. If the function contains two powers of the input, track both powers as the input increases by one. You can evaluate consecutive outputs in Desmos to check the multiplier. The multiplier tells you what remains, not what was lost.

Hints

Hint 1
A percent decrease compares the new output with the previous one. If $p\%$ is lost, $100\%-p\%$ remains. In the question’s comparison, which function value is the previous output?
Hint 2
As the input increases by $1$ , one exponent rises and the other falls. Both powers change, so account for both before deciding how much of the previous output remains.
Hint 3
The statement holds for any positive integer $n$ , so you may use $n=1$ and compare $f(1)$ with $f(0)$ . Their ratio is the fraction of the previous output that remains.

Step-by-step

Find the one-step multiplier
Step 1
Translate “less than” into a ratio
The previous output is $f(n-1)$ . A ratio compares the new output with that previous output by dividing. If the new output is $p\%$ less, it keeps $1-\frac{p}{100}$ of the previous one, so:
$\frac{f(n)}{f(n-1)}=1-\frac{p}{100}.$
Step 2
Track both powers for one input step
When $x$ increases by $1$ , $(0.4)^x$ is multiplied by $0.4$ . At the same time, $1-x$ decreases by $1$ , so $(2.5)^{1-x}$ is divided by $2.5$ . The $150$ doesn't change. A multiplier is the factor taking one output to the next, so for every step:
$\frac{f(n)}{f(n-1)}=\frac{0.4}{2.5}.$
Step 3
Evaluate the multiplier in Desmos
Because the statement holds for any positive integer $n$ , choose $n=1$ . Type $f(x)=150(0.4)^x(2.5)^{1-x}$ , then $\frac{f(1)}{f(0)}$ . Desmos prints $0.16$ . So each new output is $0.16$ of the previous output, not $0.4$ of it.
Step 4
Find the percent lost
The multiplier $0.16$ means $16\%$ remains, not that $16\%$ was lost. Subtract what remains from the whole:
$100\%-16\%=84\%.$
So $f(n)$ is $84\%$ less than $f(n-1)$ , and $p=84$ . Choice A.

Question 9·200 Super-Hard SAT Math Questions·Advanced Math

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