The function is defined by For any positive integer , the value of is % less than the value of . Which choice is the value of ?

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Super-Hard SAT Math Question 9 | Free SAT Prep | Aniko

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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 $p$ ?

When a function is written as a product with exponents, first look for bases that are reciprocals (like $0.4$ and $2.5$ ). Rewrite them as fractions and use exponent rules to combine everything into the form $C\cdot r^x$ . Then the percent change from $n-1$ to $n$ is determined only by the ratio $r$ : if $f(n)=r\,f(n-1)$ , the percent decrease is $(1-r)\cdot 100$ .

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Rewrite the decimals as fractions

Convert $0.4$ and $2.5$ to fractions. Check whether one is the reciprocal of the other.

Combine the exponential factors

Use $a^{1-x}=a\cdot a^{-x}$ and the rule $a^m\cdot a^n=a^{m+n}$ to rewrite $f(x)$ as a constant times one base raised to $x$ .

Compare consecutive inputs

Find the constant ratio $\frac{f(n)}{f(n-1)}$ . Then convert that multiplier into a percent decrease by computing $1-$ (multiplier).

Desmos Guide

Enter the function

In Desmos, enter

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

Use a table to compare consecutive values

Click Add Item > Table. In the first column, type a few integers like $1,2,3$ .

In the second column, enter $f(x_1)$ to generate $f(1), f(2), f(3)$ .

Find the common ratio and convert to a percent decrease

Compute the ratio $\frac{f(2)}{f(1)}$ (and check it matches $\frac{f(3)}{f(2)}$ ).

Then compute $1-$ (that ratio), and multiply by $100$ to get $p$ .

Step-by-step Explanation

Rewrite using fractions and exponent rules

Note that $0.4=\frac{2}{5}$ and $2.5=\frac{5}{2}$ .

Then

\begin{aligned} (0.4)^x(2.5)^{1-x} &=\left(\frac{2}{5}\right)^x\left(\frac{5}{2}\right)^{1-x} \\ &=\left(\frac{5}{2}\right)\left(\frac{2}{5}\right)^x\left(\frac{5}{2}\right)^{-x}. \end{aligned}

Since $\left(\frac{5}{2}\right)^{-x}=\left(\frac{2}{5}\right)^x$ , the product becomes

\left(\frac{5}{2}\right)\left(\frac{2}{5}\right)^x\left(\frac{2}{5}\right)^x =\left(\frac{5}{2}\right)\left(\frac{2}{5}\right)^{2x}.

Identify the common ratio from $f(n-1)$ to $f(n)$

f(x)=150\left(\frac{5}{2}\right)\left(\frac{2}{5}\right)^{2x}.

When $x$ increases by $1$ , the factor $\left(\frac{2}{5}\right)^{2x}$ is multiplied by

\left(\frac{2}{5}\right)^2=\frac{4}{25}=0.16.

Therefore,

\frac{f(n)}{f(n-1)}=0.16.

Convert the ratio to a percent decrease

From $\frac{f(n)}{f(n-1)}=0.16$ , we have $f(n)=0.16\,f(n-1)$ .

That means $f(n)$ is 16% of $f(n-1)$ , so it is $1-0.16=0.84$ , or 84%, less than $f(n-1)$ .

Therefore, $p=84$ , and the correct choice is $84$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation