Question 21·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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$a$ is 62.5% more than $b$ , and $b$ is 0.32% of $c$ .

If $a$ is 0.13% of $d$ , then $d$ is $k$ % of $c$ . Which choice must $k$ equal?

A chain of percent relationships can use a different base in each sentence, so identify what each percent is taken of. When no amount is fixed, choose 100 for the quantity in the final comparison. Then use Desmos percent entries to follow the chain and a one-variable regression to work backward. The key trap is treating “more than” as “of” and leaving out the original 100%.

Hints

Hint 1
Because the question gives only percent comparisons, scaling all the amounts together won’t change the answer. Choose $c=100$ . What will $d$ equal if it is $k\%$ of 100?
Hint 2
The phrase “62.5% more than” keeps all of $b$ and adds $62.5\%$ of $b$ . What percent of $b$ is $a$ altogether?
Hint 3
In “ $a$ is $0.13\%$ of $d$ ,” $d$ is the base, the amount the percent is taken of. Put the unknown $d$ after “of” when you ask Desmos to solve.

Step-by-step

Choose 100 and follow the percent chain
Step 1
Make the requested percent readable
Choose $c=100$ . The question gives only percent comparisons, so scaling all the amounts together won’t change $k$ . Since $k\%$ of 100 is $k$ , the number you find for $d$ will also be $k$ .
Step 2
Find $b$ from $c$
The base of “ $0.32\%$ of $c$ ” is $c$ . Type $0.32\%\operatorname{of}100$ in Desmos. It prints $0.32$ , so $b=0.32$ .
Step 3
Turn “more than” into the full percent
More than keeps the original $100\%$ of $b$ before adding the increase. Add the two percents: $100\%+62.5\%=162.5\%$ . So $a$ is $162.5\%$ of $b$ , not $62.5\%$ of $b$ .
Step 4
Find $a$
Type $162.5\%\operatorname{of}0.32$ beneath the first Desmos line. It prints $0.52$ , so $a=0.52$ .
Step 5
Put the unknown base after “of”
The problem says $a$ is $0.13\%$ of $d$ , and you found $a=0.52$ . So the equation is $0.13\%\text{ of }d=0.52$ . Here $d$ is the base. Taking $0.13\%$ of $0.52$ instead would go in the wrong direction.
Step 6
Solve for $d$ and read $k$
Type $0.13\%\operatorname{of}x_{1}\sim0.52$ . Use $x_{1}$ for the unknown $d$ and $\sim$ to tell Desmos to solve. Under PARAMETERS, Desmos shows $x_{1}=400$ . With $c=100$ , that means $d$ is $400\%$ of $c$ , so $k=400$ . Choice D.

Question 21·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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