In a lab, the concentrations of three solutions , , and are parts per million, parts per million, and parts per million, respectively. If the concentration of solution is 125% of the concentration of solution , and the concentration of solution is 0.006% of the concentration of solution , which expression represents the value of in terms of ?

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Super-Hard SAT Math Question 38 | Free SAT Prep | Aniko

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Question 38·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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In a lab, the concentrations of three solutions $D$ , $E$ , and $F$ are $d$ parts per million, $e$ parts per million, and $f$ parts per million, respectively. If the concentration of solution $D$ is 125% of the concentration of solution $F$ , and the concentration of solution $F$ is 0.006% of the concentration of solution $E$ , which expression represents the value of $d+e$ in terms of $f$ ?

When percent statements link three quantities, translate each statement into an equation immediately using $x=\frac{p}{100}y$ . For a tiny percent, convert it to a fraction first to avoid misplaced decimals, then solve for the variable you need (here, solve for $e$ in terms of $f$ ). Only after both $d$ and $e$ are in terms of the same variable should you add them, using a common denominator.

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Translate each percent statement into an equation

Use “ $x$ is $p$ % of $y$ ” $\Rightarrow$ $x=\frac{p}{100}y$ .

Be careful with the very small percent

First turn 0.006% into a decimal or fraction of 1, then multiply by $e$ .

Solve for $e$ before adding

After writing $f$ as a fraction of $e$ , rearrange to get $e$ in terms of $f$ , then compute $d+e$ .

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Define a convenient value for $f$

Enter f=12 so that expressions with denominator 12 become whole numbers when evaluated.

Compute $d$ and $e$ from the percent relationships

Enter d=(125/100)*f.

Enter e=f/(0.006/100) (because $f=(0.006/100)e$ ).

Compute the total and compare to the options

Enter total=d+e.

Then enter each option’s value at f:

opt1=(200015/12)*f

opt2=(200005/12)*f

opt3=(50005/3)*f

opt4=(200015/4)*f

The correct choice is the one whose value matches total.

Step-by-step Explanation

Write $d$ in terms of $f$

“125% of $f$ ” means multiply $f$ by $\frac{125}{100}=\frac{5}{4}$ .

So,

d=\frac{5}{4}f

Convert 0.006% to a multiplier

“0.006% of $e$ ” means

0.006\% = \frac{0.006}{100}=0.00006=\frac{6}{100{,}000}=\frac{3}{50{,}000}

So “ $f$ is 0.006% of $e$ ” becomes

f=\frac{3}{50{,}000}e

Solve for $e$ in terms of $f$

From

f=\frac{3}{50{,}000}e

multiply both sides by $\frac{50{,}000}{3}$ :

e=\frac{50{,}000}{3}f

Add $d$ and $e$

Now add the expressions:

\begin{aligned} d+e &= \frac{5}{4}f+\frac{50{,}000}{3}f \\ &=\left(\frac{15}{12}+\frac{200{,}000}{12}\right)f \\ &=\frac{200{,}015}{12}f \end{aligned}

So the correct expression is $\frac{200{,}015}{12}f$ .

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