Question 78·200 Super-Hard SAT Math Questions·Advanced Math

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Which equation correctly gives $p$ in terms of $q$ and $r$ ?

\frac{m}{p}=\frac{m}{q}+\frac{m}{r}-\frac{m}{q+r}

(Assume $m\ne 0$ and the denominators are nonzero.)

The words “ $p$ in terms of $q$ and $r$ ” signal a formula rearrangement: get $p$ alone while leaving $q$ and $r$ as letters. A Desmos regression with allowed sample values can check your work, but one sample cannot prove a formula. Cancel the nonzero $m$ , combine the fractions using a common denominator, and invert only after you have one fraction equal to $1/p$ .

Hints

Hint 1
The problem gives $m\ne0$ , so you can divide the whole equation by $m$ . What does each fraction become when its numerator $m$ cancels?
Hint 2
A common denominator contains every original denominator. For $q$ , $r$ , and $q+r$ , use $qr(q+r)$ , then give each numerator the factors its denominator was missing.
Hint 3
After combining the right side, the left side is still $1/p$ , not $p$ . Once the right side is a single fraction, what operation gets $p$ alone?

Step-by-step

Combine the fractions, then invert
Step 1
Get a numerical check in Desmos
For a check, choose $m=1$ , $q=2$ , and $r=3$ . These values meet the given nonzero conditions. Type the three assignments and the original equation, changing $p$ to $p_1$ and $=$ to $\sim$ so Desmos solves for $p_1$ . Under PARAMETERS, it shows $p_1\approx1.578947$ . Keep that value to check your formula later.
Step 2
Cancel the common numerator
The given $m\ne0$ lets you divide both whole sides by $m$ . Cancel it from every term:
$\frac{1}{p}=\frac{1}{q}+\frac{1}{r}-\frac{1}{q+r}.$
Step 3
Give the fractions one denominator
The common denominator $qr(q+r)$ contains each denominator on the right. Multiply each fraction's numerator and denominator by its missing factors:
$\frac{1}{p}=\frac{r(q+r)}{qr(q+r)}+\frac{q(q+r)}{qr(q+r)}-\frac{qr}{qr(q+r)}.$
Step 4
Combine the numerators
Now the denominators match, so add and subtract only the numerators:
$\frac{1}{p}=\frac{r(q+r)+q(q+r)-qr}{qr(q+r)}.$
Step 5
Distribute without losing a term
Distribute both parentheses, keeping the $-qr$ at the end:
$\frac{1}{p}=\frac{qr+r^2+q^2+qr-qr}{qr(q+r)}.$
Step 6
Combine like terms
There are two positive $qr$ terms and one negative $qr$ term, so one positive $qr$ remains. Combine them:
$\frac{1}{p}=\frac{q^2+qr+r^2}{qr(q+r)}.$
Step 7
Invert to get p
You have $1/p$ , but the question asks for $p$ . Combine the reciprocals first; invert only the finished fraction. Invert both sides:
$p=\frac{qr(q+r)}{q^2+qr+r^2}.$
Type that fraction on the next Desmos line. It shows about $1.578947$ for your chosen values, matching $p_1$ . The algebra establishes the formula; Desmos checks it. Choice A.

Question 78·200 Super-Hard SAT Math Questions·Advanced Math

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