Which equation correctly gives in terms of and ? (Assume and the denominators are nonzero.)

Solution available with step-by-step explanation

Super-Hard SAT Math Question 78 | Free SAT Prep | Aniko

00:00

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

0 / 200

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.)

When a variable appears only in a denominator, solve by first clearing any common factor (here, cancel $m$ ), then combine the fractions on the other side using a common denominator. Be especially careful with subtraction and with expanding $(q+r)^2$ , and remember that getting $\frac{1}{p}$ is not the same as getting $p$ —take the reciprocal at the end.

Hints

Click me!

Remove the common factor

Since $m$ appears in every term and $m\ne0$ , divide both sides of the equation by $m$ .

Combine carefully

First add $\frac{1}{q}$ and $\frac{1}{r}$ , then subtract $\frac{1}{q+r}$ using a single common denominator.

Remember the final move

After you find an expression for $\frac{1}{p}$ , you still need to take the reciprocal to get $p$ .

Desmos Guide

Create sliders

Create sliders for $m$ , $q$ , and $r$ . Choose values that keep denominators nonzero (for example, avoid $q=0$ , $r=0$ , and $q+r=0$ ).

Define the right-hand side value

Define

R = m/q + m/r - m/(q+r)

This is the right-hand side of the original equation.

Test each answer choice as a candidate for $p$

For each option, define a candidate $p$ (for example, p1 = ...), then define

D1 = m/p1 - R

Repeat for the other options (D2, D3, D4).

Identify which option works for many slider settings

Move the sliders to several different valid values. The correct option is the one whose difference expression stays at $0$ (or extremely close to $0$ ) for all valid settings.

Step-by-step Explanation

Cancel the common factor $m$

Since $m\ne 0$ , divide both sides by $m$ :

\frac{1}{p}=\frac{1}{q}+\frac{1}{r}-\frac{1}{q+r}

Combine the first two fractions

Add $\frac{1}{q}$ and $\frac{1}{r}$ :

\frac{1}{q}+\frac{1}{r}=\frac{r+q}{qr}=\frac{q+r}{qr}

Subtract using a common denominator

Now subtract $\frac{1}{q+r}$ :

\frac{1}{p}=\frac{q+r}{qr}-\frac{1}{q+r}

Use the common denominator $qr(q+r)$ :

\frac{q+r}{qr}=\frac{(q+r)^2}{qr(q+r)},\qquad \frac{1}{q+r}=\frac{qr}{qr(q+r)}

\frac{1}{p}=\frac{(q+r)^2-qr}{qr(q+r)}

Simplify and take the reciprocal

Expand and simplify the numerator:

(q+r)^2-qr=q^2+2qr+r^2-qr=q^2+qr+r^2

Thus,

\frac{1}{p}=\frac{q^2+qr+r^2}{qr(q+r)}

Taking the reciprocal gives

p=\frac{qr(q+r)}{q^2+qr+r^2}

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation