Question 1036·Hard·Math

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The positive variables $a,\,b$ , and $c$ satisfy

\frac{45}{a}=\frac{30}{b}-\frac{15}{c}.

Which of the following is equivalent to $b$ ?

An equation with the requested letter in a denominator calls for rearranging a rational formula. A Desmos regression with distinct positive values for the other letters can check one case, but algebra gives a formula that works for every allowed value. Divide out any shared numerical factor, combine the fractions, then clear denominators. The trap is reporting the reciprocal of the target as the target.

Hints

Hint 1
For a numerical check, choose different positive values for $a$ and $c$ . A Desmos regression can find the corresponding $b$ , but one set of values cannot prove which formula always works.
Hint 2
All three numerators share a factor of $15$ . Divide every term by $15$ before moving the fraction with $c$ ; the smaller coefficients make the next step easier to track.
Hint 3
To add $3/a$ and $1/c$ , use the common denominator $ac$ . The first numerator becomes $3c$ , because you multiply its top and bottom by $c$ . What does the second numerator become?

Step-by-step

Check one case, then isolate $b$
Step 1
Find a sample value of $b$
Choose $a=2$ and $c=3$ : they are different positive values, so a check won't hide a difference between $a$ and $c$ . Type $a=2$ , $c=3$ , and $\frac{45}{a}\sim\frac{30}{b_1}-\frac{15}{c}$ . The regression symbol $\sim$ tells Desmos to find $b_1$ , our sample value of $b$ . Under PARAMETERS, Desmos reports $b_1\approx1.091$ . This checks one case; it doesn't give the general formula.
Step 2
Reduce the coefficients
Each term in the given equation has a numerator divisible by $15$ . Divide every term by $15$ :
$\frac{3}{a}=\frac{2}{b}-\frac{1}{c}.$
Step 3
Put the other fractions together
Add $\frac{1}{c}$ to both sides, so the fraction containing $b$ stays alone:
$\frac{3}{a}+\frac{1}{c}=\frac{2}{b}.$
Step 4
Combine the left side
Use the common denominator $ac$ . Multiply each numerator by the factor its denominator is missing. Rewrite the fractions:
$\frac{3c}{ac}+\frac{a}{ac}=\frac{2}{b}.$
Add their numerators:
$\frac{3c+a}{ac}=\frac{2}{b}.$
Step 5
Clear the denominators
Because $a$ , $b$ , and $c$ are positive, none is zero. Multiply both sides by $abc$ to remove both denominators:
$b(3c+a)=2ac.$
Step 6
Isolate $b$
Since $a$ and $c$ are positive, $3c+a$ is positive too. Divide by $3c+a$ :
$b=\frac{2ac}{3c+a}.$
So the expression equivalent to $b$ is $\frac{2ac}{3c+a}$ . Choice B.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 1036·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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