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2014 AMC 10B Problem 7

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2014 AMC 10B Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

Suppose $A>B>0$ and $A$ is $x\%$ greater than $B$ . What is $x$ ?

A)
$100\left(\frac{A-B}{B}\right)$
B)
$100\left(\frac{A+B}{B}\right)$
C)
$100\left(\frac{A+B}{A}\right)$
D)
$100\left(\frac{A-B}{A}\right)$
E)
$100\left(\frac{A}{B}\right)$

Answer

Key insight

Percent greater is measured relative to the smaller base B: x/100 = (A - B)/B.

Solution

" $A$ is $x\%$ greater than $B$ " means $A = B\left(1 + \frac{x}{100}\right)$ .

Solve for $x$ : $\frac{A}{B} = 1 + \frac{x}{100}$ , so $\frac{x}{100} = \frac{A}{B} - 1 = \frac{A - B}{B}$ , and
$x = 100\left(\frac{A - B}{B}\right).$

The answer is $\boxed{\textbf{(A)}\ 100\left(\frac{A-B}{B}\right)}$ .

Why this works

A percent change is always (change)/(original) times $100$ . The phrase "greater than $B$ " identifies $B$ as the original. Writing the definition as $A = B(1 + x/100)$ and solving is safer than recalling a formula.

Alternative approach

Test $B = 100$ , $A = 150$ : then $x$ should be $50$ . Only choice (A) gives $100 \cdot \frac{50}{100} = 50$ ; choice (D) gives $33\frac13$ .

The trap

Dividing the difference by A instead of the base B, which gives choice (D).

Common mistakes

Dividing the difference by A instead of the base B, which gives choice (D).
Choosing $100 \cdot \frac{A}{B}$ , which is "what percent of $B$ is $A$ ," not "how much greater."

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer