AMC 10 · 2014 · #7

Grade 7 arithmetic

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage

📏 Short solution 💡 1 insight

Problem

Two positive numbers A and B satisfy A ＞ B ＞ 0, and A is x% greater than B. Write x in terms of A and B, and match it to one of the given formulas.

Pick an answer.

(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)

$100\left(\frac{A}{B}\right)$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

The phrase "A is x% greater than B" is a word statement hiding one equation. Tool #4 (Introduce a Variable) is already done for us — x is named — so the work is Tool #13 (Convert to Algebra): turn the words into A = B + x/100B and solve for x. Because the choices are formulas, Tool #6 (Guess and Check) with concrete numbers gives an independent confirmation that the algebra landed on the right choice.

1STEP 1

Translate the percent phrase

"A is x% greater than B" means start at B and add x% of B, so A = B + (x/100)B.

A = B + x/100 B

2STEP 2

Isolate the percent slice

Subtract B from both sides, and the leftover is exactly the added slice: A - B = (x/100)B.

A - B = x/100 B

3STEP 3

Solve for x

Divide by B, then multiply by 100: x = 100((A-B)/B), the gap measured against the original B — choice (A).

(A-B)/B = x/100 → x = 100((A-B)/B) → (A)

Answer

100((A-B)/B)

The result compares the change A-B to the original B, which is what "greater than B" demands — the baseline is B, so B belongs in the denominator. Since A ＞ B, the numerator A-B is positive, so x ＞ 0, matching "greater." Choices that use A in the denominator (D, E, C) or add instead of subtract (B, C) measure against the wrong baseline or the wrong quantity, so they fail.

💡Key takeaway

Percent increase is the change divided by what you started with, times 100 — so it is 100 times (A-B) over B.

Translate the percent phrase
Isolate the percent slice
Solve for x

Translate the percent phrase

Isolate the percent slice

Solve for x

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