AMC 8 · 2019 · #22

Grade 7 rate-ratioalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

percentageperfect-squareslinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: percentage

📏 Medium solution 💡 3 insights

Problem

A shirt's original price is first raised by some percent x%, and then the new (higher) price is lowered by the same percent x%. After both changes, the price ends up at 84% of the original. Find x.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

This is a multiple-choice percent problem with only five candidates, and the test is cheap: for each choice x, compute the combined factor (1 + $\frac{x}{100}$ ) × (1 - $\frac{x}{100}$ ) and see whether it equals 0.84. Tool #6 (Guess and Check) directly plugs each choice into this rule, and Tool #3 (Eliminate Possibilities) discards any choice whose factor is not 0.84. We first use Tool #9 (Easier Related Problem) on a tiny example to be sure that 'up by x% then down by x%' does NOT return to the original — that builds the intuition the problem depends on.

1STEP 1

Warm up: start at $100, x = 10 → up 10% is$ 110, down 10% is $99, only 99% of the original. Equal up-then-down always dips below 100%.

100 →

110 → $99 = 99% of original

2STEP 2

State the rule: a rise multiplies by (1 + $\frac{x}{100}$ ), the following drop by (1 − $\frac{x}{100}$ ), so the final fraction of the original is their product.

combined factor = (1 +

\frac{x}{100}

) (1 -

\frac{x}{100}

)

3STEP 3

Test (A) x = 16: the factor is 1.16 × 0.84 ≈ 0.97, far above 0.84, so eliminate (A).

1.16 × 0.84 ≈ 0.974 ≠ 0.84

4STEP 4

Test (B) x = 20: the factor is 1.20 × 0.80 = 0.96, still above 0.84, so eliminate (B).

1.20 × 0.80 = 0.96 ≠ 0.84

5STEP 5

Jump to the largest, (E) x = 40: bigger x shrinks the factor, and 1.40 × 0.60 = 0.84 is an exact match — the answer is (E).

1.40 × 0.60 = 0.84 ✓ → (E) 40

Answer

The warm-up case showed x = 10 gives 0.99 (a tiny dip). The final price drops to 0.84 — a much bigger dip — so x should be much bigger than 10. x = 40 is the only choice in the right ballpark, and the direct check 1.40 × 0.60 = 0.84 is exact (no rounding). The other choices (16, 20, 28, 36) all give factors above 0.84, confirming (E) is the unique answer.

💡Key takeaway

This AMC 8 problem only needs Grade 7 percent-change reasoning — up then down by the same percent never returns to the start — that you already know!

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