Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #22
Grade 7 rate-ratioalgebraPick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 + x/100) × (1 - 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.
Try a small price first
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%.
An easier related case (x = 10) shows up-then-down by the same percent shrinks the price — Grade 7 percent reasoning.
7.RP.A.3Solve An Easier Related ProblemWrite the combined factor
State the rule: a rise multiplies by (1 + ), the following drop by (1 − ), so the final fraction of the original is their product.
Converting a percent change into a decimal multiplier is core Grade 6 rate-and-percent thinking.
Raising the price by x% and then lowering the new price by the same x% multiplies the original price by (1+x/100)(1-x/100).
▸ Why?
A rise of x% turns the original price P into P plus x% of P, and that sum collects into (1+x/100)P.
▸ Why?
Taking x% of P means taking x out of every 100 of P, so it is x/100 of P.
▸ Why?
One whole P plus x/100 of P is a whole and a part of the same P, so they share the factor P and combine into (1+x/100)P.
▸ Why?
Lowering that new price by the same x% subtracts x/100 of it, and a whole minus that same-sized part again shares the common factor, leaving (1-x/100) of the new price.
▸ Why?
The decrease acts on the already-increased price, so the original P is multiplied first by (1+x/100) and then by (1-x/100); regrouping the product lets those two factors multiply together into one factor on P.
Test 16 percent
Test (A) x = 16: the factor is 1.16 × 0.84 ≈ 0.97, far above 0.84, so eliminate (A).
Multiplying two decimals to hundredths is Grade 5; the comparison rules (A) out.
5.NBT.B.7Eliminate PossibilitiesTest 20 percent
Test (B) x = 20: the factor is 1.20 × 0.80 = 0.96, still above 0.84, so eliminate (B).
Same Grade 5 decimal multiplication; result 0.96 rules (B) out.
5.NBT.B.7Eliminate PossibilitiesTest 40 percent
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).
Guess-and-check on the largest choice hits 0.84 on the nose — Grade 5 decimal multiplication confirms it.
5.NBT.B.7Guess And CheckThis 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!
- Try a small price first
- Write the combined factor
- Test 16 percent
- Test 20 percent
- Test 40 percent
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