Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #10
Grade 6 arithmeticrate-ratioPick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The sentence packs three small ideas: a percent, a comparison, and a sum. Tool #7 (Break Into Subproblems) splits the work into three short steps — (a) recover Shea's original height from her current 60 inches, (b) find how many inches Shea grew, (c) take half of that for Ara's growth and add it to Ara's original height. Step (a) is itself a Tool #9 (Work Backwards) move: we know the post-growth value and the growth rate, and we undo the × 1.2 by dividing. No variables or equations are needed — each subproblem is one line of arithmetic.
Recover Shea's old height
Undo the growth: 60 is 120% of Shea's old height, so divide by 1.2 to recover her original 50 inches.
If 120% of the old height is 60, then 100% of the old height is 60/1.2 = 50 — divide by the growth factor to step back.
Shea's original height was 50 inches.
▸ Why?
Shea's current 60 inches is 1.2 times her original height, so we recover the original by undoing that × 1.2: dividing 60 by 1.2 gives 50.
▸ Why?
Growing 20% keeps the whole original height and adds 20% of it on top, so the new height is one whole original plus 0.2 of the original, which is 1.2 of the original.
▸ Why?
The new height is the original starting height with the grown part added on, and those two pieces cover the new height with no gap or overlap.
▸ Why?
One whole original plus 0.2 of the original is (1 + 0.2) of the original — the shared "of the original" lets the counts add to 1.2 of the original.
▸ Why?
Multiplying the original by 1.2 produced 60, and dividing undoes multiplying, so dividing 60 by 1.2 brings the original back.
Find Shea's growth
Subtract old from new to see how much Shea grew: 60 - 50 = 10 inches.
Growth equals the difference between the new and old heights — a single subtraction once both heights are known.
4.OA.A.3Identify SubproblemsHalve it for Ara's growth
Ara grew half as many inches as Shea, so halve the growth: half of 10 is 5 inches.
"Half as many" is a multiplicative comparison — multiply Shea's growth by 1/2 to get Ara's growth.
4.OA.A.2Identify SubproblemsAdd Ara's growth to 50
Ara began at the same 50 inches as Shea and grew 5, so Ara is now 55 inches tall.
New height = old height + growth — the three subproblems come together in one addition.
4.OA.A.3Identify SubproblemsA 20% growth means the new value is 1.2 times the old, so dividing by 1.2 undoes it. Once you know Shea grew 10 inches, Ara's "half as many" is just 5 inches on top of the same 50-inch starting height — answer 55, choice (E).
- Recover Shea's old height
- Find Shea's growth
- Halve it for Ara's growth
- Add Ara's growth to 50
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