AMC 8 · 2000 · #10

Grade 6 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagefraction-arithmeticmulti-digit-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: percentagemulti-digit-arithmetic

📏 Short solution 💡 3 insights

Problem

Ara and Shea used to be the same height. Since then Shea has grown 20%, and Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break Into Subproblems

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.

1STEP 1

Undo the growth: 60 is 120% of Shea's old height, so divide by 1.2 to recover her original 50 inches.

Shea's old height =

\frac{60}{1.2}

= 50 inches

2STEP 2

Subtract old from new to see how much Shea grew: 60 - 50 = 10 inches.

Shea's growth = 60 - 50 = 10 inches

3STEP 3

Ara grew half as many inches as Shea, so halve the growth: half of 10 is 5 inches.

Ara's growth =

\frac{1}{2}

× 10 = 5 inches

4STEP 4

Ara began at the same 50 inches as Shea and grew 5, so Ara is now 55 inches tall.

Ara now = 50 + 5 = 55 inches → (E)

Answer

Sanity check the size: Ara grew less than Shea, so Ara should be shorter than 60 but taller than 50. Our answer 55 sits exactly between 50 and 60, which fits. Also verify the percent step by going forward: 1.2 × 50 = 60, matching Shea's current height. The trap choices line up with common slips. (A) 48 treats Shea's 20% growth as a 20% drop from 60. (D) 54 takes 20% of Shea's 60 (= 12) and then halves it (= 6), forgetting that 20% is measured off the old height. Choosing (E) means the percent was undone correctly before halving.

💡Key takeaway

A 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).

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