AMC 8 · 2017 · #4

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

estimationplace-valuemulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Short solution 💡 2 insights

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Problem

Multiply the tiny decimal 0.000315 by the large whole number 7,928,564 and decide which of the five answer choices the product is closest to. Because the choices (210, 240, 2100, 2400, 24000) are spread across very different orders of magnitude, an exact long multiplication is unnecessary — a careful estimate will pick out the right one.

Pick an answer.

(A)

210

(B)

240

(C)

2100

(D)

2400

(E)

24000

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The exact product 0.000315 × 7,928,564 is painful to compute by hand, but the problem only asks which choice it is closest to. Tool #9 (Easier Related Problem) says: swap the ugly numbers for friendly nearby ones — 0.000315 → 0.0003 and 7,928,564 → 8,000,000 — and solve that easier version instead. Tool #3 (Eliminate Possibilities) then takes our estimate and matches it against the five widely-spaced choices; because (A)/(B) are around 200, (C)/(D) around 2000, and (E) around 24,000, a single-digit estimate is enough to land on exactly one choice.

1STEP 1

Swap each factor for the nearest one-nonzero-digit number: 0.0003 and 8,000,000.

0.000315 ≈ 0.0003 7,928,564 ≈ 8,000,000

2STEP 2

Split off the leading digits: 3 × 8 = 24, and 10⁻⁴ × 10⁶ = 10², so the estimate is 2400.

0.0003 × 8,000,000 = (3 × 8) ×

\frac{1,000,000}{10,000}

= 24 × 100 = 2400

3STEP 3

Only 2400 matches the order of magnitude — (A)/(B) are 10× too small, (E) is 10× too big, and (C) 2100 is farther.

estimate = 2400 → (D)

Answer

2400

A quick sanity check: 3 × 8 = 24, and the place-value bookkeeping gives 10⁻⁴ × 10⁶ = 10² = 100, so the product is around 24 × 100 = 2400. The exact product is 0.000315 × 7,928,564 ≈ 2497.5, which rounds to 2500 — much closer to (D) 2400 than to (C) 2100 or (E) 24,000. The answer (D) is consistent with both the rough estimate and the true value.

💡Key takeaway

This AMC 8 problem only needs Grade 5 place-value rounding and the powers-of-10 shortcut you already know!

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