Competition · AMC preparation · step 4 of 4

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

Round both factors

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

Multiply the rounded numbers

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) × 1,000,000/10,000 = 24 × 100 = 2400

Multiplying by powers of 10 just shifts the decimal point — the Grade 5 "patterns in zeros and the decimal point" standard turns the calculation into 3 × 8 = 24 with a × 100 rescue.

The friendly product 0.0003 × 8,000,000 comes out to 2400.

▸ Why?

Each factor is really one nonzero digit sitting in a place: 0.0003 is a 3 four places below one, and 8,000,000 is an 8 six places above one, so we can write them as 3 × 1/10,000 and 8 × 1,000,000.

▸ Why?

Naming a number by which place its digit occupies is exactly how our written numbers work, since every place is worth ten of the place to its right.

Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.

▸ Why?

With each factor split into a digit and a power of ten, we are free to rearrange the four pieces so the digits 3 and 8 pair up and the two powers of ten pair up.

▸ Why?

The factors in a product can be multiplied in any order without changing the result.

Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.

▸ Why?

When several numbers are multiplied, we may choose which two to combine first without changing the result.

Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.

▸ Why?

The digit pair gives 3 × 8 = 24, and combining the powers of ten (six places up, four places down) nets two places up, i.e. × 100, so the answer is 24 carried up two places, 2400.

▸ Why?

3 × 8 = 24 is a known fact: it is 8 counted three times.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

Multiplying by a power of ten just shifts a digit into higher places — up six then down four lands two places up — because each place is worth ten of the one below.

Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.

5.NBT.A.2Solve An Easier Related Problem

3STEP 3

Pick the closest choice

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!

Round both factors
Multiply the rounded numbers
Pick the closest choice

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Round both factors

Multiply the rounded numbers

Pick the closest choice

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