AMC 10 · 2005 · #5

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

fraction-multiplicationratio-proportion ratio-proportioncomplementary-counting ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Short solution 💡 2 insights

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Problem

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

Pick an answer.

(A)

$\frac15$

(B)

$\frac13$

(C)

$\frac25$

(D)

$\frac23$

(E)

$\frac45$

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Brianna spends money on several CDs that all cost the same. Spending one fifth of her money buys her one third of the CDs. Find what fraction of her money is left once she has bought every CD.

Givens: All the CDs cost the same amount each; One fifth of her money pays for one third of the CDs; She plans to buy all of the CDs; Answer choices: (A) $\frac15$, (B) $\frac13$, (C) $\frac25$, (D) $\frac23$, (E) $\frac45$

Unknowns: The fraction of her money remaining after she buys all the CDs

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #16 Change Focus / Count the Complement

Because every CD costs the same, the money spent tracks the number of CDs one-for-one — that is a rate, so Tool #8 (Analyze the Units) lets me scale the given 'one fifth pays for one third' up to the whole set without ever finding an actual dollar amount. Then the question asks for money *left*, not money *spent*, so Tool #16 (Count the Complement) finishes the job: subtract the spent fraction from the whole. The proportional link plus the complement flip are the entire problem — no equation-solving is needed.

Execute — Answer: C

#8 Analyze the Units 4.OA.A.2 Step 1

Cost scales with the number of CDs

Every CD costs the same, so twice as many CDs cost twice as much money, and three times as many cost three times as much.
The whole collection of CDs is three copies of the 'one third of the CDs' batch, so it must cost three times what that batch cost.

$$\text{all CDs} = 3 \times \left(\tfrac13 \text{ of the CDs}\right)$$

💡 Same price per CD means the money you spend rides along with how many CDs you buy.

#8 Analyze the Units 4.NF.B.4 Step 2

Scale the spent money up to 3/5

The one-third batch cost one fifth of her money.
Buying all the CDs costs three of those batches, so it costs $3$ times one fifth.
Multiply the fraction by $3$: $3 \times \tfrac15 = \tfrac35$.
So buying every CD uses up three fifths of her money.

$$3 \times \tfrac15 = \tfrac35 \text{ of her money spent}$$

💡 Three equal batches at one fifth each add up to three fifths.

#16 Change Focus / Count the Complement 4.NF.B.3 Step 3

Take the complement to find what's left

All her money is the whole, $1 = \tfrac55$.
She spends $\tfrac35$, so what remains is the whole minus the spent part.
Since the fifths share the same denominator, just subtract the numerators: $\tfrac55 - \tfrac35 = \tfrac25$.
She has two fifths of her money left, which is choice (C).

$$1 - \tfrac35 = \tfrac55 - \tfrac35 = \tfrac25 \;\Rightarrow\; \textbf{(C)}$$

💡 Money left is simply all her money minus the part she spent.

[1] #8 4.OA.A.2 Every CD costs the same, so twice as many CDs cost twice as much money, and thre

[2] #8 4.NF.B.4 The one-third batch cost one fifth of her money. Buying all the CDs costs three

[3] #16 4.NF.B.3 All her money is the whole, $1 = \tfrac55$. She spends $\tfrac35$, so what remai

Review

Reasonableness: A one-third batch of CDs eats up one fifth of the money, so all three batches take a bit more than half — three fifths ($0.6$) is a sensible 'more than half but not everything' amount to spend. That leaves two fifths ($0.4$), less than half, which matches the idea that she can afford the whole set with money to spare. The trap answer $\tfrac15$ is the cost of just one batch, and $\tfrac35$ is the money spent rather than the money left — both are pieces of the correct chain, not the final leftover.

Alternative: Pick a convenient total, say $\$15$ (a multiple of $5$). One fifth is $\$3$, which buys one third of the CDs, so all the CDs cost $3 \times \$3 = \$9$. Money left is $\$15 - \$9 = \$6$, and $\tfrac{6}{15} = \tfrac25$ — the same (C).

CCSS standards used (min grade 4)

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Recognizing that the full set of CDs is three times the one-third batch, so it costs three times as much money.)
4.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a whole number (Computing $3 \times \tfrac15 = \tfrac35$, the total fraction of money spent on all the CDs.)
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions (Subtracting like fractions $\tfrac55 - \tfrac35 = \tfrac25$ to find the money remaining.)

⭐ When every item costs the same, the money you spend grows right along with how many you buy — so scale the cost up, then subtract from the whole to see what's left.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Cost scales with the number of CDs

Scale the spent money up to 3/5

Take the complement to find what's left

Review

CCSS standards used (min grade 4)

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