AMC 10 · 2005 · #3

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
Spending one fifth of her money buys one third of the equally priced discs. Find what fraction of her money is left after buying every disc.

Pick an answer.

(A)
$\frac15$
(B)
$\frac13$
(C)
$\frac25$
(D)
$\frac23$
(E)
$\frac45$
How to solve
Strategy Analyze the Units

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.

1STEP 1

Cost scales with the number of CDs

Equal prices make the cost scale with the count.

all CDs = 3 × (1/3 of the CDs)
2STEP 2

Scale the spent money up to 3/5

Three batches make the whole, so spending is 3/5.

3 × 1/5 = 3/5 of her money spent
3STEP 3

Take the complement to find what's left

Taking the complement leaves 2/5, choice (C).

1 - 3/5 = 5/5 - 3/5 = 2/5 → (C)
Answer
2/5
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 1/5 is the cost of just one batch, and 3/5 is the money spent rather than the money left — both are pieces of the correct chain, not the final leftover.
💡Key takeaway

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.

  • Cost scales with the number of CDs
  • Scale the spent money up to 3/5
  • Take the complement to find what's left

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