AMC 10 · 2005 · #5
Grade 4 rate-ratioPick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Cost scales with the number of CDs
Same price per CD, so cost rides on count: the whole collection is three of the one-third batches, so it costs three times one batch.
Same price per CD means the money you spend rides along with how many CDs you buy.
4.OA.A.2Analyze The UnitsScale the spent money up to 3/5
One batch takes 1/5 of her money, so three batches take 3 × 1/5 = 3/5 — every CD costs three fifths of what she has.
Three equal batches at one fifth each add up to three fifths.
Three equal batches at one fifth each add up to three fifths of the money.
▸ Why?
A fifth is one of five equal shares, so three of them are three of those shares.
▸ Why?
With the price per disc fixed, the money spent rises in step with the number bought.
Take the complement to find what's left
Her whole pile is 1 = 5/5 and 3/5 goes out, so same denominators means subtract numerators: 2/5 is left — choice (C).
Money left is simply all her money minus the part she spent.
4.NF.B.3Change Focus Count The ComplementWhen 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