AMC 10 · 2014 · #6
Grade 6 rate-ratioPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
No dollar amount is given, so Tool #4 (Introduce a Variable) fixes a convenient regular price; because the count of balloons cannot depend on the price, any handy value works, and 3 makes the 1/3 discount a whole number. Tool #8 (Analyze the Units) tracks dollars per balloon and dollars per pair. Tool #7 (Identify Subproblems) turns the sale into one repeatable unit — a pair of balloons at a fixed cost — so the whole purchase is just 'how many pairs fit in the money.'
Pick a convenient price
The price never appears in the answer, so let it be 3 dollars: Orvin's money is 30 × 3 = 90 dollars.
When only ratios matter, you are free to choose the actual price; a clean number just makes the arithmetic easy.
6.RP.A.3Introduce A VariablePrice the discounted balloon
In each pair the second balloon is 1/3 off, a 1 dollar cut, so it costs 3 - 1 = 2 dollars.
Taking a fraction off means paying the rest: 1/3 off leaves 2/3 of the price.
Taking a fraction off means paying the rest, so a third off leaves two thirds of the price.
▸ Why?
The price is exactly the discount plus what is paid, so naming one names the other.
▸ Why?
A third is one of three equal shares, so the other two shares are what remains.
Cost of one pair
Buy two at a time: one full-price balloon plus one discounted one is 3 + 2 = 5 dollars per 2 balloons.
The sale repeats in blocks of two, so the natural unit to count is the pair, not the single balloon.
4.OA.A.3Identify SubproblemsFit the pairs into the money
A pair costs 5 dollars and he has 90, so he affords 90 ÷ 5 = 18 pairs with nothing left over.
Once each pair has a fixed price, the whole budget question becomes a single division.
4.OA.A.3Analyze The UnitsCount the balloons
Each pair holds 2 balloons, so 18 × 2 = 36 balloons, and the money is spent exactly: choice (C).
The saved money — one third off every second balloon — is exactly what buys the extra six balloons beyond the original thirty.
4.OA.A.3Introduce A VariableBuy balloons two at a time: every second one is a third off, so five dollars' worth of deal gets you six dollars' worth of balloons, stretching money for 30 into 36.
- Pick a convenient price
- Price the discounted balloon
- Cost of one pair
- Fit the pairs into the money
- Count the balloons