AMC 10 · 2014 · #2
Grade 6 rate-ratioPick an answer.
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
A convenient price makes everything whole.
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
The discounted item costs 2.
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?
A third is one of three equal shares, so removing it leaves the other two shares.
▸ Why?
What is taken off and what is paid together make the full price, so naming one names the other.
Cost of one pair
So one pair costs 5.
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
The money holds 18 whole pairs.
Once each pair has a fixed price, the whole budget question becomes a single division.
4.OA.A.3Analyze The UnitsCount the balloons
That is 36 items, 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