AMC 10 · 2007 · #4
Grade 6 rate-ratioPick an answer.
Each fruit behaves like a currency, and each given fact is an exchange rate between two of them. Tool #8 (Analyze the Units) is exactly the move for chaining exchange rates: convert bananas into apples, then apples into oranges, watching the fruit names cancel the way units do. Tool #4 (Introduce a Variable) names the three prices, which makes visible the assumption the whole argument rests on — cost is proportional to count, so a trade quoted for 3 bananas is still valid for 18. Tool #3 (Eliminate Possibilities) supplies a fast size check: a banana is worth less than half an orange, so the answer must be under 9.
Name the three prices
Naming the prices turns each trade into an equation.
Fixed prices mean cost climbs in step with count, so one quoted trade fixes the rate at every quantity.
6.RP.A.1Introduce A VariableReduce both trades
Both trades reduce to simpler forms.
Halving both sides of an equal-cost trade leaves it an equal-cost trade.
Halving both sides of an equal-cost trade leaves it an equal-cost trade.
▸ Why?
Doing the same thing to both sides of an equation keeps it true.
▸ Why?
With a fixed price per item the cost climbs in step with the count, so one quoted trade fixes the rate everywhere.
Trade 18 bananas for apples
The first conversion gives 12 of the middle fruit.
Six swaps that each keep the price equal stack into one big swap that keeps the price equal.
6.RP.A.3Analyze The UnitsTrade the apples for oranges
The second conversion gives 8 of the last.
Whatever the bananas can buy, the apples can buy, and the apples can be spent on oranges.
6.RP.A.3Analyze The UnitsConfirm with a single rate
Chaining the two rates directly confirms 8, choice (B).
Two conversion rates multiply into a single rate, and that rate does not care whether the counts divide evenly.
6.RP.A.2Analyze The UnitsTrade in bundles: three bananas become two apples, three apples become two oranges, and the total cost never changes along the way.
- Name the three prices
- Reduce both trades
- Trade 18 bananas for apples
- Trade the apples for oranges
- Confirm with a single rate