AMC 10 · 2008 · #3
Grade 6 rate-ratioPick an answer.
The whole problem is one exchange rate: oranges per banana. Tool #8 (Analyze the Units) says to lock onto that rate — find how many oranges a single banana is worth, and every question about bananas becomes a multiplication. Tool #9 (Solve an Easier Related Problem) is why we shrink the messy "2/3 of 10 bananas" down to the value of just one banana first. Tool #4 (Introduce a Variable) backs this up: calling one banana b oranges keeps the balance equation honest.
Simplify the given amount of bananas
Simplifying gives a plain amount of bananas on the given side.
"2/3 of 10" is just a multiplication, so replace the phrase with the single number it stands for.
5.NF.B.4Analyze The UnitsFind the value of one banana in oranges
Dividing gives the value of one banana in oranges.
Split the total orange value evenly across the bananas to see what a single banana is worth.
Splitting the total orange value evenly across the bananas shows what a single banana is worth.
▸ Why?
Sharing a total among equal items gives each one an equal share of it.
▸ Why?
With the rate fixed, any number of bananas converts by multiplying that single value.
Simplify the target amount of bananas
The target amount also simplifies.
Same move as before: turn the phrase into one clean banana count.
5.NF.B.4Solve An Easier Related ProblemScale by the rate to get the answer
Scaling by the rate gives 3, choice (C).
Once you know oranges-per-banana, any banana count converts to oranges by one multiplication.
6.RP.A.3Analyze The UnitsFind what one banana is worth in oranges first; after that, any pile of bananas becomes oranges with a single multiply.
- Simplify the given amount of bananas
- Find the value of one banana in oranges
- Simplify the target amount of bananas
- Scale by the rate to get the answer