AMC 10 · 2006 · #3
Grade 6 rate-ratioPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A ratio 3:5 is really a set of equal-sized shares: Mary gets 3 shares and Alice gets 5. Tool #4 (Introduce a Variable) names the size of one share x, so Mary =3x and Alice =5x. Tool #13 (Convert to Algebra) turns the fact 'Alice is 30' into the equation 5x=30, which pins down x. Tool #3 (Eliminate Possibilities) gives a fast reality check: Mary is the smaller side of the ratio, so she must be younger than 30, which instantly kills (E) 50.
Split the ages into equal shares
Read 3:5 as equal shares of years: if one share is x years, Mary = 3x and Alice = 5x.
A ratio just chops both ages into identical blocks — Mary gets 3 blocks, Alice gets 5.
A ratio chops both ages into identical blocks, a fixed count for each person.
▸ Why?
Each block is one equal share of the whole, so counting blocks is enough to name the amount.
▸ Why?
Both ages are built from the same block, so knowing one age sizes the block and then the other.
Use Alice's age to size one share
Alice is 5 shares and Alice is 30, so 5x = 30 and x = 6: one share is worth 6 years.
If 5 equal blocks make 30, each block must be 30 ÷ 5 = 6.
6.RP.A.3Convert To AlgebraBuild Mary's age and confirm
Mary is 3 shares of 6 years, so Mary = 18, choice (B); 18:30 reduces to 3:5, and (E) 50 was never possible.
Three blocks of 6 years each stack up to 18, and 18:30 folds right back to 3:5.
6.RP.A.3Eliminate PossibilitiesA ratio splits both amounts into equal shares — find the size of one share from the number you know, then multiply to get the one you want.
- Split the ages into equal shares
- Use Alice's age to size one share
- Build Mary's age and confirm