AMC 10 · 2014 · #3
Grade 6 rate-ratioPick an answer.
The whole trip is the one number everything else is measured against, so Tool #4 (Introduce a Variable) names it x and turns "one-third" and "one-fifth" into x/3 and x/5. Tool #7 (Identify Subproblems) splits the job cleanly: first combine the two fraction parts, then see what fraction the pavement must fill. Tool #13 (Convert to Algebra) writes the sentence "the parts add to the whole" as one equation, so the 20 miles pins down x instead of floating free.
Name the whole trip
One letter names the whole trip.
Giving the unknown whole a name lets every "fraction of the trip" become a real expression you can add up.
6.EE.B.6Introduce A VariableCombine the two fraction parts
The two fractions combine to 8/15.
Rewriting both fractions over the same denominator lets you add them into a single share of the trip.
5.NF.A.1Identify SubproblemsFind the pavement's share
What is left is the fixed stretch.
Whatever the two fractions do not cover, the pavement covers, so its fraction of the trip equals 20 miles.
Whatever the two fractions do not cover, the pavement covers, so its share is what is left of the trip.
▸ Why?
The trip is exactly its three stretches put together, so the shares must fill it completely.
▸ Why?
Rewriting both fractions over one denominator lets them be added without changing either value.
Solve for the total
Solving gives 300/7, choice (E).
Multiplying by the reciprocal strips off the 7/15 and leaves the whole trip x alone.
6.NS.A.1Introduce A VariableThe 20 miles is whatever fraction of the trip the two fractions leave over, so find that leftover fraction first.
- Name the whole trip
- Combine the two fraction parts
- Find the pavement's share
- Solve for the total