Competition · AMC preparation · step 4 of 4
AMC 8 · 2018 · #6
Grade 4 rate-ratioPick an answer.
AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a classic rate problem (distance, speed, time). Tool #8 (Analyze the Units) keeps the bookkeeping honest: miles/(minutes-per-mile) = minutes, so working with the rate "minutes per mile" lets us avoid converting to mph at all. Tool #9 (Easier Related Problem) is the elegance move: instead of computing the highway speed, first ask the smaller question "how long would 10 highway miles take?" — that gives a per-mile rate we can scale up by 5 to reach the actual 50 miles. Tool #7 (Identify Subproblems) splits the trip into the highway leg and the coastal leg, which are summed at the very end. This avoids reaching for Tool #13 (Algebra) when scaling and addition are enough.
Turn speed into time per mile
Flip "3 times as fast" into per-mile time: each highway mile takes the time of a coastal mile.
"3 times as fast" is a Grade 4 multiplicative comparison — and the speed/time inverse just flips the multiplier.
Because the highway speed is three times the coastal speed, one highway mile takes one third of the time that one coastal mile takes.
▸ Why?
Over the same single mile, speed and time are the two factors whose product is that fixed one-mile distance, so making the speed three times larger forces the time to shrink to one third.
▸ Why?
Distance covered is speed multiplied by time: each minute of driving adds one fixed helping of miles, so the miles pile up as equal groups, one group per minute.
▸ Why?
To keep that fixed one-mile product unchanged, tripling the speed must be undone by taking one third of the time, because dividing by three exactly reverses multiplying by three.
Find the time for 10 miles
Solve the easier case first: 10 highway miles take of the coastal 30 minutes, or 10 minutes.
Multiplying a whole number by the fraction 1/3 is exactly the Grade 4 "fraction times a whole number" standard.
4.NF.B.4Solve An Easier Related ProblemScale up to 50 miles
Scale up: 50 highway miles is 5 times 10 miles at the same speed, so 5 × 10 = 50 minutes.
Scaling time by the same factor as distance (at constant speed) is the Grade 4 distance-time word-problem move.
4.MD.A.2Analyze The UnitsAdd the two legs
Add the legs: highway 50 minutes plus coastal 30 minutes gives 80 minutes total.
Combining two sub-trip times into one total trip time is the addition piece of a multi-step distance-time word problem.
4.MD.A.2Identify SubproblemsThis AMC 8 problem only needs Grade 4 multiplicative comparison ("3 times as fast means the time") that you already know!
- Turn speed into time per mile
- Find the time for 10 miles
- Scale up to 50 miles
- Add the two legs
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