AMC 10 · 2018 · #2
Grade 6 rate-ratioPick an answer.
This is a rate problem (distance = speed × time), so Tool #8 (Analyze the Units) leads: convert each 30-minute leg to 1/2 hour so that mph × hours gives miles cleanly. Tool #7 (Identify Subproblems) supports it — break the 96-mile total into three leg-distances, find the two known legs, and the missing third leg is forced by subtraction. Then one more unit step turns that distance back into a speed.
Match the units
Match the units first.
A speed in mph only multiplies cleanly when the time is in hours, so turn the 30 minutes into half an hour first.
4.MD.A.1Analyze The UnitsDistance of the first two legs
Find the first two legs' distance.
At a steady speed, distance is just speed scaled by how long you drive — half an hour means half the miles-per-hour number.
6.RP.A.3Analyze The UnitsDistance left for the last leg
Subtracting gives the leftover distance.
The total distance is fixed, so whatever the first two legs do not cover is exactly what the last leg must cover.
The total distance is fixed, so whatever the first legs do not cover is exactly what the last leg must cover.
▸ Why?
The trip is exactly its legs laid end to end, so their distances add to the whole.
▸ Why?
At a steady speed each leg's distance is the speed multiplied by its own time.
Turn distance back into speed
Dividing by the leftover time gives 67 miles per hour.
If you cover 33.5 miles in half an hour, in a whole hour you would cover twice as much — so the unit rate is 67 mph.
6.RP.A.2Analyze The UnitsTurn every 30-minute leg into half an hour, add up the miles you know, and whatever distance is missing from 96 tells you the last leg's speed.
- Match the units
- Distance of the first two legs
- Distance left for the last leg
- Turn distance back into speed