AMC 10 · 2014 · #15
Grade 7 rate-ratioPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The whole problem turns on the distance still left after the first hour, so Tool #4 (Introduce a Variable) names that leftover distance d and lets me write its travel time at each speed. Tool #8 (Analyze the Units) keeps the rates honest: miles divided by miles-per-hour gives hours, so d/35 and d/50 are times I can compare. Tool #13 (Convert to Algebra) turns the phrase "from 1 hour late to 30 minutes early" into a single equation about saved time.
Read off the two speeds
35 miles in one hour means 35 mph; add 15 and the rest of the road is driven at 50 mph.
Driving 35 miles in exactly one hour is what "35 mph" means, so the first hour hands us both the speed and the miles.
6.RP.A.3Analyze The UnitsName the leftover distance
Let d be the miles left after that first hour: the same d takes d/35 hours at 35 mph but d/50 hours at 50 mph.
One letter for the unknown leftover distance lets both travel times be written from the single rule time = distance / speed.
6.EE.B.6Introduce A VariableTurn the timing into an equation
1 hour late turning into half an hour early saves 1.5 hours, so d/35-d/50=1.5.
Going from late to early is nothing more than saved minutes, so the gap between the two travel times equals that saving.
Going from late to early is nothing more than saved minutes, so the gap between the two times is that saving.
▸ Why?
At a steady speed each travel time is the distance divided by its own speed.
▸ Why?
Both times are measured against the same appointment, so shifting that reference leaves the gap unchanged.
Solve for the leftover distance
Over the common denominator 350 the difference is 3d/350, so 3d/350=3/2 leaves d=175 miles.
Once both times share a denominator, the equation collapses to a single fraction equal to a number, and one step frees d.
6.EE.B.7Introduce A VariableAdd back the first hour
That 175 is only the road after hour one, so add the first 35 miles: 35+175=210 miles, choice (C).
The variable only tracked the road after the first hour, so the first-hour miles still have to be added on.
6.RP.A.3Analyze The UnitsGoing from late to early just means saved time, so compare how long the leftover road takes at each speed, solve for that leftover distance, then add back the first-hour miles.
- Read off the two speeds
- Name the leftover distance
- Turn the timing into an equation
- Solve for the leftover distance
- Add back the first hour