AMC 10 · 2014 · #11
Grade 7 rate-ratioPick an answer.
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
The second speed is the first plus 15.
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
One letter names the distance still to go.
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
The two timings differ by 1.5 hours.
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 travel times equals that saving.
▸ Why?
At a steady speed the time is the distance divided by the speed, so each plan gives one such time.
▸ Why?
Both times are measured from the same departure, so the shared start cancels and only the gap survives.
Solve for the leftover distance
Solving gives the leftover distance 175.
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
Adding the first hour gives 210, 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