AMC 10 · 2017 · #13
Grade 7 rate-ratioPick an answer.
Everything here is distance, speed and time, and those three are locked together by distance = speed × time — so Tool #8 (Analyze the Units) is the engine. The useful consequence of that formula is this: over a fixed distance, speed and time move in opposite directions, so halving the speed doubles the time. That turns a pair of times into a ratio of speeds without ever solving an equation. First use Tool #7 (Identify Subproblems) to cut the trip into its two legs, since only the second leg is affected by the storm. Then use Tool #15 (Organize Information in More Ways) to lay the snowy day's timetable next to the usual day's timetable leg by leg — the comparison is where the ratio appears. The one number carrying real units, 20 miles per hour, is what converts that unit-free ratio into an actual speed. Tool #4 (Introduce a Variable) is held in reserve: naming the distance d and writing one equation also works, and it is used below as the cross-check.
Split the trip at the storm
The first leg takes 60 minutes as usual.
At an unchanged speed, a third of the road always eats exactly a third of the clock.
6.RP.A.3Identify SubproblemsTime the snowy leg two ways
The second leg is timed two ways.
Putting the snowy timetable beside the normal timetable isolates exactly what the storm cost her.
6.RP.A.3Organize Information In More WaysTurn the time ratio into a speed ratio
The time ratio flips into a speed ratio.
Same road, more time means less speed by exactly the same factor — that is what a constant product forces.
On the same road, more time means less speed by exactly the same factor.
▸ Why?
At a steady pace the distance is the speed multiplied by the time, and here that distance is fixed.
▸ Why?
With the product fixed, scaling one factor up scales the other down by the reciprocal.
Anchor the ratio with the 20 mph drop
The stated drop anchors the usual speed.
Ratios alone fix only proportions; one measured quantity is needed to set the scale.
7.RP.A.3Analyze The UnitsConvert speed and time to distance
Speed times time gives 135, choice (B).
Multiplying a per-hour rate by a count of hours cancels the hours and leaves plain miles.
6.RP.A.3Analyze The UnitsOn the same stretch of road, whatever factor stretches the time shrinks the speed by that same factor — so 216 minutes instead of 120 means she drove at 5/9 speed, and the missing 4/9 being 20 miles per hour pins her usual speed at 45.
- Split the trip at the storm
- Time the snowy leg two ways
- Turn the time ratio into a speed ratio
- Anchor the ratio with the 20 mph drop
- Convert speed and time to distance