Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #9
Grade 6 rate-ratio
Pick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks for a rate in miles per hour, which screams Tool #8 (Analyze the Units): mph = miles/hours, so we just need one number for total miles and one number for total hours. Tool #3 (Draw a Picture) is already done for us — the graph is the picture — but we have to read it correctly: only the final point (7, 35) matters for the average, because everything in between is just "how she got there". The intermediate bends are a distractor that tempts students to average the segment speeds, which is wrong.
Read the total time
The graph's rightmost point sits at x = 7 on the hours axis, so the whole ride lasted 7 hours.
Reading the x-coordinate of a point in the first quadrant to recover "how much time" is a Grade 5 coordinate-plane skill.
5.G.A.2Eliminate PossibilitiesRead the total distance
The same endpoint sits at y = 35 on the miles axis, so Carmen rode a total of 35 miles.
Reading the y-coordinate of the endpoint gives the cumulative distance — the same Grade 5 coordinate-plane idea.
5.G.A.2Eliminate PossibilitiesDivide distance by time
Average speed is total distance over total time; the units already match, so 35 miles over 7 hours gives 5 mph.
Computing a unit rate (miles per hour) from a total distance and a total time is core Grade 6 rate reasoning.
For the whole ride, the average speed is the total distance, 35 miles, divided by the total time, 7 hours — a single pace measured in miles per hour.
▸ Why?
Average speed replaces Carmen's changing pace with one steady pace that would still cover all 35 miles in all 7 hours, so it is just how many miles belong to each single hour.
▸ Why?
At a steady pace, every one of the 7 hours adds the same number of miles, so the 7 hours stack up as 7 equal groups of that per-hour amount to make the whole 35 miles — distance is pace times time.
▸ Why?
Since that per-hour pace multiplied by 7 gives 35, the pace itself is recovered by dividing 35 by 7, because division undoes multiplication.
Average speed always means total distance ÷ total time — not the average of the speeds along the way!
- Read the total time
- Read the total distance
- Divide distance by time
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