Problem
Average distance per hour from total time
A bus drives 180 km going 90 km every hour, then drives 255 km going 85 km every hour. We want one number: how many kilometers, on average, it covered each hour for the whole trip.
Ratios
Your answer
How to solve
Strategy Identify Subproblems — The trip has two parts, so first find the time each part takes (a subproblem), then add up the total distance and total time. Watching the units (km and hours) shows that average speed is total km divided by total hours, which keeps us from wrongly averaging 90 and 85 directly.
1STEP 1
Time for the first part
The bus covers 180 km while going 90 km each hour, so divide the distance by the speed to find how many hours that part takes.
180 ÷ 90 = 2 hours
If you go 90 km in one hour, then 180 km is just two of those hours.
6.RP.A.3Identify Subproblems2STEP 2
Time for the second part
The bus covers 255 km while going 85 km each hour, so divide again to find the time for this part.
255 ÷ 85 = 3 hours
85 km fits into 255 km exactly three times, so it takes three hours.
6.RP.A.3Identify Subproblems3STEP 3
Add the totals
Add the two distances to get the whole trip distance, and add the two times to get the whole trip time.
180 + 255 = 435 km, 2 + 3 = 5 hours
The whole journey is just its parts put end to end, both in distance and in time.
6.RP.A.3Identify Subproblems4STEP 4
Find the average per hour
Average speed is total distance divided by total time. The units km per hour tell us to divide kilometers by hours.
435 ÷ 5 = 87 km per hour
Sharing the 435 km equally over the 5 hours gives 87 km in each hour.
6.RP.A.2Analyze The UnitsThe average speed is the whole distance shared over the whole time, not the average of the two speeds.
Why?
Asking how far the bus goes in an hour on average means spreading the whole journey evenly over every hour it lasted.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The two legs last different lengths of time, so averaging the two speeds would wrongly give the short leg as much weight as the long one.
Why?
The whole trip is the two legs put together, in distance and in time alike, with nothing left out of either total.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
87 km
The answer 87 km/h sits between the two speeds 85 and 90, which makes sense. It is closer to 85 because the bus spent more time (3 hours) at 85 km/h than at 90 km/h (2 hours). The units work out to km per hour, exactly what was asked.
Takeaway
Average speed is the whole distance shared evenly over the whole time, not just the average of the two speeds.
- Time for the first part
- Time for the second part
- Add the totals
- Find the average per hour
Where next?
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