Speed is distance per unit time

4.MD.A.23.OA.A.3 · adapt · grade 4

Archetype: Multiplicative Comparison and Unit Rate · step in a 7-type progression

Mia's family drove to her grandmother's house, which is $120 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $60 \text{ miles}$ per hour, and on the way home they drove at a speed of $60 \text{ miles}$ every 30 minutes. For the whole round trip, how many miles per hour did they travel on average?

Show solution

Understand

Mia's family drives 120 miles to grandma's at 60 miles per hour, then drives the same 120 miles home at a speed of 60 miles every 30 minutes. I need the average speed in miles per hour for the whole round trip, which is the total distance divided by the total time.

Givens

One-way distance is 120 miles, so the round trip is 240 miles.
Going there: 60 miles per hour.
Coming home: 60 miles every 30 minutes.

Unknowns

The average speed for the entire round trip, in miles per hour.

Constraints

Average speed = total distance / total time (NOT the average of the two speeds).
30 minutes = half an hour; 60 miles per 30 minutes = 120 miles per hour.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

Speed is a rate (miles per hour), so watching the units tells me to first convert 'miles per 30 minutes' into miles per hour, then find each leg's time, and finally divide total miles by total hours. Splitting the trip into the two legs keeps each time calculation simple.

Execute

#7 Identify Subproblems 3.OA.A.3

Going there is 120 miles at 60 miles per hour. Divide distance by speed to get the time.

120 \div 60 = 2 \text{ hours}

If you cover 60 miles each hour, 120 miles is just two of those hours - a simple division fact.

#8 Analyze the Units 4.MD.A.2

Coming home is 60 miles every 30 minutes. Since 30 minutes is half an hour, in a full hour they cover twice as far: 60 + 60 = 120 miles per hour.

60 \text{ mi} / 30 \text{ min} = 120 \text{ mi/h}

Two equal 30-minute halves make one hour, so the per-hour distance is double the per-half-hour distance.

#7 Identify Subproblems 3.OA.A.3

Going home is 120 miles at 120 miles per hour, so it takes 1 hour.

120 \div 120 = 1 \text{ hour}

Covering 120 miles in an hour means 120 miles takes exactly one hour.

#8 Analyze the Units 4.MD.A.2

The whole round trip is 120 + 120 = 240 miles, and the total time is 2 + 1 = 3 hours. Average speed is total distance divided by total time.

240 \div 3 = 80 \text{ mi/h}

Average speed asks 'how many miles per hour overall,' so divide all the miles by all the hours.

Answer: 80 miles per hour

Review

The two speeds are 60 mph and 120 mph, so the average must lie between them; 80 mph is between 60 and 120, and it leans toward the slower speed because more time was spent driving slowly (2 hours vs 1 hour), which is exactly what we expect.

Guess and check (tool 6): the wrong 'average of speeds' (60+120)/2 = 90 would assume equal times, but the slow leg takes longer, so the true average must be below 90 - confirming 80 mph is sensible.

Standards · min grade 4

3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing distance by speed to find each leg's travel time.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Converting 60 miles per 30 minutes to a per-hour rate and dividing total distance by total time.

💡 Average speed is all the miles divided by all the hours - not just the average of the two speeds!