Practice · Speed is distance per unit time · Sensim Math

Variant 1 answer: 80 miles per hour

Mia's family drove to her grandmother's house, which is $240 \text{ miles}$ 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 $120 \text{ miles}$ every 60 minutes. For the whole round trip, how many miles per hour did they travel on average?

Show solution

Understand

Mia's family drives 240 miles to grandma's at 60 miles per hour, then drives the same 240 miles home at a speed of 120 miles every 60 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 240 miles, so the round trip is 480 miles.
Going there: 60 miles per hour.
Coming home: 120 miles every 60 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).
60 minutes = 1 hour; 120 miles per 60 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 some 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 240 miles at 60 miles per hour. Divide distance by speed to get the time.

240 \div 60 = 4 \text{ hours}

If you cover 60 miles each hour, 240 miles takes 4 of those hours - a simple division fact.

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

Coming home is 120 miles every 60 minutes. Since 60 minutes is 1 of an hour, scaling to a full hour gives 120 miles per hour.

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

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 240 miles at 120 miles per hour, so it takes 2 hour(s).

240 \div 120 = 2 \text{ hours}

Covering 120 miles in an hour means 240 miles takes 2 hour(s).

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

The whole round trip is 240 + 240 = 480 miles, and the total time is 4 + 2 = 6 hours. Average speed is total distance divided by total time.

480 \div 6 = 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 them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (60+120)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 80 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 2 answer: 48 miles per hour

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 $40 \text{ miles}$ per hour, and on the way home they drove at a speed of $60 \text{ miles}$ every 60 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 40 miles per hour, then drives the same 120 miles home at a speed of 60 miles every 60 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: 40 miles per hour.
Coming home: 60 miles every 60 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).
60 minutes = 1 hour; 60 miles per 60 minutes = 60 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 some 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 40 miles per hour. Divide distance by speed to get the time.

120 \div 40 = 3 \text{ hours}

If you cover 40 miles each hour, 120 miles takes 3 of those hours - a simple division fact.

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

Coming home is 60 miles every 60 minutes. Since 60 minutes is 1 of an hour, scaling to a full hour gives 60 miles per hour.

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

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 120 miles at 60 miles per hour, so it takes 2 hour(s).

120 \div 60 = 2 \text{ hours}

Covering 60 miles in an hour means 120 miles takes 2 hour(s).

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

The whole round trip is 120 + 120 = 240 miles, and the total time is 3 + 2 = 5 hours. Average speed is total distance divided by total time.

240 \div 5 = 48 \text{ mi/h}

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

Answer: 48 miles per hour

Review

The two speeds are 40 mph and 60 mph, so the average must lie between them; 48 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (40+60)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 48 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 3 answer: 80 miles per hour

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 = 1/2 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 some 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 takes 2 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 1/2 of an hour, scaling to a full hour gives 120 miles per hour.

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

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

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

120 \div 120 = 1 \text{ hours}

Covering 120 miles in an hour means 120 miles takes 1 hour(s).

#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 them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (60+120)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 80 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 4 answer: 80 miles per hour

Mia's family drove to her grandmother's house, which is $200 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $50 \text{ miles}$ per hour, and on the way home they drove at a speed of $100 \text{ miles}$ 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 200 miles to grandma's at 50 miles per hour, then drives the same 200 miles home at a speed of 100 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 200 miles, so the round trip is 400 miles.
Going there: 50 miles per hour.
Coming home: 100 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 = 1/2 hour; 100 miles per 30 minutes = 200 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 some 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 200 miles at 50 miles per hour. Divide distance by speed to get the time.

200 \div 50 = 4 \text{ hours}

If you cover 50 miles each hour, 200 miles takes 4 of those hours - a simple division fact.

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

Coming home is 100 miles every 30 minutes. Since 30 minutes is 1/2 of an hour, scaling to a full hour gives 200 miles per hour.

100 \text{ mi} / 30 \text{ min} = 200 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 200 miles at 200 miles per hour, so it takes 1 hour(s).

200 \div 200 = 1 \text{ hours}

Covering 200 miles in an hour means 200 miles takes 1 hour(s).

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

The whole round trip is 200 + 200 = 400 miles, and the total time is 4 + 1 = 5 hours. Average speed is total distance divided by total time.

400 \div 5 = 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 50 mph and 200 mph, so the average must lie between them; 80 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (50+200)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 80 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 5 answer: 64 miles per hour

Mia's family drove to her grandmother's house, which is $160 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $40 \text{ miles}$ per hour, and on the way home they drove at a speed of $80 \text{ miles}$ 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 160 miles to grandma's at 40 miles per hour, then drives the same 160 miles home at a speed of 80 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 160 miles, so the round trip is 320 miles.
Going there: 40 miles per hour.
Coming home: 80 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 = 1/2 hour; 80 miles per 30 minutes = 160 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 some 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 160 miles at 40 miles per hour. Divide distance by speed to get the time.

160 \div 40 = 4 \text{ hours}

If you cover 40 miles each hour, 160 miles takes 4 of those hours - a simple division fact.

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

Coming home is 80 miles every 30 minutes. Since 30 minutes is 1/2 of an hour, scaling to a full hour gives 160 miles per hour.

80 \text{ mi} / 30 \text{ min} = 160 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 160 miles at 160 miles per hour, so it takes 1 hour(s).

160 \div 160 = 1 \text{ hours}

Covering 160 miles in an hour means 160 miles takes 1 hour(s).

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

The whole round trip is 160 + 160 = 320 miles, and the total time is 4 + 1 = 5 hours. Average speed is total distance divided by total time.

320 \div 5 = 64 \text{ mi/h}

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

Answer: 64 miles per hour

Review

The two speeds are 40 mph and 160 mph, so the average must lie between them; 64 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (40+160)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 64 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 6 answer: 90 miles per hour

Mia's family drove to her grandmother's house, which is $180 \text{ miles}$ 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 $90 \text{ miles}$ 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 180 miles to grandma's at 60 miles per hour, then drives the same 180 miles home at a speed of 90 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 180 miles, so the round trip is 360 miles.
Going there: 60 miles per hour.
Coming home: 90 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 = 1/2 hour; 90 miles per 30 minutes = 180 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 some 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 180 miles at 60 miles per hour. Divide distance by speed to get the time.

180 \div 60 = 3 \text{ hours}

If you cover 60 miles each hour, 180 miles takes 3 of those hours - a simple division fact.

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

Coming home is 90 miles every 30 minutes. Since 30 minutes is 1/2 of an hour, scaling to a full hour gives 180 miles per hour.

90 \text{ mi} / 30 \text{ min} = 180 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 180 miles at 180 miles per hour, so it takes 1 hour(s).

180 \div 180 = 1 \text{ hours}

Covering 180 miles in an hour means 180 miles takes 1 hour(s).

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

The whole round trip is 180 + 180 = 360 miles, and the total time is 3 + 1 = 4 hours. Average speed is total distance divided by total time.

360 \div 4 = 90 \text{ mi/h}

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

Answer: 90 miles per hour

Review

The two speeds are 60 mph and 180 mph, so the average must lie between them; 90 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (60+180)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 90 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 7 answer: 75 miles per hour

Mia's family drove to her grandmother's house, which is $150 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $50 \text{ miles}$ per hour, and on the way home they drove at a speed of $75 \text{ miles}$ 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 150 miles to grandma's at 50 miles per hour, then drives the same 150 miles home at a speed of 75 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 150 miles, so the round trip is 300 miles.
Going there: 50 miles per hour.
Coming home: 75 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 = 1/2 hour; 75 miles per 30 minutes = 150 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 some 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 150 miles at 50 miles per hour. Divide distance by speed to get the time.

150 \div 50 = 3 \text{ hours}

If you cover 50 miles each hour, 150 miles takes 3 of those hours - a simple division fact.

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

Coming home is 75 miles every 30 minutes. Since 30 minutes is 1/2 of an hour, scaling to a full hour gives 150 miles per hour.

75 \text{ mi} / 30 \text{ min} = 150 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 150 miles at 150 miles per hour, so it takes 1 hour(s).

150 \div 150 = 1 \text{ hours}

Covering 150 miles in an hour means 150 miles takes 1 hour(s).

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

The whole round trip is 150 + 150 = 300 miles, and the total time is 3 + 1 = 4 hours. Average speed is total distance divided by total time.

300 \div 4 = 75 \text{ mi/h}

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

Answer: 75 miles per hour

Review

The two speeds are 50 mph and 150 mph, so the average must lie between them; 75 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (50+150)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 75 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 8 answer: 48 miles per hour

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 $30 \text{ miles}$ 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 30 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: 30 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 = 1/2 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 some 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 30 miles per hour. Divide distance by speed to get the time.

120 \div 30 = 4 \text{ hours}

If you cover 30 miles each hour, 120 miles takes 4 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 1/2 of an hour, scaling to a full hour gives 120 miles per hour.

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

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

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

120 \div 120 = 1 \text{ hours}

Covering 120 miles in an hour means 120 miles takes 1 hour(s).

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

The whole round trip is 120 + 120 = 240 miles, and the total time is 4 + 1 = 5 hours. Average speed is total distance divided by total time.

240 \div 5 = 48 \text{ mi/h}

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

Answer: 48 miles per hour

Review

The two speeds are 30 mph and 120 mph, so the average must lie between them; 48 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (30+120)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 48 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 9 answer: 45 miles per hour

Mia's family drove to her grandmother's house, which is $90 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $30 \text{ miles}$ per hour, and on the way home they drove at a speed of $90 \text{ miles}$ every 60 minutes. For the whole round trip, how many miles per hour did they travel on average?

Show solution

Understand

Mia's family drives 90 miles to grandma's at 30 miles per hour, then drives the same 90 miles home at a speed of 90 miles every 60 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 90 miles, so the round trip is 180 miles.
Going there: 30 miles per hour.
Coming home: 90 miles every 60 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).
60 minutes = 1 hour; 90 miles per 60 minutes = 90 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 some 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 90 miles at 30 miles per hour. Divide distance by speed to get the time.

90 \div 30 = 3 \text{ hours}

If you cover 30 miles each hour, 90 miles takes 3 of those hours - a simple division fact.

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

Coming home is 90 miles every 60 minutes. Since 60 minutes is 1 of an hour, scaling to a full hour gives 90 miles per hour.

90 \text{ mi} / 60 \text{ min} = 90 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 90 miles at 90 miles per hour, so it takes 1 hour(s).

90 \div 90 = 1 \text{ hours}

Covering 90 miles in an hour means 90 miles takes 1 hour(s).

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

The whole round trip is 90 + 90 = 180 miles, and the total time is 3 + 1 = 4 hours. Average speed is total distance divided by total time.

180 \div 4 = 45 \text{ mi/h}

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

Answer: 45 miles per hour

Review

The two speeds are 30 mph and 90 mph, so the average must lie between them; 45 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (30+90)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 45 mph comes from total miles over total hours.

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 a per-minute rate 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!

Variant 10 answer: 105 miles per hour

Mia's family drove to her grandmother's house, which is $210 \text{ miles}$ from their home, and then drove back. On the way there they drove at a speed of $70 \text{ miles}$ per hour, and on the way home they drove at a speed of $105 \text{ miles}$ 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 210 miles to grandma's at 70 miles per hour, then drives the same 210 miles home at a speed of 105 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 210 miles, so the round trip is 420 miles.
Going there: 70 miles per hour.
Coming home: 105 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 = 1/2 hour; 105 miles per 30 minutes = 210 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 some 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 210 miles at 70 miles per hour. Divide distance by speed to get the time.

210 \div 70 = 3 \text{ hours}

If you cover 70 miles each hour, 210 miles takes 3 of those hours - a simple division fact.

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

Coming home is 105 miles every 30 minutes. Since 30 minutes is 1/2 of an hour, scaling to a full hour gives 210 miles per hour.

105 \text{ mi} / 30 \text{ min} = 210 \text{ mi/h}

Stretching the per-minute rate up to one full hour gives the miles-per-hour speed.

#7 Identify Subproblems 3.OA.A.3

Going home is 210 miles at 210 miles per hour, so it takes 1 hour(s).

210 \div 210 = 1 \text{ hours}

Covering 210 miles in an hour means 210 miles takes 1 hour(s).

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

The whole round trip is 210 + 210 = 420 miles, and the total time is 3 + 1 = 4 hours. Average speed is total distance divided by total time.

420 \div 4 = 105 \text{ mi/h}

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

Answer: 105 miles per hour

Review

The two speeds are 70 mph and 210 mph, so the average must lie between them; 105 mph is between them, and it leans toward the speed the family spent more time driving, which is exactly what we expect.

Guess and check: the wrong 'average of speeds' (70+210)/2 would assume equal times, but the legs take different times, so the true average differs - confirming 105 mph comes from total miles over total hours.

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 a per-minute rate 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!

Speed is distance per unit time · 10 practice problems

Generated variants — 10

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