Practice · Average distance per hour from total time · Sensim Math

Variant 1 easy

A bus first travels $120 \text{ km}$ at a steady pace of $60 \text{ km}$ per hour, then travels another $45 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 120 km at 60 km/h, then 45 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 120 km at 60 km per hour.
Second leg: 45 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

120 \div 60 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

45 \div 45 = 1\ \text{h}

The second leg takes 1 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

120 + 45 = 165\ \text{km},\quad 2 + 1 = 3\ \text{h}

165 km in 3 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

165 \div 3 = 55

The bus averaged 55 km per hour.

Answer: 55 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 60, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 105/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 1.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 2 easy

A bus first travels $130 \text{ km}$ at a steady pace of $65 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 130 km at 65 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 130 km at 65 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

130 \div 65 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

130 + 135 = 265\ \text{km},\quad 2 + 3 = 5\ \text{h}

265 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

265 \div 5 = 53

The bus averaged 53 km per hour.

Answer: 53 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 65, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 55, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 3 easy

A bus first travels $110 \text{ km}$ at a steady pace of $55 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 110 km at 55 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 110 km at 55 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

110 \div 55 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

110 + 135 = 245\ \text{km},\quad 2 + 3 = 5\ \text{h}

245 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

245 \div 5 = 49

The bus averaged 49 km per hour.

Answer: 49 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 55, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 50, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 4 easy

A bus first travels $100 \text{ km}$ at a steady pace of $50 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 100 km at 50 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 100 km at 50 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

100 \div 50 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

100 + 135 = 235\ \text{km},\quad 2 + 3 = 5\ \text{h}

235 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

235 \div 5 = 47

The bus averaged 47 km per hour.

Answer: 47 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 50, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 95/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 5 medium

A bus first travels $140 \text{ km}$ at a steady pace of $70 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 140 km at 70 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 140 km at 70 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

140 \div 70 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

140 + 135 = 275\ \text{km},\quad 2 + 3 = 5\ \text{h}

275 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

275 \div 5 = 55

The bus averaged 55 km per hour.

Answer: 55 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 70, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 115/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 6 medium

A bus first travels $90 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour, then travels another $150 \text{ km}$ at a steady pace of $50 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 90 km at 45 km/h, then 150 km at 50 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 90 km at 45 km per hour.
Second leg: 150 km at 50 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

90 \div 45 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

150 \div 50 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

90 + 150 = 240\ \text{km},\quad 2 + 3 = 5\ \text{h}

240 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

240 \div 5 = 48

The bus averaged 48 km per hour.

Answer: 48 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 50, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 95/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 7 medium

A bus first travels $150 \text{ km}$ at a steady pace of $75 \text{ km}$ per hour, then travels another $45 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 150 km at 75 km/h, then 45 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 150 km at 75 km per hour.
Second leg: 45 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

150 \div 75 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

45 \div 45 = 1\ \text{h}

The second leg takes 1 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

150 + 45 = 195\ \text{km},\quad 2 + 1 = 3\ \text{h}

195 km in 3 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

195 \div 3 = 65

The bus averaged 65 km per hour.

Answer: 65 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 75, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 60, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 1.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 8 medium

A bus first travels $160 \text{ km}$ at a steady pace of $80 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 160 km at 80 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 160 km at 80 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

160 \div 80 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

160 + 135 = 295\ \text{km},\quad 2 + 3 = 5\ \text{h}

295 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

295 \div 5 = 59

The bus averaged 59 km per hour.

Answer: 59 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 80, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 125/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 9 hard

A bus first travels $170 \text{ km}$ at a steady pace of $85 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 170 km at 85 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 170 km at 85 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

170 \div 85 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

170 + 135 = 305\ \text{km},\quad 2 + 3 = 5\ \text{h}

305 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

305 \div 5 = 61

The bus averaged 61 km per hour.

Answer: 61 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 85, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 65, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 10 hard

A bus first travels $190 \text{ km}$ at a steady pace of $95 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 190 km at 95 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 190 km at 95 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

190 \div 95 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

190 + 135 = 325\ \text{km},\quad 2 + 3 = 5\ \text{h}

325 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

325 \div 5 = 65

The bus averaged 65 km per hour.

Answer: 65 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 95, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 70, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 11 hard

A bus first travels $200 \text{ km}$ at a steady pace of $100 \text{ km}$ per hour, then travels another $135 \text{ km}$ at a steady pace of $45 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 200 km at 100 km/h, then 135 km at 45 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 200 km at 100 km per hour.
Second leg: 135 km at 45 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

200 \div 100 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

135 \div 45 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

200 + 135 = 335\ \text{km},\quad 2 + 3 = 5\ \text{h}

335 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

335 \div 5 = 67

The bus averaged 67 km per hour.

Answer: 67 km

4 · Reviewdoes it hold up?

The answer sits between 45 and 100, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 145/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Variant 12 hard

A bus first travels $180 \text{ km}$ at a steady pace of $90 \text{ km}$ per hour, then travels another $255 \text{ km}$ at a steady pace of $85 \text{ km}$ per hour. On average, how many kilometers did the bus travel per hour over the whole trip?

Your answer

Show solution

1 · Understandwhat's really being asked

A bus covers 180 km at 90 km/h, then 255 km at 85 km/h. We need the average kilometres per hour for the trip as a whole.

Givens

First leg: 180 km at 90 km per hour.
Second leg: 255 km at 85 km per hour.

Unknowns

The average speed over the whole trip.

Constraints

Each leg is at a steady speed, but the two speeds differ.

2 · Planchoose the strategy

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

Average speed means total distance divided by total time, so find each leg's time first. The units say so: kilometres per hour needs a kilometre total and an hour total, not two speeds averaged.

3 · Execute4 carry out the plan

1Time for the first part

#7 Identify Subproblems 6.RP.A.3

Distance divided by speed gives time.

180 \div 90 = 2\ \text{h}

The first leg takes 2 hours.

2Time for the second part

#7 Identify Subproblems 6.RP.A.3

Same again with the second leg's numbers.

255 \div 85 = 3\ \text{h}

The second leg takes 3 hours.

3Add the totals

#7 Identify Subproblems 6.RP.A.3

The whole trip is both distances and both times.

180 + 255 = 435\ \text{km},\quad 2 + 3 = 5\ \text{h}

435 km in 5 hours.

4Find the average per hour

#8 Analyze the Units 6.RP.A.2

Divide the total distance by the total time.

435 \div 5 = 87

The bus averaged 87 km per hour.

Answer: 87 km

4 · Reviewdoes it hold up?

The answer sits between 85 and 90, as any average of the two speeds must, and leans towards the speed the bus held for longer.

Another way: Averaging the two speeds gives 175/2, which is wrong: it treats both legs as taking equal time when in fact one took 2 hours and the other 3.

Standardsmin grade 6

6.RP.A.2 Understand the concept of a unit rate and use rate language — Reading the answer as a unit rate: kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems — Turning each distance and speed into a time, then totalling both.

💡Takeaway. Average speed is total distance over total time. Averaging the two speeds only works if the bus spent equal time at each.

Average distance per hour from total time · 12 practice problems

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