Practice · The shared distance links two different rates · Sensim Math

Variant 1 easy answer:

39\frac{13}{30}

km

A taxi travels $45\frac{1}{2}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $4$ hours $20$ minutes, but in $5$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 45 and 1/2 km an hour. A truck has to cover whatever the taxi covers in 4 hours 20 minutes, but it has 5 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 45 and 1/2 km per hour.
The distance is what the taxi covers in 4 hours 20 minutes.
The truck takes 5 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

20 minutes is 20 sixtieths of an hour.

4\ \text{h}\ 20\ \text{min} = 4\frac{1}{3}\ \text{h}

4 and 1/3 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

45\frac{1}{2} \times 4\frac{1}{3} = 197\frac{1}{6}\ \text{km}

Both vehicles cover 197 and 1/6 km.

3Spread it over 5 hours

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

The truck covers the same distance in 5 hours, so each hour it does one 5th of it.

197\frac{1}{6} \div 5 = 39\frac{13}{30}\ \text{km}

The truck needs 39 and 13/30 km every hour.

Answer:

39\frac{13}{30}

km

4 · Reviewdoes it hold up?

5 hours at 39 and 13/30 km comes back to 197 and 1/6 km, the distance we started from. The truck is slower than the taxi, which fits: it has 5 hours for a 4 and 1/3-hour trip.

Another way: Comparing the times first works too: 4 and 1/3 hours against 5 means the truck's speed is $\frac{13}{15}$ of the taxi's, and $45\frac{1}{2}$ times that is the same $39\frac{13}{30}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 2 easy answer:

42\frac{7}{12}

km

A taxi travels $48\frac{2}{3}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $3$ hours $30$ minutes, but in $4$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 48 and 2/3 km an hour. A truck has to cover whatever the taxi covers in 3 hours 30 minutes, but it has 4 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 48 and 2/3 km per hour.
The distance is what the taxi covers in 3 hours 30 minutes.
The truck takes 4 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

30 minutes is 30 sixtieths of an hour.

3\ \text{h}\ 30\ \text{min} = 3\frac{1}{2}\ \text{h}

3 and 1/2 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

48\frac{2}{3} \times 3\frac{1}{2} = 170\frac{1}{3}\ \text{km}

Both vehicles cover 170 and 1/3 km.

3Spread it over 4 hours

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

The truck covers the same distance in 4 hours, so each hour it does one 4th of it.

170\frac{1}{3} \div 4 = 42\frac{7}{12}\ \text{km}

The truck needs 42 and 7/12 km every hour.

Answer:

42\frac{7}{12}

km

4 · Reviewdoes it hold up?

4 hours at 42 and 7/12 km comes back to 170 and 1/3 km, the distance we started from. The truck is slower than the taxi, which fits: it has 4 hours for a 3 and 1/2-hour trip.

Another way: Comparing the times first works too: 3 and 1/2 hours against 4 means the truck's speed is $\frac{7}{8}$ of the taxi's, and $48\frac{2}{3}$ times that is the same $42\frac{7}{12}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 3 easy answer:

38\frac{27}{40}

km

A taxi travels $54\frac{3}{5}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $2$ hours $50$ minutes, but in $4$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 54 and 3/5 km an hour. A truck has to cover whatever the taxi covers in 2 hours 50 minutes, but it has 4 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 54 and 3/5 km per hour.
The distance is what the taxi covers in 2 hours 50 minutes.
The truck takes 4 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

50 minutes is 50 sixtieths of an hour.

2\ \text{h}\ 50\ \text{min} = 2\frac{5}{6}\ \text{h}

2 and 5/6 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

54\frac{3}{5} \times 2\frac{5}{6} = 154\frac{7}{10}\ \text{km}

Both vehicles cover 154 and 7/10 km.

3Spread it over 4 hours

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

The truck covers the same distance in 4 hours, so each hour it does one 4th of it.

154\frac{7}{10} \div 4 = 38\frac{27}{40}\ \text{km}

The truck needs 38 and 27/40 km every hour.

Answer:

38\frac{27}{40}

km

4 · Reviewdoes it hold up?

4 hours at 38 and 27/40 km comes back to 154 and 7/10 km, the distance we started from. The truck is slower than the taxi, which fits: it has 4 hours for a 2 and 5/6-hour trip.

Another way: Comparing the times first works too: 2 and 5/6 hours against 4 means the truck's speed is $\frac{17}{24}$ of the taxi's, and $54\frac{3}{5}$ times that is the same $38\frac{27}{40}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 4 easy answer:

52\frac{15}{16}

km

A taxi travels $60\frac{1}{2}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $1$ hours $45$ minutes, but in $2$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 60 and 1/2 km an hour. A truck has to cover whatever the taxi covers in 1 hours 45 minutes, but it has 2 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 60 and 1/2 km per hour.
The distance is what the taxi covers in 1 hours 45 minutes.
The truck takes 2 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

45 minutes is 45 sixtieths of an hour.

1\ \text{h}\ 45\ \text{min} = 1\frac{3}{4}\ \text{h}

1 and 3/4 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

60\frac{1}{2} \times 1\frac{3}{4} = 105\frac{7}{8}\ \text{km}

Both vehicles cover 105 and 7/8 km.

3Spread it over 2 hours

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

The truck covers the same distance in 2 hours, so each hour it does one 2th of it.

105\frac{7}{8} \div 2 = 52\frac{15}{16}\ \text{km}

The truck needs 52 and 15/16 km every hour.

Answer:

52\frac{15}{16}

km

4 · Reviewdoes it hold up?

2 hours at 52 and 15/16 km comes back to 105 and 7/8 km, the distance we started from. The truck is slower than the taxi, which fits: it has 2 hours for a 1 and 3/4-hour trip.

Another way: Comparing the times first works too: 1 and 3/4 hours against 2 means the truck's speed is $\frac{7}{8}$ of the taxi's, and $60\frac{1}{2}$ times that is the same $52\frac{15}{16}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 5 medium answer:

51\frac{79}{144}

km

A taxi travels $63\frac{4}{9}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $3$ hours $15$ minutes, but in $4$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 63 and 4/9 km an hour. A truck has to cover whatever the taxi covers in 3 hours 15 minutes, but it has 4 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 63 and 4/9 km per hour.
The distance is what the taxi covers in 3 hours 15 minutes.
The truck takes 4 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

15 minutes is 15 sixtieths of an hour.

3\ \text{h}\ 15\ \text{min} = 3\frac{1}{4}\ \text{h}

3 and 1/4 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

63\frac{4}{9} \times 3\frac{1}{4} = 206\frac{7}{36}\ \text{km}

Both vehicles cover 206 and 7/36 km.

3Spread it over 4 hours

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

The truck covers the same distance in 4 hours, so each hour it does one 4th of it.

206\frac{7}{36} \div 4 = 51\frac{79}{144}\ \text{km}

The truck needs 51 and 79/144 km every hour.

Answer:

51\frac{79}{144}

km

4 · Reviewdoes it hold up?

4 hours at 51 and 79/144 km comes back to 206 and 7/36 km, the distance we started from. The truck is slower than the taxi, which fits: it has 4 hours for a 3 and 1/4-hour trip.

Another way: Comparing the times first works too: 3 and 1/4 hours against 4 means the truck's speed is $\frac{13}{16}$ of the taxi's, and $63\frac{4}{9}$ times that is the same $51\frac{79}{144}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 6 medium answer:

47\frac{49}{54}

km

A taxi travels $66\frac{1}{3}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $2$ hours $10$ minutes, but in $3$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 66 and 1/3 km an hour. A truck has to cover whatever the taxi covers in 2 hours 10 minutes, but it has 3 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 66 and 1/3 km per hour.
The distance is what the taxi covers in 2 hours 10 minutes.
The truck takes 3 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

10 minutes is 10 sixtieths of an hour.

2\ \text{h}\ 10\ \text{min} = 2\frac{1}{6}\ \text{h}

2 and 1/6 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

66\frac{1}{3} \times 2\frac{1}{6} = 143\frac{13}{18}\ \text{km}

Both vehicles cover 143 and 13/18 km.

3Spread it over 3 hours

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

The truck covers the same distance in 3 hours, so each hour it does one 3th of it.

143\frac{13}{18} \div 3 = 47\frac{49}{54}\ \text{km}

The truck needs 47 and 49/54 km every hour.

Answer:

47\frac{49}{54}

km

4 · Reviewdoes it hold up?

3 hours at 47 and 49/54 km comes back to 143 and 13/18 km, the distance we started from. The truck is slower than the taxi, which fits: it has 3 hours for a 2 and 1/6-hour trip.

Another way: Comparing the times first works too: 2 and 1/6 hours against 3 means the truck's speed is $\frac{13}{18}$ of the taxi's, and $66\frac{1}{3}$ times that is the same $47\frac{49}{54}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 7 medium answer:

53\frac{37}{90}

km

A taxi travels $72\frac{5}{6}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $3$ hours $40$ minutes, but in $5$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 72 and 5/6 km an hour. A truck has to cover whatever the taxi covers in 3 hours 40 minutes, but it has 5 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 72 and 5/6 km per hour.
The distance is what the taxi covers in 3 hours 40 minutes.
The truck takes 5 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

40 minutes is 40 sixtieths of an hour.

3\ \text{h}\ 40\ \text{min} = 3\frac{2}{3}\ \text{h}

3 and 2/3 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

72\frac{5}{6} \times 3\frac{2}{3} = 267\frac{1}{18}\ \text{km}

Both vehicles cover 267 and 1/18 km.

3Spread it over 5 hours

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

The truck covers the same distance in 5 hours, so each hour it does one 5th of it.

267\frac{1}{18} \div 5 = 53\frac{37}{90}\ \text{km}

The truck needs 53 and 37/90 km every hour.

Answer:

53\frac{37}{90}

km

4 · Reviewdoes it hold up?

5 hours at 53 and 37/90 km comes back to 267 and 1/18 km, the distance we started from. The truck is slower than the taxi, which fits: it has 5 hours for a 3 and 2/3-hour trip.

Another way: Comparing the times first works too: 3 and 2/3 hours against 5 means the truck's speed is $\frac{11}{15}$ of the taxi's, and $72\frac{5}{6}$ times that is the same $53\frac{37}{90}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 8 medium answer:

58\frac{11}{12}

km

A taxi travels $75\frac{3}{4}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $2$ hours $20$ minutes, but in $3$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 75 and 3/4 km an hour. A truck has to cover whatever the taxi covers in 2 hours 20 minutes, but it has 3 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 75 and 3/4 km per hour.
The distance is what the taxi covers in 2 hours 20 minutes.
The truck takes 3 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

20 minutes is 20 sixtieths of an hour.

2\ \text{h}\ 20\ \text{min} = 2\frac{1}{3}\ \text{h}

2 and 1/3 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

75\frac{3}{4} \times 2\frac{1}{3} = 176\frac{3}{4}\ \text{km}

Both vehicles cover 176 and 3/4 km.

3Spread it over 3 hours

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

The truck covers the same distance in 3 hours, so each hour it does one 3th of it.

176\frac{3}{4} \div 3 = 58\frac{11}{12}\ \text{km}

The truck needs 58 and 11/12 km every hour.

Answer:

58\frac{11}{12}

km

4 · Reviewdoes it hold up?

3 hours at 58 and 11/12 km comes back to 176 and 3/4 km, the distance we started from. The truck is slower than the taxi, which fits: it has 3 hours for a 2 and 1/3-hour trip.

Another way: Comparing the times first works too: 2 and 1/3 hours against 3 means the truck's speed is $\frac{7}{9}$ of the taxi's, and $75\frac{3}{4}$ times that is the same $58\frac{11}{12}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 9 hard answer:

71\frac{7}{15}

km

A taxi travels $80\frac{2}{5}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $2$ hours $40$ minutes, but in $3$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 80 and 2/5 km an hour. A truck has to cover whatever the taxi covers in 2 hours 40 minutes, but it has 3 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 80 and 2/5 km per hour.
The distance is what the taxi covers in 2 hours 40 minutes.
The truck takes 3 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

40 minutes is 40 sixtieths of an hour.

2\ \text{h}\ 40\ \text{min} = 2\frac{2}{3}\ \text{h}

2 and 2/3 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

80\frac{2}{5} \times 2\frac{2}{3} = 214\frac{2}{5}\ \text{km}

Both vehicles cover 214 and 2/5 km.

3Spread it over 3 hours

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

The truck covers the same distance in 3 hours, so each hour it does one 3th of it.

214\frac{2}{5} \div 3 = 71\frac{7}{15}\ \text{km}

The truck needs 71 and 7/15 km every hour.

Answer:

71\frac{7}{15}

km

4 · Reviewdoes it hold up?

3 hours at 71 and 7/15 km comes back to 214 and 2/5 km, the distance we started from. The truck is slower than the taxi, which fits: it has 3 hours for a 2 and 2/3-hour trip.

Another way: Comparing the times first works too: 2 and 2/3 hours against 3 means the truck's speed is $\frac{8}{9}$ of the taxi's, and $80\frac{2}{5}$ times that is the same $71\frac{7}{15}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 10 hard answer:

77\frac{11}{42}

km

A taxi travels $84\frac{2}{7}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $1$ hours $50$ minutes, but in $2$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 84 and 2/7 km an hour. A truck has to cover whatever the taxi covers in 1 hours 50 minutes, but it has 2 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 84 and 2/7 km per hour.
The distance is what the taxi covers in 1 hours 50 minutes.
The truck takes 2 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

50 minutes is 50 sixtieths of an hour.

1\ \text{h}\ 50\ \text{min} = 1\frac{5}{6}\ \text{h}

1 and 5/6 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

84\frac{2}{7} \times 1\frac{5}{6} = 154\frac{11}{21}\ \text{km}

Both vehicles cover 154 and 11/21 km.

3Spread it over 2 hours

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

The truck covers the same distance in 2 hours, so each hour it does one 2th of it.

154\frac{11}{21} \div 2 = 77\frac{11}{42}\ \text{km}

The truck needs 77 and 11/42 km every hour.

Answer:

77\frac{11}{42}

km

4 · Reviewdoes it hold up?

2 hours at 77 and 11/42 km comes back to 154 and 11/21 km, the distance we started from. The truck is slower than the taxi, which fits: it has 2 hours for a 1 and 5/6-hour trip.

Another way: Comparing the times first works too: 1 and 5/6 hours against 2 means the truck's speed is $\frac{11}{12}$ of the taxi's, and $84\frac{2}{7}$ times that is the same $77\frac{11}{42}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 11 hard answer:

56\frac{13}{32}

km

A taxi travels $90\frac{1}{4}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $1$ hours $15$ minutes, but in $2$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 90 and 1/4 km an hour. A truck has to cover whatever the taxi covers in 1 hours 15 minutes, but it has 2 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 90 and 1/4 km per hour.
The distance is what the taxi covers in 1 hours 15 minutes.
The truck takes 2 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

15 minutes is 15 sixtieths of an hour.

1\ \text{h}\ 15\ \text{min} = 1\frac{1}{4}\ \text{h}

1 and 1/4 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

90\frac{1}{4} \times 1\frac{1}{4} = 112\frac{13}{16}\ \text{km}

Both vehicles cover 112 and 13/16 km.

3Spread it over 2 hours

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

The truck covers the same distance in 2 hours, so each hour it does one 2th of it.

112\frac{13}{16} \div 2 = 56\frac{13}{32}\ \text{km}

The truck needs 56 and 13/32 km every hour.

Answer:

56\frac{13}{32}

km

4 · Reviewdoes it hold up?

2 hours at 56 and 13/32 km comes back to 112 and 13/16 km, the distance we started from. The truck is slower than the taxi, which fits: it has 2 hours for a 1 and 1/4-hour trip.

Another way: Comparing the times first works too: 1 and 1/4 hours against 2 means the truck's speed is $\frac{5}{8}$ of the taxi's, and $90\frac{1}{4}$ times that is the same $56\frac{13}{32}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

Variant 12 hard answer:

77\frac{61}{96}

km

A taxi travels $96\frac{3}{8}\ \text{km}$ in one hour at a steady speed. A truck wants to cover the distance the taxi travels in $2$ hours $25$ minutes, but in $3$ hours at a steady speed. How many kilometers must the truck travel each hour?

Show solution

1 · Understandwhat's really being asked

A taxi does 96 and 3/8 km an hour. A truck has to cover whatever the taxi covers in 2 hours 25 minutes, but it has 3 hours to do it. We want the truck's kilometres per hour.

Givens

The taxi's steady speed is 96 and 3/8 km per hour.
The distance is what the taxi covers in 2 hours 25 minutes.
The truck takes 3 hours to cover it.

Unknowns

How far the truck must travel in one hour.

Constraints

Both vehicles hold a steady speed, so distance is speed times time.

2 · Planchoose the strategy

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

The two speeds are not linked to each other, only to the distance they share. Find that distance from the taxi, then hand it to the truck. The minutes must become a fraction of an hour first, since the rate is per hour.

3 · Execute3 carry out the plan

1Write the time as hours

#8 Analyze the Units 6.RP.A.3

25 minutes is 25 sixtieths of an hour.

2\ \text{h}\ 25\ \text{min} = 2\frac{5}{12}\ \text{h}

2 and 5/12 hours, all in one number.

2Find the distance

#7 Identify Subproblems 7.RP.A.1

The taxi's speed times that time.

96\frac{3}{8} \times 2\frac{5}{12} = 232\frac{29}{32}\ \text{km}

Both vehicles cover 232 and 29/32 km.

3Spread it over 3 hours

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

The truck covers the same distance in 3 hours, so each hour it does one 3th of it.

232\frac{29}{32} \div 3 = 77\frac{61}{96}\ \text{km}

The truck needs 77 and 61/96 km every hour.

Answer:

77\frac{61}{96}

km

4 · Reviewdoes it hold up?

3 hours at 77 and 61/96 km comes back to 232 and 29/32 km, the distance we started from. The truck is slower than the taxi, which fits: it has 3 hours for a 2 and 5/12-hour trip.

Another way: Comparing the times first works too: 2 and 5/12 hours against 3 means the truck's speed is $\frac{29}{36}$ of the taxi's, and $96\frac{3}{8}$ times that is the same $77\frac{61}{96}$ .

Standardsmin grade 7

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading the answer as kilometres per one hour.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning minutes into a fraction of an hour before using a rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Multiplying a fractional rate by a fractional time.

💡Takeaway. When two speeds have nothing in common, find the distance they share and travel through it.

The shared distance links two different rates · 12 practice problems

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