← Express smaller units as larger using fractions · Multiplicative Comparison and Unit Rate

Express smaller units as larger using fractions · 12 practice problems

5.NF.A.15.MD.A.15.NF.B.6

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Variant 1 easy

There are two faucets that pour out water at a steady rate of $\dfrac{1}{4}\text{ L}$ and $3\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 12 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 1 over 4 litres a minute and the other at 3 and 1 over 2 litres a minute. They run for 3 minutes and 12 seconds. We need the total collected.

Givens

First tap: $\frac{1}{4}$ L per minute.
Second tap: $3\frac{1}{2}$ L per minute.
Both run for 3 minutes 12 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{1}{4} + \frac{7}{2} = \frac{15}{4}

Together they deliver

\frac{15}{4}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 12 seconds is 12 sixtieths of a minute.

3 + \frac{12}{60} = 3 + \frac{1}{5} = \frac{16}{5}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{15}{4} \times \frac{16}{5} = 12

They collect 12 L.

Answer: 12 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 3 L a minute for a bit over 3 minutes, so a total near 12 L is the right size.

Another way: Work each tap separately over the $\frac{16}{5}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 12 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 2 easy

There are two faucets that pour out water at a steady rate of $\dfrac{1}{5}\text{ L}$ and $2\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 20 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 1 over 5 litres a minute and the other at 2 and 1 over 2 litres a minute. They run for 3 minutes and 20 seconds. We need the total collected.

Givens

First tap: $\frac{1}{5}$ L per minute.
Second tap: $2\frac{1}{2}$ L per minute.
Both run for 3 minutes 20 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{1}{5} + \frac{5}{2} = \frac{27}{10}

Together they deliver

\frac{27}{10}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 20 seconds is 20 sixtieths of a minute.

3 + \frac{20}{60} = 3 + \frac{1}{3} = \frac{10}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{27}{10} \times \frac{10}{3} = 9

They collect 9 L.

Answer: 9 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 2 L a minute for a bit over 3 minutes, so a total near 9 L is the right size.

Another way: Work each tap separately over the $\frac{10}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 3 easy

There are two faucets that pour out water at a steady rate of $\dfrac{7}{8}\text{ L}$ and $1\dfrac{3}{4}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 5 minutes 20 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 7 over 8 litres a minute and the other at 1 and 3 over 4 litres a minute. They run for 5 minutes and 20 seconds. We need the total collected.

Givens

First tap: $\frac{7}{8}$ L per minute.
Second tap: $1\frac{3}{4}$ L per minute.
Both run for 5 minutes 20 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{7}{8} + \frac{7}{4} = \frac{21}{8}

Together they deliver

\frac{21}{8}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 20 seconds is 20 sixtieths of a minute.

5 + \frac{20}{60} = 5 + \frac{1}{3} = \frac{16}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{21}{8} \times \frac{16}{3} = 14

They collect 14 L.

Answer: 14 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 2 L a minute for a bit over 5 minutes, so a total near 14 L is the right size.

Another way: Work each tap separately over the $\frac{16}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 4 medium

There are two faucets that pour out water at a steady rate of $\dfrac{2}{5}\text{ L}$ and $3\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 20 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 2 over 5 litres a minute and the other at 3 and 1 over 2 litres a minute. They run for 3 minutes and 20 seconds. We need the total collected.

Givens

First tap: $\frac{2}{5}$ L per minute.
Second tap: $3\frac{1}{2}$ L per minute.
Both run for 3 minutes 20 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{2}{5} + \frac{7}{2} = \frac{39}{10}

Together they deliver

\frac{39}{10}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 20 seconds is 20 sixtieths of a minute.

3 + \frac{20}{60} = 3 + \frac{1}{3} = \frac{10}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{39}{10} \times \frac{10}{3} = 13

They collect 13 L.

Answer: 13 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 3 L a minute for a bit over 3 minutes, so a total near 13 L is the right size.

Another way: Work each tap separately over the $\frac{10}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 5 easy

There are two faucets that pour out water at a steady rate of $\dfrac{3}{4}\text{ L}$ and $1\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 5 minutes 20 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 3 over 4 litres a minute and the other at 1 and 1 over 2 litres a minute. They run for 5 minutes and 20 seconds. We need the total collected.

Givens

First tap: $\frac{3}{4}$ L per minute.
Second tap: $1\frac{1}{2}$ L per minute.
Both run for 5 minutes 20 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{3}{4} + \frac{3}{2} = \frac{9}{4}

Together they deliver

\frac{9}{4}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 20 seconds is 20 sixtieths of a minute.

5 + \frac{20}{60} = 5 + \frac{1}{3} = \frac{16}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{9}{4} \times \frac{16}{3} = 12

They collect 12 L.

Answer: 12 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 2 L a minute for a bit over 5 minutes, so a total near 12 L is the right size.

Another way: Work each tap separately over the $\frac{16}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 6 medium

There are two faucets that pour out water at a steady rate of $\dfrac{4}{5}\text{ L}$ and $2\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 20 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 4 over 5 litres a minute and the other at 2 and 1 over 2 litres a minute. They run for 3 minutes and 20 seconds. We need the total collected.

Givens

First tap: $\frac{4}{5}$ L per minute.
Second tap: $2\frac{1}{2}$ L per minute.
Both run for 3 minutes 20 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{4}{5} + \frac{5}{2} = \frac{33}{10}

Together they deliver

\frac{33}{10}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 20 seconds is 20 sixtieths of a minute.

3 + \frac{20}{60} = 3 + \frac{1}{3} = \frac{10}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{33}{10} \times \frac{10}{3} = 11

They collect 11 L.

Answer: 11 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 3 L a minute for a bit over 3 minutes, so a total near 11 L is the right size.

Another way: Work each tap separately over the $\frac{10}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 7 medium

There are two faucets that pour out water at a steady rate of $\dfrac{1}{3}\text{ L}$ and $1\dfrac{2}{3}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 4 minutes 30 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 1 over 3 litres a minute and the other at 1 and 2 over 3 litres a minute. They run for 4 minutes and 30 seconds. We need the total collected.

Givens

First tap: $\frac{1}{3}$ L per minute.
Second tap: $1\frac{2}{3}$ L per minute.
Both run for 4 minutes 30 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{1}{3} + \frac{5}{3} = 2

Together they deliver 2 L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 30 seconds is 30 sixtieths of a minute.

4 + \frac{30}{60} = 4 + \frac{1}{2} = \frac{9}{2}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

2 \times \frac{9}{2} = 9

They collect 9 L.

Answer: 9 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 2 L a minute for a bit over 4 minutes, so a total near 9 L is the right size.

Another way: Work each tap separately over the $\frac{9}{2}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 30 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 8 hard

There are two faucets that pour out water at a steady rate of $\dfrac{2}{3}\text{ L}$ and $3\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 36 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 2 over 3 litres a minute and the other at 3 and 1 over 2 litres a minute. They run for 3 minutes and 36 seconds. We need the total collected.

Givens

First tap: $\frac{2}{3}$ L per minute.
Second tap: $3\frac{1}{2}$ L per minute.
Both run for 3 minutes 36 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{2}{3} + \frac{7}{2} = \frac{25}{6}

Together they deliver

\frac{25}{6}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 36 seconds is 36 sixtieths of a minute.

3 + \frac{36}{60} = 3 + \frac{3}{5} = \frac{18}{5}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{25}{6} \times \frac{18}{5} = 15

They collect 15 L.

Answer: 15 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 4 L a minute for a bit over 3 minutes, so a total near 15 L is the right size.

Another way: Work each tap separately over the $\frac{18}{5}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 36 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 9 hard

There are two faucets that pour out water at a steady rate of $\dfrac{5}{6}\text{ L}$ and $2\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 36 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 5 over 6 litres a minute and the other at 2 and 1 over 2 litres a minute. They run for 3 minutes and 36 seconds. We need the total collected.

Givens

First tap: $\frac{5}{6}$ L per minute.
Second tap: $2\frac{1}{2}$ L per minute.
Both run for 3 minutes 36 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{5}{6} + \frac{5}{2} = \frac{10}{3}

Together they deliver

\frac{10}{3}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 36 seconds is 36 sixtieths of a minute.

3 + \frac{36}{60} = 3 + \frac{3}{5} = \frac{18}{5}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{10}{3} \times \frac{18}{5} = 12

They collect 12 L.

Answer: 12 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 3 L a minute for a bit over 3 minutes, so a total near 12 L is the right size.

Another way: Work each tap separately over the $\frac{18}{5}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 36 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 10 medium

There are two faucets that pour out water at a steady rate of $\dfrac{1}{2}\text{ L}$ and $3\dfrac{2}{3}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 3 minutes 36 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 1 over 2 litres a minute and the other at 3 and 2 over 3 litres a minute. They run for 3 minutes and 36 seconds. We need the total collected.

Givens

First tap: $\frac{1}{2}$ L per minute.
Second tap: $3\frac{2}{3}$ L per minute.
Both run for 3 minutes 36 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{1}{2} + \frac{11}{3} = \frac{25}{6}

Together they deliver

\frac{25}{6}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 36 seconds is 36 sixtieths of a minute.

3 + \frac{36}{60} = 3 + \frac{3}{5} = \frac{18}{5}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{25}{6} \times \frac{18}{5} = 15

They collect 15 L.

Answer: 15 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 4 L a minute for a bit over 3 minutes, so a total near 15 L is the right size.

Another way: Work each tap separately over the $\frac{18}{5}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 36 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 11 hard

There are two faucets that pour out water at a steady rate of $\dfrac{3}{5}\text{ L}$ and $1\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 6 minutes 40 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 3 over 5 litres a minute and the other at 1 and 1 over 2 litres a minute. They run for 6 minutes and 40 seconds. We need the total collected.

Givens

First tap: $\frac{3}{5}$ L per minute.
Second tap: $1\frac{1}{2}$ L per minute.
Both run for 6 minutes 40 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{3}{5} + \frac{3}{2} = \frac{21}{10}

Together they deliver

\frac{21}{10}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 40 seconds is 40 sixtieths of a minute.

6 + \frac{40}{60} = 6 + \frac{2}{3} = \frac{20}{3}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{21}{10} \times \frac{20}{3} = 14

They collect 14 L.

Answer: 14 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 2 L a minute for a bit over 6 minutes, so a total near 14 L is the right size.

Another way: Work each tap separately over the $\frac{20}{3}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 40 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Variant 12 hard

There are two faucets that pour out water at a steady rate of $\dfrac{1}{6}\text{ L}$ and $1\dfrac{1}{2}\text{ L}$ per minute. If you turn on both faucets at the same time and collect water for 4 minutes 48 seconds, how many liters of water can you collect in all?

Your answer

Show solution

1 · Understandwhat's really being asked

Two taps run at the same time, one at 1 over 6 litres a minute and the other at 1 and 1 over 2 litres a minute. They run for 4 minutes and 48 seconds. We need the total collected.

Givens

First tap: $\frac{1}{6}$ L per minute.
Second tap: $1\frac{1}{2}$ L per minute.
Both run for 4 minutes 48 seconds.

Unknowns

The total litres collected.

Constraints

The rates are per minute, so the time must be in minutes too.

2 · Planchoose the strategy

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

Two taps into one container add up, so combine the rates first. Then make the time match those units -- litres-per-minute times minutes gives litres, but only if the seconds are converted first.

3 · Execute3 carry out the plan

1Combine the two per-minute rates

#7 Identify Subproblems 5.NF.A.1

Both pour into the same container, so their rates add. Give them a common denominator first.

\frac{1}{6} + \frac{3}{2} = \frac{5}{3}

Together they deliver

\frac{5}{3}

L each minute.

2Write the time as minutes

#8 Analyze the Units 5.MD.A.1

A minute is 60 seconds, so 48 seconds is 48 sixtieths of a minute.

4 + \frac{48}{60} = 4 + \frac{4}{5} = \frac{24}{5}\ \text{min}

Now the time is measured in the same unit the rate uses.

3Multiply rate by time

#8 Analyze the Units 5.NF.B.6

Litres per minute times minutes leaves litres -- the minutes cancel.

\frac{5}{3} \times \frac{24}{5} = 8

They collect 8 L.

Answer: 8 L

4 · Reviewdoes it hold up?

Round to check: the taps together give a bit over 1 L a minute for a bit over 4 minutes, so a total near 8 L is the right size.

Another way: Work each tap separately over the $\frac{24}{5}$ minutes and add the two totals; combining the rates first just saves one multiplication.

Standardsmin grade 5

5.NF.A.1 Add and subtract fractions with unlike denominators — Adding the two per-minute rates over a common denominator.
5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 48 seconds into a fraction of a minute.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the combined rate by the time in minutes.

💡Takeaway. A rate tells you which unit the time must be in. Convert the seconds first, and the multiplication takes care of itself.

Where next?

← Back to the problem: Express smaller units as larger using fractions