← Call the whole job 1 and every day becomes a fraction of it · Multiplicative Comparison and Unit Rate

Call the whole job 1 and every day becomes a fraction of it · 12 practice problems

7.RP.A.16.NS.A.16.RP.A.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 4 days

Working alone, Jaewoo takes $5$ days to do $\frac{1}{2}$ of a job, and Seohee takes $5$ days to do $\frac{3}{4}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 5 days and Seohee does 3/4 of it in 5 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 5 days.
Seohee: 3/4 of the job in 5 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 5 days.

\frac{1}{2} \div 5 = \frac{1}{10}

Jaewoo does 1/10 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

3/4 of the job spread over 5 days.

\frac{3}{4} \div 5 = \frac{3}{20}

Seohee does 3/20 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{10} + \frac{3}{20} = \frac{1}{4}

Together they do 1/4 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{4} = 4

They finish in 4 days.

Answer: 4 days

4 · Reviewdoes it hold up?

4 days of 1/4 of the job is exactly 1 whole job. And the pair is quicker than Seohee alone, who would need $6\frac{2}{3}$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 4 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 2 easy answer: 4 days

Working alone, Jaewoo takes $5$ days to do $\frac{1}{4}$ of a job, and Seohee takes $2$ days to do $\frac{2}{5}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/4 of a job in 5 days and Seohee does 2/5 of it in 2 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/4 of the job in 5 days.
Seohee: 2/5 of the job in 2 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/4 of the job spread over 5 days.

\frac{1}{4} \div 5 = \frac{1}{20}

Jaewoo does 1/20 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

2/5 of the job spread over 2 days.

\frac{2}{5} \div 2 = \frac{1}{5}

Seohee does 1/5 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{20} + \frac{1}{5} = \frac{1}{4}

Together they do 1/4 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{4} = 4

They finish in 4 days.

Answer: 4 days

4 · Reviewdoes it hold up?

4 days of 1/4 of the job is exactly 1 whole job. And the pair is quicker than Seohee alone, who would need $5$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 4 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 3 easy answer: 4 days

Working alone, Jaewoo takes $3$ days to do $\frac{1}{2}$ of a job, and Seohee takes $6$ days to do $\frac{1}{2}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 3 days and Seohee does 1/2 of it in 6 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 3 days.
Seohee: 1/2 of the job in 6 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 3 days.

\frac{1}{2} \div 3 = \frac{1}{6}

Jaewoo does 1/6 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 6 days.

\frac{1}{2} \div 6 = \frac{1}{12}

Seohee does 1/12 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{6} + \frac{1}{12} = \frac{1}{4}

Together they do 1/4 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{4} = 4

They finish in 4 days.

Answer: 4 days

4 · Reviewdoes it hold up?

4 days of 1/4 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $6$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 4 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 4 easy answer: 3 days

Working alone, Jaewoo takes $4$ days to do $\frac{1}{2}$ of a job, and Seohee takes $4$ days to do $\frac{5}{6}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 4 days and Seohee does 5/6 of it in 4 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 4 days.
Seohee: 5/6 of the job in 4 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 4 days.

\frac{1}{2} \div 4 = \frac{1}{8}

Jaewoo does 1/8 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

5/6 of the job spread over 4 days.

\frac{5}{6} \div 4 = \frac{5}{24}

Seohee does 5/24 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{8} + \frac{5}{24} = \frac{1}{3}

Together they do 1/3 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{3} = 3

They finish in 3 days.

Answer: 3 days

4 · Reviewdoes it hold up?

3 days of 1/3 of the job is exactly 1 whole job. And the pair is quicker than Seohee alone, who would need $4\frac{4}{5}$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 3 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 5 medium answer: 6 days

Working alone, Jaewoo takes $7$ days to do $\frac{1}{2}$ of a job, and Seohee takes $7$ days to do $\frac{2}{3}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 7 days and Seohee does 2/3 of it in 7 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 7 days.
Seohee: 2/3 of the job in 7 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 7 days.

\frac{1}{2} \div 7 = \frac{1}{14}

Jaewoo does 1/14 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

2/3 of the job spread over 7 days.

\frac{2}{3} \div 7 = \frac{2}{21}

Seohee does 2/21 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{14} + \frac{2}{21} = \frac{1}{6}

Together they do 1/6 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{6} = 6

They finish in 6 days.

Answer: 6 days

4 · Reviewdoes it hold up?

6 days of 1/6 of the job is exactly 1 whole job. And the pair is quicker than Seohee alone, who would need $10\frac{1}{2}$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 6 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 6 medium answer: 6 days

Working alone, Jaewoo takes $4$ days to do $\frac{1}{2}$ of a job, and Seohee takes $8$ days to do $\frac{1}{3}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 4 days and Seohee does 1/3 of it in 8 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 4 days.
Seohee: 1/3 of the job in 8 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 4 days.

\frac{1}{2} \div 4 = \frac{1}{8}

Jaewoo does 1/8 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/3 of the job spread over 8 days.

\frac{1}{3} \div 8 = \frac{1}{24}

Seohee does 1/24 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{8} + \frac{1}{24} = \frac{1}{6}

Together they do 1/6 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{6} = 6

They finish in 6 days.

Answer: 6 days

4 · Reviewdoes it hold up?

6 days of 1/6 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $8$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 6 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 7 medium answer: 7 days

Working alone, Jaewoo takes $4$ days to do $\frac{1}{2}$ of a job, and Seohee takes $8$ days to do $\frac{1}{7}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 4 days and Seohee does 1/7 of it in 8 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 4 days.
Seohee: 1/7 of the job in 8 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 4 days.

\frac{1}{2} \div 4 = \frac{1}{8}

Jaewoo does 1/8 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/7 of the job spread over 8 days.

\frac{1}{7} \div 8 = \frac{1}{56}

Seohee does 1/56 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{8} + \frac{1}{56} = \frac{1}{7}

Together they do 1/7 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{7} = 7

They finish in 7 days.

Answer: 7 days

4 · Reviewdoes it hold up?

7 days of 1/7 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $8$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 7 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 8 medium answer: 9 days

Working alone, Jaewoo takes $6$ days to do $\frac{1}{2}$ of a job, and Seohee takes $9$ days to do $\frac{1}{4}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 6 days and Seohee does 1/4 of it in 9 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 6 days.
Seohee: 1/4 of the job in 9 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 6 days.

\frac{1}{2} \div 6 = \frac{1}{12}

Jaewoo does 1/12 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/4 of the job spread over 9 days.

\frac{1}{4} \div 9 = \frac{1}{36}

Seohee does 1/36 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{12} + \frac{1}{36} = \frac{1}{9}

Together they do 1/9 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{9} = 9

They finish in 9 days.

Answer: 9 days

4 · Reviewdoes it hold up?

9 days of 1/9 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $12$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 9 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 9 hard answer: 8 days

Working alone, Jaewoo takes $5$ days to do $\frac{1}{2}$ of a job, and Seohee takes $10$ days to do $\frac{1}{4}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 5 days and Seohee does 1/4 of it in 10 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 5 days.
Seohee: 1/4 of the job in 10 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 5 days.

\frac{1}{2} \div 5 = \frac{1}{10}

Jaewoo does 1/10 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/4 of the job spread over 10 days.

\frac{1}{4} \div 10 = \frac{1}{40}

Seohee does 1/40 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{10} + \frac{1}{40} = \frac{1}{8}

Together they do 1/8 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{8} = 8

They finish in 8 days.

Answer: 8 days

4 · Reviewdoes it hold up?

8 days of 1/8 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $10$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 8 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 10 hard answer: 5 days

Working alone, Jaewoo takes $3$ days to do $\frac{1}{2}$ of a job, and Seohee takes $10$ days to do $\frac{1}{3}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 3 days and Seohee does 1/3 of it in 10 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 3 days.
Seohee: 1/3 of the job in 10 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 3 days.

\frac{1}{2} \div 3 = \frac{1}{6}

Jaewoo does 1/6 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/3 of the job spread over 10 days.

\frac{1}{3} \div 10 = \frac{1}{30}

Seohee does 1/30 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{6} + \frac{1}{30} = \frac{1}{5}

Together they do 1/5 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{5} = 5

They finish in 5 days.

Answer: 5 days

4 · Reviewdoes it hold up?

5 days of 1/5 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $6$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 5 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 11 hard answer: 7 days

Working alone, Jaewoo takes $5$ days to do $\frac{1}{2}$ of a job, and Seohee takes $10$ days to do $\frac{3}{7}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 5 days and Seohee does 3/7 of it in 10 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 5 days.
Seohee: 3/7 of the job in 10 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 5 days.

\frac{1}{2} \div 5 = \frac{1}{10}

Jaewoo does 1/10 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

3/7 of the job spread over 10 days.

\frac{3}{7} \div 10 = \frac{3}{70}

Seohee does 3/70 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{10} + \frac{3}{70} = \frac{1}{7}

Together they do 1/7 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{7} = 7

They finish in 7 days.

Answer: 7 days

4 · Reviewdoes it hold up?

7 days of 1/7 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $10$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 7 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.

Variant 12 hard answer: 10 days

Working alone, Jaewoo takes $6$ days to do $\frac{1}{2}$ of a job, and Seohee takes $10$ days to do $\frac{1}{6}$ of the same job. If they work on it together, how many days will it take to finish? (Each person does the same amount of work every day.)

Show solution

1 · Understandwhat's really being asked

Jaewoo does 1/2 of a job in 6 days and Seohee does 1/6 of it in 10 days. Working together, we want how many days the whole job takes.

Givens

Jaewoo: 1/2 of the job in 6 days.
Seohee: 1/6 of the job in 10 days.
Each of them works at the same pace every day.

Unknowns

The number of days the two of them need together.

Constraints

The size of the job is never given in any unit.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #9 Solve an Easier Related Problem#7 Identify Subproblems

The job has no unit, so give it one: call the whole thing 1. Then a day of work is a fraction of the job, two people's days add, and the finish time is 1 divided by what they do in a day.

3 · Execute4 carry out the plan

1Jaewoo's day

#7 Identify Subproblems 6.NS.A.1

1/2 of the job spread over 6 days.

\frac{1}{2} \div 6 = \frac{1}{12}

Jaewoo does 1/12 of the job a day.

2Seohee's day

#7 Identify Subproblems 6.NS.A.1

1/6 of the job spread over 10 days.

\frac{1}{6} \div 10 = \frac{1}{60}

Seohee does 1/60 of the job a day.

3Add the two days together

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

Working side by side, a day of the pair is a day of each.

\frac{1}{12} + \frac{1}{60} = \frac{1}{10}

Together they do 1/10 of the job a day.

4Turn the daily share into days

#8 Analyze the Units 7.RP.A.1

The whole job is 1, so divide 1 by what they finish in a day.

1 \div \frac{1}{10} = 10

They finish in 10 days.

Answer: 10 days

4 · Reviewdoes it hold up?

10 days of 1/10 of the job is exactly 1 whole job. And the pair is quicker than Jaewoo alone, who would need $12$ days -- adding a worker cannot slow the job down.

Another way: Scaling to a made-up job size works too: call the job 10 units, and the pair does 1 units a day.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing each worker's fraction of the job by their days.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Adding two daily shares of the same whole.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Turning a fractional rate back into a number of days.

💡Takeaway. When nobody tells you how big the job is, call it 1. Everything after that is a fraction of the same thing.