← 4-2 · Add per-day work fractions to find combined output · Track a Quantity Through Changes

Add per-day work fractions to find combined output · 10 practice problems

4.NF.B.3

Generated variants — 10

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

Variant 1 answer: 8 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{1}{12}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{2}{12}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 1/12 of the fence and each of Mrs. Diaz's days adds 2/12. We want the day on which the whole fence (12/12) is finished.

Givens

Mr. Diaz paints 1/12 of the fence each of his days.
Mrs. Diaz paints 2/12 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 12/12.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 12/12.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (1/12 + 2/12 = 3/12), so I look for that repeating pattern, list the running total day by day, and use the unit 1/12 of the fence to know when 12/12 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 1/12 + 2/12 = 3/12 of the fence. Each Mr+Mrs pair contributes the same 3/12.

\dfrac{1}{12}+\dfrac{2}{12}=\dfrac{3}{12}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 6 days there have been 3 complete Mr+Mrs pairs, so the painted amount is 3 x 3/12 = 9/12.

3\times\dfrac{3}{12}=\dfrac{9}{12}

Repeating a fixed fraction 3 times reaches 9/12, still short of the whole 12/12.

#2 Make a Systematic List 4.NF.B.3

Day 7 is Mr. Diaz's turn, adding 1/12: 9/12 + 1/12 = 10/12. Not finished yet (10/12 < 12/12).

\dfrac{9}{12}+\dfrac{1}{12}=\dfrac{10}{12}

Listing the next turn keeps the running total honest -- we are 2/12 short.

#8 Analyze the Units 4.NF.B.3

Day 8 is Mrs. Diaz's turn, adding 2/12: 10/12 + 2/12 = 12/12 = 1, the whole fence. The fence is finished on day 8.

\dfrac{10}{12}+\dfrac{2}{12}=\dfrac{12}{12}=1

Counting in units of 1/12, reaching 12/12 means the whole job is done.

Answer: 8 days

Review

Together they do 3/12 every 2 days, so a rough estimate is 12/12 divided by 3/12 = about 4.00 pairs of days. The day-by-day count lands exactly on 12/12 at day 8, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/12 (tool 8): keep adding 1 then 2 to the numerator until it first reaches 12; the number of additions is the number of days, 8.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 12 and tracking the running total up to 12/12.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 12ths until you reach a whole!

Variant 2 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{4}{21}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{3}{21}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 4/21 of the fence and each of Mrs. Diaz's days adds 3/21. We want the day on which the whole fence (21/21) is finished.

Givens

Mr. Diaz paints 4/21 of the fence each of his days.
Mrs. Diaz paints 3/21 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 21/21.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 21/21.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (4/21 + 3/21 = 7/21), so I look for that repeating pattern, list the running total day by day, and use the unit 1/21 of the fence to know when 21/21 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 4/21 + 3/21 = 7/21 of the fence. Each Mr+Mrs pair contributes the same 7/21.

\dfrac{4}{21}+\dfrac{3}{21}=\dfrac{7}{21}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 7/21 = 14/21.

2\times\dfrac{7}{21}=\dfrac{14}{21}

Repeating a fixed fraction 2 times reaches 14/21, still short of the whole 21/21.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 4/21: 14/21 + 4/21 = 18/21. Not finished yet (18/21 < 21/21).

\dfrac{14}{21}+\dfrac{4}{21}=\dfrac{18}{21}

Listing the next turn keeps the running total honest -- we are 3/21 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 3/21: 18/21 + 3/21 = 21/21 = 1, the whole fence. The fence is finished on day 6.

\dfrac{18}{21}+\dfrac{3}{21}=\dfrac{21}{21}=1

Counting in units of 1/21, reaching 21/21 means the whole job is done.

Answer: 6 days

Review

Together they do 7/21 every 2 days, so a rough estimate is 21/21 divided by 7/21 = about 3.00 pairs of days. The day-by-day count lands exactly on 21/21 at day 6, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/21 (tool 8): keep adding 4 then 3 to the numerator until it first reaches 21; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 21 and tracking the running total up to 21/21.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!

Variant 3 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{2}{9}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{1}{9}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 2/9 of the fence and each of Mrs. Diaz's days adds 1/9. We want the day on which the whole fence (9/9) is finished.

Givens

Mr. Diaz paints 2/9 of the fence each of his days.
Mrs. Diaz paints 1/9 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 9/9.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 9/9.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (2/9 + 1/9 = 3/9), so I look for that repeating pattern, list the running total day by day, and use the unit 1/9 of the fence to know when 9/9 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 2/9 + 1/9 = 3/9 of the fence. Each Mr+Mrs pair contributes the same 3/9.

\dfrac{2}{9}+\dfrac{1}{9}=\dfrac{3}{9}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 3/9 = 6/9.

2\times\dfrac{3}{9}=\dfrac{6}{9}

Repeating a fixed fraction 2 times reaches 6/9, still short of the whole 9/9.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 2/9: 6/9 + 2/9 = 8/9. Not finished yet (8/9 < 9/9).

\dfrac{6}{9}+\dfrac{2}{9}=\dfrac{8}{9}

Listing the next turn keeps the running total honest -- we are 1/9 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 1/9: 8/9 + 1/9 = 9/9 = 1, the whole fence. The fence is finished on day 6.

\dfrac{8}{9}+\dfrac{1}{9}=\dfrac{9}{9}=1

Counting in units of 1/9, reaching 9/9 means the whole job is done.

Answer: 6 days

Review

Together they do 3/9 every 2 days, so a rough estimate is 9/9 divided by 3/9 = about 3.00 pairs of days. The day-by-day count lands exactly on 9/9 at day 6, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/9 (tool 8): keep adding 2 then 1 to the numerator until it first reaches 9; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 9 and tracking the running total up to 9/9.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 9ths until you reach a whole!

Variant 4 answer: 8 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{3}{20}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{2}{20}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 3/20 of the fence and each of Mrs. Diaz's days adds 2/20. We want the day on which the whole fence (20/20) is finished.

Givens

Mr. Diaz paints 3/20 of the fence each of his days.
Mrs. Diaz paints 2/20 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 20/20.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 20/20.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (3/20 + 2/20 = 5/20), so I look for that repeating pattern, list the running total day by day, and use the unit 1/20 of the fence to know when 20/20 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 3/20 + 2/20 = 5/20 of the fence. Each Mr+Mrs pair contributes the same 5/20.

\dfrac{3}{20}+\dfrac{2}{20}=\dfrac{5}{20}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 6 days there have been 3 complete Mr+Mrs pairs, so the painted amount is 3 x 5/20 = 15/20.

3\times\dfrac{5}{20}=\dfrac{15}{20}

Repeating a fixed fraction 3 times reaches 15/20, still short of the whole 20/20.

#2 Make a Systematic List 4.NF.B.3

Day 7 is Mr. Diaz's turn, adding 3/20: 15/20 + 3/20 = 18/20. Not finished yet (18/20 < 20/20).

\dfrac{15}{20}+\dfrac{3}{20}=\dfrac{18}{20}

Listing the next turn keeps the running total honest -- we are 2/20 short.

#8 Analyze the Units 4.NF.B.3

Day 8 is Mrs. Diaz's turn, adding 2/20: 18/20 + 2/20 = 20/20 = 1, the whole fence. The fence is finished on day 8.

\dfrac{18}{20}+\dfrac{2}{20}=\dfrac{20}{20}=1

Counting in units of 1/20, reaching 20/20 means the whole job is done.

Answer: 8 days

Review

Together they do 5/20 every 2 days, so a rough estimate is 20/20 divided by 5/20 = about 4.00 pairs of days. The day-by-day count lands exactly on 20/20 at day 8, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/20 (tool 8): keep adding 3 then 2 to the numerator until it first reaches 20; the number of additions is the number of days, 8.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 20 and tracking the running total up to 20/20.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 20ths until you reach a whole!

Variant 5 answer: 4 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{3}{14}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{4}{14}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 3/14 of the fence and each of Mrs. Diaz's days adds 4/14. We want the day on which the whole fence (14/14) is finished.

Givens

Mr. Diaz paints 3/14 of the fence each of his days.
Mrs. Diaz paints 4/14 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 14/14.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 14/14.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (3/14 + 4/14 = 7/14), so I look for that repeating pattern, list the running total day by day, and use the unit 1/14 of the fence to know when 14/14 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 3/14 + 4/14 = 7/14 of the fence. Each Mr+Mrs pair contributes the same 7/14.

\dfrac{3}{14}+\dfrac{4}{14}=\dfrac{7}{14}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 2 days there have been 1 complete Mr+Mrs pairs, so the painted amount is 1 x 7/14 = 7/14.

1\times\dfrac{7}{14}=\dfrac{7}{14}

Repeating a fixed fraction 1 times reaches 7/14, still short of the whole 14/14.

#2 Make a Systematic List 4.NF.B.3

Day 3 is Mr. Diaz's turn, adding 3/14: 7/14 + 3/14 = 10/14. Not finished yet (10/14 < 14/14).

\dfrac{7}{14}+\dfrac{3}{14}=\dfrac{10}{14}

Listing the next turn keeps the running total honest -- we are 4/14 short.

#8 Analyze the Units 4.NF.B.3

Day 4 is Mrs. Diaz's turn, adding 4/14: 10/14 + 4/14 = 14/14 = 1, the whole fence. The fence is finished on day 4.

\dfrac{10}{14}+\dfrac{4}{14}=\dfrac{14}{14}=1

Counting in units of 1/14, reaching 14/14 means the whole job is done.

Answer: 4 days

Review

Together they do 7/14 every 2 days, so a rough estimate is 14/14 divided by 7/14 = about 2.00 pairs of days. The day-by-day count lands exactly on 14/14 at day 4, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/14 (tool 8): keep adding 3 then 4 to the numerator until it first reaches 14; the number of additions is the number of days, 4.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 14 and tracking the running total up to 14/14.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 14ths until you reach a whole!

Variant 6 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{2}{21}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{5}{21}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 2/21 of the fence and each of Mrs. Diaz's days adds 5/21. We want the day on which the whole fence (21/21) is finished.

Givens

Mr. Diaz paints 2/21 of the fence each of his days.
Mrs. Diaz paints 5/21 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 21/21.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 21/21.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (2/21 + 5/21 = 7/21), so I look for that repeating pattern, list the running total day by day, and use the unit 1/21 of the fence to know when 21/21 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 2/21 + 5/21 = 7/21 of the fence. Each Mr+Mrs pair contributes the same 7/21.

\dfrac{2}{21}+\dfrac{5}{21}=\dfrac{7}{21}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 7/21 = 14/21.

2\times\dfrac{7}{21}=\dfrac{14}{21}

Repeating a fixed fraction 2 times reaches 14/21, still short of the whole 21/21.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 2/21: 14/21 + 2/21 = 16/21. Not finished yet (16/21 < 21/21).

\dfrac{14}{21}+\dfrac{2}{21}=\dfrac{16}{21}

Listing the next turn keeps the running total honest -- we are 5/21 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 5/21: 16/21 + 5/21 = 21/21 = 1, the whole fence. The fence is finished on day 6.

\dfrac{16}{21}+\dfrac{5}{21}=\dfrac{21}{21}=1

Counting in units of 1/21, reaching 21/21 means the whole job is done.

Answer: 6 days

Review

Track the running total in units of 1/21 (tool 8): keep adding 2 then 5 to the numerator until it first reaches 21; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 21 and tracking the running total up to 21/21.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!

Variant 7 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{4}{15}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{1}{15}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 4/15 of the fence and each of Mrs. Diaz's days adds 1/15. We want the day on which the whole fence (15/15) is finished.

Givens

Mr. Diaz paints 4/15 of the fence each of his days.
Mrs. Diaz paints 1/15 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 15/15.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 15/15.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (4/15 + 1/15 = 5/15), so I look for that repeating pattern, list the running total day by day, and use the unit 1/15 of the fence to know when 15/15 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 4/15 + 1/15 = 5/15 of the fence. Each Mr+Mrs pair contributes the same 5/15.

\dfrac{4}{15}+\dfrac{1}{15}=\dfrac{5}{15}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 5/15 = 10/15.

2\times\dfrac{5}{15}=\dfrac{10}{15}

Repeating a fixed fraction 2 times reaches 10/15, still short of the whole 15/15.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 4/15: 10/15 + 4/15 = 14/15. Not finished yet (14/15 < 15/15).

\dfrac{10}{15}+\dfrac{4}{15}=\dfrac{14}{15}

Listing the next turn keeps the running total honest -- we are 1/15 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 1/15: 14/15 + 1/15 = 15/15 = 1, the whole fence. The fence is finished on day 6.

\dfrac{14}{15}+\dfrac{1}{15}=\dfrac{15}{15}=1

Counting in units of 1/15, reaching 15/15 means the whole job is done.

Answer: 6 days

Review

Together they do 5/15 every 2 days, so a rough estimate is 15/15 divided by 5/15 = about 3.00 pairs of days. The day-by-day count lands exactly on 15/15 at day 6, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/15 (tool 8): keep adding 4 then 1 to the numerator until it first reaches 15; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 15 and tracking the running total up to 15/15.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 15ths until you reach a whole!

Variant 8 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{2}{15}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{3}{15}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 2/15 of the fence and each of Mrs. Diaz's days adds 3/15. We want the day on which the whole fence (15/15) is finished.

Givens

Mr. Diaz paints 2/15 of the fence each of his days.
Mrs. Diaz paints 3/15 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 15/15.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 15/15.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (2/15 + 3/15 = 5/15), so I look for that repeating pattern, list the running total day by day, and use the unit 1/15 of the fence to know when 15/15 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 2/15 + 3/15 = 5/15 of the fence. Each Mr+Mrs pair contributes the same 5/15.

\dfrac{2}{15}+\dfrac{3}{15}=\dfrac{5}{15}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 5/15 = 10/15.

2\times\dfrac{5}{15}=\dfrac{10}{15}

Repeating a fixed fraction 2 times reaches 10/15, still short of the whole 15/15.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 2/15: 10/15 + 2/15 = 12/15. Not finished yet (12/15 < 15/15).

\dfrac{10}{15}+\dfrac{2}{15}=\dfrac{12}{15}

Listing the next turn keeps the running total honest -- we are 3/15 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 3/15: 12/15 + 3/15 = 15/15 = 1, the whole fence. The fence is finished on day 6.

\dfrac{12}{15}+\dfrac{3}{15}=\dfrac{15}{15}=1

Counting in units of 1/15, reaching 15/15 means the whole job is done.

Answer: 6 days

Review

Track the running total in units of 1/15 (tool 8): keep adding 2 then 3 to the numerator until it first reaches 15; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 15 and tracking the running total up to 15/15.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 15ths until you reach a whole!

Variant 9 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{5}{21}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{2}{21}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 5/21 of the fence and each of Mrs. Diaz's days adds 2/21. We want the day on which the whole fence (21/21) is finished.

Givens

Mr. Diaz paints 5/21 of the fence each of his days.
Mrs. Diaz paints 2/21 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 21/21.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 21/21.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (5/21 + 2/21 = 7/21), so I look for that repeating pattern, list the running total day by day, and use the unit 1/21 of the fence to know when 21/21 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 5/21 + 2/21 = 7/21 of the fence. Each Mr+Mrs pair contributes the same 7/21.

\dfrac{5}{21}+\dfrac{2}{21}=\dfrac{7}{21}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 7/21 = 14/21.

2\times\dfrac{7}{21}=\dfrac{14}{21}

Repeating a fixed fraction 2 times reaches 14/21, still short of the whole 21/21.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 5/21: 14/21 + 5/21 = 19/21. Not finished yet (19/21 < 21/21).

\dfrac{14}{21}+\dfrac{5}{21}=\dfrac{19}{21}

Listing the next turn keeps the running total honest -- we are 2/21 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 2/21: 19/21 + 2/21 = 21/21 = 1, the whole fence. The fence is finished on day 6.

\dfrac{19}{21}+\dfrac{2}{21}=\dfrac{21}{21}=1

Counting in units of 1/21, reaching 21/21 means the whole job is done.

Answer: 6 days

Review

Track the running total in units of 1/21 (tool 8): keep adding 5 then 2 to the numerator until it first reaches 21; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 21 and tracking the running total up to 21/21.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!

Variant 10 answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints $\dfrac{3}{12}$ of the whole fence, and each day Mrs. Diaz paints $\dfrac{1}{12}$ of the whole fence. Mr. Diaz starts first, and then they take turns painting one day each, alternating between them. In how many days can they finish painting the fence?

Show solution

Understand

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 3/12 of the fence and each of Mrs. Diaz's days adds 1/12. We want the day on which the whole fence (12/12) is finished.

Givens

Mr. Diaz paints 3/12 of the fence each of his days.
Mrs. Diaz paints 1/12 of the fence each of her days.
Mr. Diaz starts, then they alternate one day each.
The whole fence is 1 = 12/12.

Unknowns

The number of days needed to finish the fence.

Constraints

Days alternate strictly: Mr, Mrs, Mr, Mrs, ...
Painting stops once the running total reaches 12/12.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#8 Analyze the Units

Every Mr+Mrs pair of days adds the same amount (3/12 + 1/12 = 4/12), so I look for that repeating pattern, list the running total day by day, and use the unit 1/12 of the fence to know when 12/12 is reached.

Execute

#5 Look for a Pattern 4.NF.B.3

One day of Mr. Diaz plus one day of Mrs. Diaz adds 3/12 + 1/12 = 4/12 of the fence. Each Mr+Mrs pair contributes the same 4/12.

\dfrac{3}{12}+\dfrac{1}{12}=\dfrac{4}{12}

Adding fractions with the same denominator just adds the numerators -- a Grade 4 same-denominator add.

#5 Look for a Pattern 4.NF.B.3

After 4 days there have been 2 complete Mr+Mrs pairs, so the painted amount is 2 x 4/12 = 8/12.

2\times\dfrac{4}{12}=\dfrac{8}{12}

Repeating a fixed fraction 2 times reaches 8/12, still short of the whole 12/12.

#2 Make a Systematic List 4.NF.B.3

Day 5 is Mr. Diaz's turn, adding 3/12: 8/12 + 3/12 = 11/12. Not finished yet (11/12 < 12/12).

\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}

Listing the next turn keeps the running total honest -- we are 1/12 short.

#8 Analyze the Units 4.NF.B.3

Day 6 is Mrs. Diaz's turn, adding 1/12: 11/12 + 1/12 = 12/12 = 1, the whole fence. The fence is finished on day 6.

\dfrac{11}{12}+\dfrac{1}{12}=\dfrac{12}{12}=1

Counting in units of 1/12, reaching 12/12 means the whole job is done.

Answer: 6 days

Review

Together they do 4/12 every 2 days, so a rough estimate is 12/12 divided by 4/12 = about 3.00 pairs of days. The day-by-day count lands exactly on 12/12 at day 6, so the magnitude and units (fractions of one fence) check out.

Track the running total in units of 1/12 (tool 8): keep adding 3 then 1 to the numerator until it first reaches 12; the number of additions is the number of days, 6.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the per-day fractions with denominator 12 and tracking the running total up to 12/12.

💡 This only needs Grade 4 same-denominator fraction adding -- count in 12ths until you reach a whole!