← 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 easy answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 29\dfrac{2}{9} of the whole fence, and each day Mrs. Diaz paints 19\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 29\frac{2}{9} of the fence and each of Mrs. Diaz's days adds 19\frac{1}{9}. We want the day on which the whole fence (99\frac{9}{9}) is finished.

Givens
  • Mr. Diaz paints 29\frac{2}{9} of the fence each of his days.
  • Mrs. Diaz paints 19\frac{1}{9} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 99\frac{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 99\frac{9}{9}.
2 · Planchoose the strategy

#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 (29\frac{2}{9} + 19\frac{1}{9} = 39\frac{3}{9}), so I look for that repeating pattern, list the running total day by day, and use the unit 19\frac{1}{9} of the fence to know when 99\frac{9}{9} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 29\frac{2}{9} + 19\frac{1}{9} = 39\frac{3}{9} of the fence. Each Mr+Mrs pair contributes the same 39\frac{3}{9}.
29+19=39\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.

2Total after 4 days (2 pairs)

#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 39\frac{3}{9} = 69\frac{6}{9}.
2×39=692\times\dfrac{3}{9}=\dfrac{6}{9}
Repeating a fixed fraction 2 times reaches 69\frac{6}{9}, still short of the whole 99\frac{9}{9}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 29\frac{2}{9}: 69\frac{6}{9} + 29\frac{2}{9} = 89\frac{8}{9}. Not finished yet (89\frac{8}{9} < 99\frac{9}{9}).
69+29=89\dfrac{6}{9}+\dfrac{2}{9}=\dfrac{8}{9}
Listing the next turn keeps the running total honest -- we are 19\frac{1}{9} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 19\frac{1}{9}: 89\frac{8}{9} + 19\frac{1}{9} = 99\frac{9}{9} = 1, the whole fence. The fence is finished on day 6.
89+19=99=1\dfrac{8}{9}+\dfrac{1}{9}=\dfrac{9}{9}=1
Counting in units of 19\frac{1}{9}, reaching 99\frac{9}{9} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 39\frac{3}{9} every 2 days, so a rough estimate is 99\frac{9}{9} divided by 39\frac{3}{9} = about 3.00 pairs of days. The day-by-day count lands exactly on 99\frac{9}{9} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 19\frac{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.

Standardsmin 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 $\frac{9}{9}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 9ths until you reach a whole!
Variant 2 easy answer: 8 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 112\dfrac{1}{12} of the whole fence, and each day Mrs. Diaz paints 212\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 112\frac{1}{12} of the fence and each of Mrs. Diaz's days adds 212\frac{2}{12}. We want the day on which the whole fence (1212\frac{12}{12}) is finished.

Givens
  • Mr. Diaz paints 112\frac{1}{12} of the fence each of his days.
  • Mrs. Diaz paints 212\frac{2}{12} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 1212\frac{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 1212\frac{12}{12}.
2 · Planchoose the strategy

#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 (112+212=312\frac{1}{12} + \frac{2}{12} = \frac{3}{12}), so I look for that repeating pattern, list the running total day by day, and use the unit 112\frac{1}{12} of the fence to know when 1212\frac{12}{12} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 112+212=312\frac{1}{12} + \frac{2}{12} = \frac{3}{12} of the fence. Each Mr+Mrs pair contributes the same 312\frac{3}{12}.
112+212=312\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.

2Total after 6 days (3 pairs)

#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 312\frac{3}{12} = 912\frac{9}{12}.
3×312=9123\times\dfrac{3}{12}=\dfrac{9}{12}
Repeating a fixed fraction 3 times reaches 912\frac{9}{12}, still short of the whole 1212\frac{12}{12}.

3Day 7 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 7 is Mr. Diaz's turn, adding 112\frac{1}{12}: 912+112=1012\frac{9}{12} + \frac{1}{12} = \frac{10}{12}. Not finished yet (1012<1212\frac{10}{12} < \frac{12}{12}).
912+112=1012\dfrac{9}{12}+\dfrac{1}{12}=\dfrac{10}{12}
Listing the next turn keeps the running total honest -- we are 212\frac{2}{12} short.

4Day 8 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 8 is Mrs. Diaz's turn, adding 212\frac{2}{12}: 1012+212=1212=1\frac{10}{12} + \frac{2}{12} = \frac{12}{12} = 1, the whole fence. The fence is finished on day 8.
1012+212=1212=1\dfrac{10}{12}+\dfrac{2}{12}=\dfrac{12}{12}=1
Counting in units of 112\frac{1}{12}, reaching 1212\frac{12}{12} means the whole job is done.
Answer: 8 days
4 · Reviewdoes it hold up?

Together they do 312\frac{3}{12} every 2 days, so a rough estimate is 1212\frac{12}{12} divided by 312\frac{3}{12} = about 4.00 pairs of days. The day-by-day count lands exactly on 1212\frac{12}{12} at day 8, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 112\frac{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.

Standardsmin 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 $\frac{12}{12}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 12ths until you reach a whole!
Variant 3 easy answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 312\dfrac{3}{12} of the whole fence, and each day Mrs. Diaz paints 112\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 312\frac{3}{12} of the fence and each of Mrs. Diaz's days adds 112\frac{1}{12}. We want the day on which the whole fence (1212\frac{12}{12}) is finished.

Givens
  • Mr. Diaz paints 312\frac{3}{12} of the fence each of his days.
  • Mrs. Diaz paints 112\frac{1}{12} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 1212\frac{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 1212\frac{12}{12}.
2 · Planchoose the strategy

#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 (312\frac{3}{12} + 112\frac{1}{12} = 412\frac{4}{12}), so I look for that repeating pattern, list the running total day by day, and use the unit 112\frac{1}{12} of the fence to know when 1212\frac{12}{12} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 312\frac{3}{12} + 112\frac{1}{12} = 412\frac{4}{12} of the fence. Each Mr+Mrs pair contributes the same 412\frac{4}{12}.
312+112=412\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.

2Total after 4 days (2 pairs)

#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 412\frac{4}{12} = 812\frac{8}{12}.
2×412=8122\times\dfrac{4}{12}=\dfrac{8}{12}
Repeating a fixed fraction 2 times reaches 812\frac{8}{12}, still short of the whole 1212\frac{12}{12}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 312\frac{3}{12}: 812\frac{8}{12} + 312\frac{3}{12} = 1112\frac{11}{12}. Not finished yet (1112\frac{11}{12} < 1212\frac{12}{12}).
812+312=1112\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}
Listing the next turn keeps the running total honest -- we are 112\frac{1}{12} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 112\frac{1}{12}: 1112\frac{11}{12} + 112\frac{1}{12} = 1212\frac{12}{12} = 1, the whole fence. The fence is finished on day 6.
1112+112=1212=1\dfrac{11}{12}+\dfrac{1}{12}=\dfrac{12}{12}=1
Counting in units of 112\frac{1}{12}, reaching 1212\frac{12}{12} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 412\frac{4}{12} every 2 days, so a rough estimate is 1212\frac{12}{12} divided by 412\frac{4}{12} = about 3.00 pairs of days. The day-by-day count lands exactly on 1212\frac{12}{12} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 112\frac{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.

Standardsmin 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 $\frac{12}{12}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 12ths until you reach a whole!
Variant 4 medium answer: 4 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 314\dfrac{3}{14} of the whole fence, and each day Mrs. Diaz paints 414\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 314\frac{3}{14} of the fence and each of Mrs. Diaz's days adds 414\frac{4}{14}. We want the day on which the whole fence (1414\frac{14}{14}) is finished.

Givens
  • Mr. Diaz paints 314\frac{3}{14} of the fence each of his days.
  • Mrs. Diaz paints 414\frac{4}{14} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 1414\frac{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 1414\frac{14}{14}.
2 · Planchoose the strategy

#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 (314\frac{3}{14} + 414\frac{4}{14} = 714\frac{7}{14}), so I look for that repeating pattern, list the running total day by day, and use the unit 114\frac{1}{14} of the fence to know when 1414\frac{14}{14} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 314\frac{3}{14} + 414\frac{4}{14} = 714\frac{7}{14} of the fence. Each Mr+Mrs pair contributes the same 714\frac{7}{14}.
314+414=714\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.

2Total after 2 days (1 pair)

#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 714\frac{7}{14} = 714\frac{7}{14}.
1×714=7141\times\dfrac{7}{14}=\dfrac{7}{14}
Repeating a fixed fraction 1 times reaches 714\frac{7}{14}, still short of the whole 1414\frac{14}{14}.

3Day 3 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 3 is Mr. Diaz's turn, adding 314\frac{3}{14}: 714\frac{7}{14} + 314\frac{3}{14} = 1014\frac{10}{14}. Not finished yet (1014\frac{10}{14} < 1414\frac{14}{14}).
714+314=1014\dfrac{7}{14}+\dfrac{3}{14}=\dfrac{10}{14}
Listing the next turn keeps the running total honest -- we are 414\frac{4}{14} short.

4Day 4 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 4 is Mrs. Diaz's turn, adding 414\frac{4}{14}: 1014\frac{10}{14} + 414\frac{4}{14} = 1414\frac{14}{14} = 1, the whole fence. The fence is finished on day 4.
1014+414=1414=1\dfrac{10}{14}+\dfrac{4}{14}=\dfrac{14}{14}=1
Counting in units of 114\frac{1}{14}, reaching 1414\frac{14}{14} means the whole job is done.
Answer: 4 days
4 · Reviewdoes it hold up?

Together they do 714\frac{7}{14} every 2 days, so a rough estimate is 1414\frac{14}{14} divided by 714\frac{7}{14} = about 2.00 pairs of days. The day-by-day count lands exactly on 1414\frac{14}{14} at day 4, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 114\frac{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.

Standardsmin 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 $\frac{14}{14}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 14ths until you reach a whole!
Variant 5 medium answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 415\dfrac{4}{15} of the whole fence, and each day Mrs. Diaz paints 115\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 415\frac{4}{15} of the fence and each of Mrs. Diaz's days adds 115\frac{1}{15}. We want the day on which the whole fence (1515\frac{15}{15}) is finished.

Givens
  • Mr. Diaz paints 415\frac{4}{15} of the fence each of his days.
  • Mrs. Diaz paints 115\frac{1}{15} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 1515\frac{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 1515\frac{15}{15}.
2 · Planchoose the strategy

#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 (415\frac{4}{15} + 115\frac{1}{15} = 515\frac{5}{15}), so I look for that repeating pattern, list the running total day by day, and use the unit 115\frac{1}{15} of the fence to know when 1515\frac{15}{15} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 415\frac{4}{15} + 115\frac{1}{15} = 515\frac{5}{15} of the fence. Each Mr+Mrs pair contributes the same 515\frac{5}{15}.
415+115=515\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.

2Total after 4 days (2 pairs)

#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 515\frac{5}{15} = 1015\frac{10}{15}.
2×515=10152\times\dfrac{5}{15}=\dfrac{10}{15}
Repeating a fixed fraction 2 times reaches 1015\frac{10}{15}, still short of the whole 1515\frac{15}{15}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 415\frac{4}{15}: 1015\frac{10}{15} + 415\frac{4}{15} = 1415\frac{14}{15}. Not finished yet (1415\frac{14}{15} < 1515\frac{15}{15}).
1015+415=1415\dfrac{10}{15}+\dfrac{4}{15}=\dfrac{14}{15}
Listing the next turn keeps the running total honest -- we are 115\frac{1}{15} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 115\frac{1}{15}: 1415\frac{14}{15} + 115\frac{1}{15} = 1515\frac{15}{15} = 1, the whole fence. The fence is finished on day 6.
1415+115=1515=1\dfrac{14}{15}+\dfrac{1}{15}=\dfrac{15}{15}=1
Counting in units of 115\frac{1}{15}, reaching 1515\frac{15}{15} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 515\frac{5}{15} every 2 days, so a rough estimate is 1515\frac{15}{15} divided by 515\frac{5}{15} = about 3.00 pairs of days. The day-by-day count lands exactly on 1515\frac{15}{15} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 115\frac{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.

Standardsmin 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 $\frac{15}{15}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 15ths until you reach a whole!
Variant 6 medium answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 215\dfrac{2}{15} of the whole fence, and each day Mrs. Diaz paints 315\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 215\frac{2}{15} of the fence and each of Mrs. Diaz's days adds 315\frac{3}{15}. We want the day on which the whole fence (1515\frac{15}{15}) is finished.

Givens
  • Mr. Diaz paints 215\frac{2}{15} of the fence each of his days.
  • Mrs. Diaz paints 315\frac{3}{15} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 1515\frac{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 1515\frac{15}{15}.
2 · Planchoose the strategy

#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 (215\frac{2}{15} + 315\frac{3}{15} = 515\frac{5}{15}), so I look for that repeating pattern, list the running total day by day, and use the unit 115\frac{1}{15} of the fence to know when 1515\frac{15}{15} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 215\frac{2}{15} + 315\frac{3}{15} = 515\frac{5}{15} of the fence. Each Mr+Mrs pair contributes the same 515\frac{5}{15}.
215+315=515\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.

2Total after 4 days (2 pairs)

#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 515\frac{5}{15} = 1015\frac{10}{15}.
2×515=10152\times\dfrac{5}{15}=\dfrac{10}{15}
Repeating a fixed fraction 2 times reaches 1015\frac{10}{15}, still short of the whole 1515\frac{15}{15}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 215\frac{2}{15}: 1015\frac{10}{15} + 215\frac{2}{15} = 1215\frac{12}{15}. Not finished yet (1215\frac{12}{15} < 1515\frac{15}{15}).
1015+215=1215\dfrac{10}{15}+\dfrac{2}{15}=\dfrac{12}{15}
Listing the next turn keeps the running total honest -- we are 315\frac{3}{15} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 315\frac{3}{15}: 1215\frac{12}{15} + 315\frac{3}{15} = 1515\frac{15}{15} = 1, the whole fence. The fence is finished on day 6.
1215+315=1515=1\dfrac{12}{15}+\dfrac{3}{15}=\dfrac{15}{15}=1
Counting in units of 115\frac{1}{15}, reaching 1515\frac{15}{15} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 515\frac{5}{15} every 2 days, so a rough estimate is 1515\frac{15}{15} divided by 515\frac{5}{15} = about 3.00 pairs of days. The day-by-day count lands exactly on 1515\frac{15}{15} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 115\frac{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.

Standardsmin 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 $\frac{15}{15}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 15ths until you reach a whole!
Variant 7 medium answer: 8 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 320\dfrac{3}{20} of the whole fence, and each day Mrs. Diaz paints 220\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 320\frac{3}{20} of the fence and each of Mrs. Diaz's days adds 220\frac{2}{20}. We want the day on which the whole fence (2020\frac{20}{20}) is finished.

Givens
  • Mr. Diaz paints 320\frac{3}{20} of the fence each of his days.
  • Mrs. Diaz paints 220\frac{2}{20} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 2020\frac{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 2020\frac{20}{20}.
2 · Planchoose the strategy

#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 (320\frac{3}{20} + 220\frac{2}{20} = 520\frac{5}{20}), so I look for that repeating pattern, list the running total day by day, and use the unit 120\frac{1}{20} of the fence to know when 2020\frac{20}{20} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 320\frac{3}{20} + 220\frac{2}{20} = 520\frac{5}{20} of the fence. Each Mr+Mrs pair contributes the same 520\frac{5}{20}.
320+220=520\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.

2Total after 6 days (3 pairs)

#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 520\frac{5}{20} = 1520\frac{15}{20}.
3×520=15203\times\dfrac{5}{20}=\dfrac{15}{20}
Repeating a fixed fraction 3 times reaches 1520\frac{15}{20}, still short of the whole 2020\frac{20}{20}.

3Day 7 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 7 is Mr. Diaz's turn, adding 320\frac{3}{20}: 1520\frac{15}{20} + 320\frac{3}{20} = 1820\frac{18}{20}. Not finished yet (1820\frac{18}{20} < 2020\frac{20}{20}).
1520+320=1820\dfrac{15}{20}+\dfrac{3}{20}=\dfrac{18}{20}
Listing the next turn keeps the running total honest -- we are 220\frac{2}{20} short.

4Day 8 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 8 is Mrs. Diaz's turn, adding 220\frac{2}{20}: 1820\frac{18}{20} + 220\frac{2}{20} = 2020\frac{20}{20} = 1, the whole fence. The fence is finished on day 8.
1820+220=2020=1\dfrac{18}{20}+\dfrac{2}{20}=\dfrac{20}{20}=1
Counting in units of 120\frac{1}{20}, reaching 2020\frac{20}{20} means the whole job is done.
Answer: 8 days
4 · Reviewdoes it hold up?

Together they do 520\frac{5}{20} every 2 days, so a rough estimate is 2020\frac{20}{20} divided by 520\frac{5}{20} = about 4.00 pairs of days. The day-by-day count lands exactly on 2020\frac{20}{20} at day 8, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 120\frac{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.

Standardsmin 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 $\frac{20}{20}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 20ths until you reach a whole!
Variant 8 hard answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 421\dfrac{4}{21} of the whole fence, and each day Mrs. Diaz paints 321\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 421\frac{4}{21} of the fence and each of Mrs. Diaz's days adds 321\frac{3}{21}. We want the day on which the whole fence (2121\frac{21}{21}) is finished.

Givens
  • Mr. Diaz paints 421\frac{4}{21} of the fence each of his days.
  • Mrs. Diaz paints 321\frac{3}{21} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 2121\frac{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 2121\frac{21}{21}.
2 · Planchoose the strategy

#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 (421\frac{4}{21} + 321\frac{3}{21} = 721\frac{7}{21}), so I look for that repeating pattern, list the running total day by day, and use the unit 121\frac{1}{21} of the fence to know when 2121\frac{21}{21} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 421\frac{4}{21} + 321\frac{3}{21} = 721\frac{7}{21} of the fence. Each Mr+Mrs pair contributes the same 721\frac{7}{21}.
421+321=721\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.

2Total after 4 days (2 pairs)

#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 721\frac{7}{21} = 1421\frac{14}{21}.
2×721=14212\times\dfrac{7}{21}=\dfrac{14}{21}
Repeating a fixed fraction 2 times reaches 1421\frac{14}{21}, still short of the whole 2121\frac{21}{21}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 421\frac{4}{21}: 1421\frac{14}{21} + 421\frac{4}{21} = 1821\frac{18}{21}. Not finished yet (1821\frac{18}{21} < 2121\frac{21}{21}).
1421+421=1821\dfrac{14}{21}+\dfrac{4}{21}=\dfrac{18}{21}
Listing the next turn keeps the running total honest -- we are 321\frac{3}{21} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 321\frac{3}{21}: 1821\frac{18}{21} + 321\frac{3}{21} = 2121\frac{21}{21} = 1, the whole fence. The fence is finished on day 6.
1821+321=2121=1\dfrac{18}{21}+\dfrac{3}{21}=\dfrac{21}{21}=1
Counting in units of 121\frac{1}{21}, reaching 2121\frac{21}{21} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 721\frac{7}{21} every 2 days, so a rough estimate is 2121\frac{21}{21} divided by 721\frac{7}{21} = about 3.00 pairs of days. The day-by-day count lands exactly on 2121\frac{21}{21} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 121\frac{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.

Standardsmin 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 $\frac{21}{21}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!
Variant 9 hard answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 221\dfrac{2}{21} of the whole fence, and each day Mrs. Diaz paints 521\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 221\frac{2}{21} of the fence and each of Mrs. Diaz's days adds 521\frac{5}{21}. We want the day on which the whole fence (2121\frac{21}{21}) is finished.

Givens
  • Mr. Diaz paints 221\frac{2}{21} of the fence each of his days.
  • Mrs. Diaz paints 521\frac{5}{21} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 2121\frac{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 2121\frac{21}{21}.
2 · Planchoose the strategy

#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 (221\frac{2}{21} + 521\frac{5}{21} = 721\frac{7}{21}), so I look for that repeating pattern, list the running total day by day, and use the unit 121\frac{1}{21} of the fence to know when 2121\frac{21}{21} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 221\frac{2}{21} + 521\frac{5}{21} = 721\frac{7}{21} of the fence. Each Mr+Mrs pair contributes the same 721\frac{7}{21}.
221+521=721\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.

2Total after 4 days (2 pairs)

#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 721\frac{7}{21} = 1421\frac{14}{21}.
2×721=14212\times\dfrac{7}{21}=\dfrac{14}{21}
Repeating a fixed fraction 2 times reaches 1421\frac{14}{21}, still short of the whole 2121\frac{21}{21}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 221\frac{2}{21}: 1421\frac{14}{21} + 221\frac{2}{21} = 1621\frac{16}{21}. Not finished yet (1621\frac{16}{21} < 2121\frac{21}{21}).
1421+221=1621\dfrac{14}{21}+\dfrac{2}{21}=\dfrac{16}{21}
Listing the next turn keeps the running total honest -- we are 521\frac{5}{21} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 521\frac{5}{21}: 1621\frac{16}{21} + 521\frac{5}{21} = 2121\frac{21}{21} = 1, the whole fence. The fence is finished on day 6.
1621+521=2121=1\dfrac{16}{21}+\dfrac{5}{21}=\dfrac{21}{21}=1
Counting in units of 121\frac{1}{21}, reaching 2121\frac{21}{21} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 721\frac{7}{21} every 2 days, so a rough estimate is 2121\frac{21}{21} divided by 721\frac{7}{21} = about 3.00 pairs of days. The day-by-day count lands exactly on 2121\frac{21}{21} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 121\frac{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.

Standardsmin 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 $\frac{21}{21}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!
Variant 10 hard answer: 6 days

Mr. Diaz and Mrs. Diaz are painting a fence together. Each day Mr. Diaz paints 521\dfrac{5}{21} of the whole fence, and each day Mrs. Diaz paints 221\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
1 · Understandwhat's really being asked

Two people paint a fence by taking turns one day at a time, Mr. Diaz first. Each of Mr. Diaz's days adds 521\frac{5}{21} of the fence and each of Mrs. Diaz's days adds 221\frac{2}{21}. We want the day on which the whole fence (2121\frac{21}{21}) is finished.

Givens
  • Mr. Diaz paints 521\frac{5}{21} of the fence each of his days.
  • Mrs. Diaz paints 221\frac{2}{21} of the fence each of her days.
  • Mr. Diaz starts, then they alternate one day each.
  • The whole fence is 1 = 2121\frac{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 2121\frac{21}{21}.
2 · Planchoose the strategy

#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 (521\frac{5}{21} + 221\frac{2}{21} = 721\frac{7}{21}), so I look for that repeating pattern, list the running total day by day, and use the unit 121\frac{1}{21} of the fence to know when 2121\frac{21}{21} is reached.

3 · Execute4 carry out the plan

1Amount in one full pair of days

#5 Look for a Pattern 4.NF.B.3
One day of Mr. Diaz plus one day of Mrs. Diaz adds 521\frac{5}{21} + 221\frac{2}{21} = 721\frac{7}{21} of the fence. Each Mr+Mrs pair contributes the same 721\frac{7}{21}.
521+221=721\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.

2Total after 4 days (2 pairs)

#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 721\frac{7}{21} = 1421\frac{14}{21}.
2×721=14212\times\dfrac{7}{21}=\dfrac{14}{21}
Repeating a fixed fraction 2 times reaches 1421\frac{14}{21}, still short of the whole 2121\frac{21}{21}.

3Day 5 (Mr. Diaz)

#2 Make a Systematic List 4.NF.B.3
Day 5 is Mr. Diaz's turn, adding 521\frac{5}{21}: 1421\frac{14}{21} + 521\frac{5}{21} = 1921\frac{19}{21}. Not finished yet (1921\frac{19}{21} < 2121\frac{21}{21}).
1421+521=1921\dfrac{14}{21}+\dfrac{5}{21}=\dfrac{19}{21}
Listing the next turn keeps the running total honest -- we are 221\frac{2}{21} short.

4Day 6 (Mrs. Diaz) finishes

#8 Analyze the Units 4.NF.B.3
Day 6 is Mrs. Diaz's turn, adding 221\frac{2}{21}: 1921\frac{19}{21} + 221\frac{2}{21} = 2121\frac{21}{21} = 1, the whole fence. The fence is finished on day 6.
1921+221=2121=1\dfrac{19}{21}+\dfrac{2}{21}=\dfrac{21}{21}=1
Counting in units of 121\frac{1}{21}, reaching 2121\frac{21}{21} means the whole job is done.
Answer: 6 days
4 · Reviewdoes it hold up?

Together they do 721\frac{7}{21} every 2 days, so a rough estimate is 2121\frac{21}{21} divided by 721\frac{7}{21} = about 3.00 pairs of days. The day-by-day count lands exactly on 2121\frac{21}{21} at day 6, so the magnitude and units (fractions of one fence) check out.

Another way: Track the running total in units of 121\frac{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.

Standardsmin 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 $\frac{21}{21}$.
💡Takeaway. This only needs Grade 4 same-denominator fraction adding -- count in 21ths until you reach a whole!