Add per-day work fractions to find combined output

4.NF.B.3 · take · grade 4

Archetype: Track a Quantity Through Changes · step in a 7-type progression

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 three complete Mr+Mrs pairs, so the painted amount is 3 × 5/20 = 15/20.

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

Repeating a fixed fraction is just adding it three times — 15/20 is 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 ÷ 5/20 = 4 pairs = 8 days. The day-by-day count lands exactly on 20/20 at day 8, so the magnitude is right and units (fractions of one fence) check out.

Convert to algebra (tool 13): after n pairs the total is 5n/20; set 5n/20 = 20/20 to get n = 4 pairs = 8 days. The pattern method is simpler and avoids needing the last day to be a partial day here.

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 twentieths until you reach a whole!