← 3-2 · Add poured water to find total capacity · Track a Quantity Through Changes

Add poured water to find total capacity · 12 practice problems

3.MD.A.2

Generated variants — 12

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

Variant 1 answer: 8 fl oz

Mrs. Carter has a jug holding $9\ \text{cups}\ 6\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $4$ more times. If the jug now holds $13\ \text{cups}\ 6\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 9 cups 6 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 4 times. The jug then holds 13 cups 6 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 9 cups 6 fl oz.
Ending amount in the jug is 13 cups 6 fl oz.
A full small cup is poured in 4 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 4 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 4 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 9 cups 6 fl oz = 9 x 8 + 6 = 78 fl oz, and the end 13 cups 6 fl oz = 13 x 8 + 6 = 110 fl oz.

9 \times 8 + 6 = 78,\quad 13 \times 8 + 6 = 110

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 78 fl oz to 110 fl oz, so the 4 pours added 110 - 78 = 32 fl oz altogether.

110 - 78 = 32\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 4 pours were equal full cups, so each cup holds 32 divided by 4 = 8 fl oz.

32 \div 4 = 8\ \text{fl oz}

Sharing the 32 fl oz into 4 equal pours gives the size of one cup.

Answer: 8 fl oz

Review

4 cups of 8 fl oz add 4 x 8 = 32 fl oz, and 78 + 32 = 110 fl oz = 13 cups 6 fl oz, which matches the final amount, so 8 fl oz is correct.

Make a systematic list (tool 2): after each pour of 8 fl oz the jug holds 86, 94, 102, 110 fl oz, reaching 110 fl oz after the 4th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 4 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 2 answer: 8 fl oz

Mrs. Carter has a jug holding $7\ \text{cups}\ 4\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $2$ more times. If the jug now holds $9\ \text{cups}\ 4\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 7 cups 4 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 2 times. The jug then holds 9 cups 4 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 7 cups 4 fl oz.
Ending amount in the jug is 9 cups 4 fl oz.
A full small cup is poured in 2 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 2 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 2 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 7 cups 4 fl oz = 7 x 8 + 4 = 60 fl oz, and the end 9 cups 4 fl oz = 9 x 8 + 4 = 76 fl oz.

7 \times 8 + 4 = 60,\quad 9 \times 8 + 4 = 76

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 60 fl oz to 76 fl oz, so the 2 pours added 76 - 60 = 16 fl oz altogether.

76 - 60 = 16\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 2 pours were equal full cups, so each cup holds 16 divided by 2 = 8 fl oz.

16 \div 2 = 8\ \text{fl oz}

Sharing the 16 fl oz into 2 equal pours gives the size of one cup.

Answer: 8 fl oz

Review

2 cups of 8 fl oz add 2 x 8 = 16 fl oz, and 60 + 16 = 76 fl oz = 9 cups 4 fl oz, which matches the final amount, so 8 fl oz is correct.

Make a systematic list (tool 2): after each pour of 8 fl oz the jug holds 68, 76 fl oz, reaching 76 fl oz after the 2th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 2 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 3 answer: 8 fl oz

Mrs. Carter has a jug holding $20\ \text{cups}\ 1\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $3$ more times. If the jug now holds $23\ \text{cups}\ 1\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 20 cups 1 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 3 times. The jug then holds 23 cups 1 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 20 cups 1 fl oz.
Ending amount in the jug is 23 cups 1 fl oz.
A full small cup is poured in 3 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 3 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 3 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 20 cups 1 fl oz = 20 x 8 + 1 = 161 fl oz, and the end 23 cups 1 fl oz = 23 x 8 + 1 = 185 fl oz.

20 \times 8 + 1 = 161,\quad 23 \times 8 + 1 = 185

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 161 fl oz to 185 fl oz, so the 3 pours added 185 - 161 = 24 fl oz altogether.

185 - 161 = 24\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 3 pours were equal full cups, so each cup holds 24 divided by 3 = 8 fl oz.

24 \div 3 = 8\ \text{fl oz}

Sharing the 24 fl oz into 3 equal pours gives the size of one cup.

Answer: 8 fl oz

Review

3 cups of 8 fl oz add 3 x 8 = 24 fl oz, and 161 + 24 = 185 fl oz = 23 cups 1 fl oz, which matches the final amount, so 8 fl oz is correct.

Make a systematic list (tool 2): after each pour of 8 fl oz the jug holds 169, 177, 185 fl oz, reaching 185 fl oz after the 3th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 3 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 4 answer: 2 fl oz

Mrs. Carter has a jug holding $4\ \text{cups}\ 2\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $8$ more times. If the jug now holds $6\ \text{cups}\ 2\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 4 cups 2 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 8 times. The jug then holds 6 cups 2 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 4 cups 2 fl oz.
Ending amount in the jug is 6 cups 2 fl oz.
A full small cup is poured in 8 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 8 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 8 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 4 cups 2 fl oz = 4 x 8 + 2 = 34 fl oz, and the end 6 cups 2 fl oz = 6 x 8 + 2 = 50 fl oz.

4 \times 8 + 2 = 34,\quad 6 \times 8 + 2 = 50

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 34 fl oz to 50 fl oz, so the 8 pours added 50 - 34 = 16 fl oz altogether.

50 - 34 = 16\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 8 pours were equal full cups, so each cup holds 16 divided by 8 = 2 fl oz.

16 \div 8 = 2\ \text{fl oz}

Sharing the 16 fl oz into 8 equal pours gives the size of one cup.

Answer: 2 fl oz

Review

8 cups of 2 fl oz add 8 x 2 = 16 fl oz, and 34 + 16 = 50 fl oz = 6 cups 2 fl oz, which matches the final amount, so 2 fl oz is correct.

Make a systematic list (tool 2): after each pour of 2 fl oz the jug holds 36, 38, 40, 42, 44, 46, 48, 50 fl oz, reaching 50 fl oz after the 8th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 8 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 5 answer: 4 fl oz

Mrs. Carter has a jug holding $6\ \text{cups}\ 3\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $6$ more times. If the jug now holds $9\ \text{cups}\ 3\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 6 cups 3 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 6 times. The jug then holds 9 cups 3 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 6 cups 3 fl oz.
Ending amount in the jug is 9 cups 3 fl oz.
A full small cup is poured in 6 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 6 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 6 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 6 cups 3 fl oz = 6 x 8 + 3 = 51 fl oz, and the end 9 cups 3 fl oz = 9 x 8 + 3 = 75 fl oz.

6 \times 8 + 3 = 51,\quad 9 \times 8 + 3 = 75

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 51 fl oz to 75 fl oz, so the 6 pours added 75 - 51 = 24 fl oz altogether.

75 - 51 = 24\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 6 pours were equal full cups, so each cup holds 24 divided by 6 = 4 fl oz.

24 \div 6 = 4\ \text{fl oz}

Sharing the 24 fl oz into 6 equal pours gives the size of one cup.

Answer: 4 fl oz

Review

6 cups of 4 fl oz add 6 x 4 = 24 fl oz, and 51 + 24 = 75 fl oz = 9 cups 3 fl oz, which matches the final amount, so 4 fl oz is correct.

Make a systematic list (tool 2): after each pour of 4 fl oz the jug holds 55, 59, 63, 67, 71, 75 fl oz, reaching 75 fl oz after the 6th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 6 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 6 answer: 4 fl oz

Mrs. Carter has a jug holding $3\ \text{cups}\ 7\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $6$ more times. If the jug now holds $6\ \text{cups}\ 7\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 3 cups 7 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 6 times. The jug then holds 6 cups 7 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 3 cups 7 fl oz.
Ending amount in the jug is 6 cups 7 fl oz.
A full small cup is poured in 6 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 6 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 6 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 3 cups 7 fl oz = 3 x 8 + 7 = 31 fl oz, and the end 6 cups 7 fl oz = 6 x 8 + 7 = 55 fl oz.

3 \times 8 + 7 = 31,\quad 6 \times 8 + 7 = 55

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 31 fl oz to 55 fl oz, so the 6 pours added 55 - 31 = 24 fl oz altogether.

55 - 31 = 24\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 6 pours were equal full cups, so each cup holds 24 divided by 6 = 4 fl oz.

24 \div 6 = 4\ \text{fl oz}

Sharing the 24 fl oz into 6 equal pours gives the size of one cup.

Answer: 4 fl oz

Review

6 cups of 4 fl oz add 6 x 4 = 24 fl oz, and 31 + 24 = 55 fl oz = 6 cups 7 fl oz, which matches the final amount, so 4 fl oz is correct.

Make a systematic list (tool 2): after each pour of 4 fl oz the jug holds 35, 39, 43, 47, 51, 55 fl oz, reaching 55 fl oz after the 6th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 6 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 7 answer: 4 fl oz

Mrs. Carter has a jug holding $10\ \text{cups}\ 5\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $8$ more times. If the jug now holds $14\ \text{cups}\ 5\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 10 cups 5 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 8 times. The jug then holds 14 cups 5 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 10 cups 5 fl oz.
Ending amount in the jug is 14 cups 5 fl oz.
A full small cup is poured in 8 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 8 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 8 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 10 cups 5 fl oz = 10 x 8 + 5 = 85 fl oz, and the end 14 cups 5 fl oz = 14 x 8 + 5 = 117 fl oz.

10 \times 8 + 5 = 85,\quad 14 \times 8 + 5 = 117

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 85 fl oz to 117 fl oz, so the 8 pours added 117 - 85 = 32 fl oz altogether.

117 - 85 = 32\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 8 pours were equal full cups, so each cup holds 32 divided by 8 = 4 fl oz.

32 \div 8 = 4\ \text{fl oz}

Sharing the 32 fl oz into 8 equal pours gives the size of one cup.

Answer: 4 fl oz

Review

8 cups of 4 fl oz add 8 x 4 = 32 fl oz, and 85 + 32 = 117 fl oz = 14 cups 5 fl oz, which matches the final amount, so 4 fl oz is correct.

Make a systematic list (tool 2): after each pour of 4 fl oz the jug holds 89, 93, 97, 101, 105, 109, 113, 117 fl oz, reaching 117 fl oz after the 8th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 8 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 8 answer: 4 fl oz

Mrs. Carter has a jug holding $8\ \text{cups}\ 0\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $4$ more times. If the jug now holds $10\ \text{cups}\ 0\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 8 cups 0 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 4 times. The jug then holds 10 cups 0 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 8 cups 0 fl oz.
Ending amount in the jug is 10 cups 0 fl oz.
A full small cup is poured in 4 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 4 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 4 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 8 cups 0 fl oz = 8 x 8 + 0 = 64 fl oz, and the end 10 cups 0 fl oz = 10 x 8 + 0 = 80 fl oz.

8 \times 8 + 0 = 64,\quad 10 \times 8 + 0 = 80

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 64 fl oz to 80 fl oz, so the 4 pours added 80 - 64 = 16 fl oz altogether.

80 - 64 = 16\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 4 pours were equal full cups, so each cup holds 16 divided by 4 = 4 fl oz.

16 \div 4 = 4\ \text{fl oz}

Sharing the 16 fl oz into 4 equal pours gives the size of one cup.

Answer: 4 fl oz

Review

4 cups of 4 fl oz add 4 x 4 = 16 fl oz, and 64 + 16 = 80 fl oz = 10 cups 0 fl oz, which matches the final amount, so 4 fl oz is correct.

Make a systematic list (tool 2): after each pour of 4 fl oz the jug holds 68, 72, 76, 80 fl oz, reaching 80 fl oz after the 4th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 4 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 9 answer: 3 fl oz

Mrs. Carter has a jug holding $11\ \text{cups}\ 2\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $5$ more times. If the jug now holds $13\ \text{cups}\ 1\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 11 cups 2 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 5 times. The jug then holds 13 cups 1 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 11 cups 2 fl oz.
Ending amount in the jug is 13 cups 1 fl oz.
A full small cup is poured in 5 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 5 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 5 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 11 cups 2 fl oz = 11 x 8 + 2 = 90 fl oz, and the end 13 cups 1 fl oz = 13 x 8 + 1 = 105 fl oz.

11 \times 8 + 2 = 90,\quad 13 \times 8 + 1 = 105

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 90 fl oz to 105 fl oz, so the 5 pours added 105 - 90 = 15 fl oz altogether.

105 - 90 = 15\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 5 pours were equal full cups, so each cup holds 15 divided by 5 = 3 fl oz.

15 \div 5 = 3\ \text{fl oz}

Sharing the 15 fl oz into 5 equal pours gives the size of one cup.

Answer: 3 fl oz

Review

5 cups of 3 fl oz add 5 x 3 = 15 fl oz, and 90 + 15 = 105 fl oz = 13 cups 1 fl oz, which matches the final amount, so 3 fl oz is correct.

Make a systematic list (tool 2): after each pour of 3 fl oz the jug holds 93, 96, 99, 102, 105 fl oz, reaching 105 fl oz after the 5th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 5 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 10 answer: 4 fl oz

Mrs. Carter has a jug holding $12\ \text{cups}\ 0\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $6$ more times. If the jug now holds $15\ \text{cups}\ 0\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 12 cups 0 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 6 times. The jug then holds 15 cups 0 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 12 cups 0 fl oz.
Ending amount in the jug is 15 cups 0 fl oz.
A full small cup is poured in 6 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 6 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 6 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 12 cups 0 fl oz = 12 x 8 + 0 = 96 fl oz, and the end 15 cups 0 fl oz = 15 x 8 + 0 = 120 fl oz.

12 \times 8 + 0 = 96,\quad 15 \times 8 + 0 = 120

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 96 fl oz to 120 fl oz, so the 6 pours added 120 - 96 = 24 fl oz altogether.

120 - 96 = 24\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 6 pours were equal full cups, so each cup holds 24 divided by 6 = 4 fl oz.

24 \div 6 = 4\ \text{fl oz}

Sharing the 24 fl oz into 6 equal pours gives the size of one cup.

Answer: 4 fl oz

Review

6 cups of 4 fl oz add 6 x 4 = 24 fl oz, and 96 + 24 = 120 fl oz = 15 cups 0 fl oz, which matches the final amount, so 4 fl oz is correct.

Make a systematic list (tool 2): after each pour of 4 fl oz the jug holds 100, 104, 108, 112, 116, 120 fl oz, reaching 120 fl oz after the 6th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 6 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 11 answer: 2 fl oz

Mrs. Carter has a jug holding $15\ \text{cups}\ 0\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $12$ more times. If the jug now holds $18\ \text{cups}\ 0\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 15 cups 0 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 12 times. The jug then holds 18 cups 0 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 15 cups 0 fl oz.
Ending amount in the jug is 18 cups 0 fl oz.
A full small cup is poured in 12 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 12 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 12 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 15 cups 0 fl oz = 15 x 8 + 0 = 120 fl oz, and the end 18 cups 0 fl oz = 18 x 8 + 0 = 144 fl oz.

15 \times 8 + 0 = 120,\quad 18 \times 8 + 0 = 144

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 120 fl oz to 144 fl oz, so the 12 pours added 144 - 120 = 24 fl oz altogether.

144 - 120 = 24\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 12 pours were equal full cups, so each cup holds 24 divided by 12 = 2 fl oz.

24 \div 12 = 2\ \text{fl oz}

Sharing the 24 fl oz into 12 equal pours gives the size of one cup.

Answer: 2 fl oz

Review

12 cups of 2 fl oz add 12 x 2 = 24 fl oz, and 120 + 24 = 144 fl oz = 18 cups 0 fl oz, which matches the final amount, so 2 fl oz is correct.

Make a systematic list (tool 2): after each pour of 2 fl oz the jug holds 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144 fl oz, reaching 144 fl oz after the 12th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 12 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!

Variant 12 answer: 6 fl oz

Mrs. Carter has a jug holding $5\ \text{cups}\ 1\ \text{fl oz}$ of lemonade. She fills a small cup to the brim with lemonade and pours it into the jug $3$ more times. If the jug now holds $7\ \text{cups}\ 5\ \text{fl oz}$ of lemonade, what is the capacity of the small cup, in fluid ounces? (Note: $1\ \text{cup} = 8\ \text{fl oz}$ .)

Show solution

Understand

A jug starts with 5 cups 1 fl oz of lemonade. A small cup of lemonade is filled to the brim and poured in 3 times. The jug then holds 7 cups 5 fl oz. Find the capacity of the small cup in fluid ounces, where 1 cup = 8 fl oz.

Givens

Starting amount in the jug is 5 cups 1 fl oz.
Ending amount in the jug is 7 cups 5 fl oz.
A full small cup is poured in 3 times.
1 cup = 8 fl oz.

Unknowns

The capacity of the small cup in fluid ounces.

Constraints

The small cup is filled to the brim each time, so all 3 pours are equal.

Plan

#8 Analyze the Units · also uses: #11 Work Backwards

Convert both amounts to fluid ounces, subtract to find the total amount poured in, then divide that by 3 equal pours to get the small cup's capacity.

Execute

#8 Analyze the Units 3.MD.A.2

Since 1 cup = 8 fl oz, the start 5 cups 1 fl oz = 5 x 8 + 1 = 41 fl oz, and the end 7 cups 5 fl oz = 7 x 8 + 5 = 61 fl oz.

5 \times 8 + 1 = 41,\quad 7 \times 8 + 5 = 61

Using one unit (fluid ounces) makes the added amount easy to find.

#11 Work Backwards 3.MD.A.2

The jug grew from 41 fl oz to 61 fl oz, so the 3 pours added 61 - 41 = 20 fl oz altogether.

61 - 41 = 20\ \text{fl oz}

The increase in the jug is exactly what the small cup added in total.

#8 Analyze the Units 3.OA.A.3

All 3 pours were equal full cups, so each cup holds 20 divided by 3 = 6 fl oz.

20 \div 3 = 6\ \text{fl oz}

Sharing the 20 fl oz into 3 equal pours gives the size of one cup.

Answer: 6 fl oz

Review

3 cups of 6 fl oz add 3 x 6 = 20 fl oz, and 41 + 20 = 61 fl oz = 7 cups 5 fl oz, which matches the final amount, so 6 fl oz is correct.

Make a systematic list (tool 2): after each pour of 6 fl oz the jug holds 47, 53, 59 fl oz, reaching 61 fl oz after the 3th pour.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting cup-and-ounce volumes and finding the added amount.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the total added volume by 3 equal pours.

💡 This only needs Grade 3 sense: find how much the jug grew, then split it into equal cups!