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← 3-2 · Set up one-unknown equation for capacities · Find Two Unknowns from Sum and Difference

Set up one-unknown equation for capacities · 10 practice problems

3.MD.A.23.OA.D.8

Generated variants — 10

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

Variant 1 answer: 32 fl oz

A total of 7272 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 88 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

72 fl oz of cooking oil is split between two bottles whose capacities differ by 8 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 72 fl oz when both are full.
  • The two capacities differ by 8 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 8 fl oz more than the smaller one.
  • The two amounts add up to 72 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 8 fl oz of the bigger bottle, the two bottles would be equal. Remove that 8 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 8 fl oz more than the smaller. Take those 8 fl oz away from the total of 72 fl oz, leaving 72 - 8 = 64 fl oz to share as two equal smaller bottles.
72−8=64 fl oz72 - 8 = 64\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 64 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 64 divided by 2 = 32 fl oz.
64÷2=32 fl oz64 \div 2 = 32\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 32 fl oz

Review

Smaller bottle 32 fl oz and larger bottle 32 + 8 = 40 fl oz add to 32 + 40 = 72 fl oz, and they differ by 8 fl oz, so 32 fl oz checks out.

Guess and check (tool 6): try smaller = 32, larger = 40, sum 72, difference 8; both conditions fit, confirming 32 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 8 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 2 answer: 22 fl oz

A total of 4848 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 44 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

48 fl oz of cooking oil is split between two bottles whose capacities differ by 4 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 48 fl oz when both are full.
  • The two capacities differ by 4 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 4 fl oz more than the smaller one.
  • The two amounts add up to 48 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 4 fl oz of the bigger bottle, the two bottles would be equal. Remove that 4 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 4 fl oz more than the smaller. Take those 4 fl oz away from the total of 48 fl oz, leaving 48 - 4 = 44 fl oz to share as two equal smaller bottles.
48−4=44 fl oz48 - 4 = 44\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 44 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 44 divided by 2 = 22 fl oz.
44÷2=22 fl oz44 \div 2 = 22\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 22 fl oz

Review

Smaller bottle 22 fl oz and larger bottle 22 + 4 = 26 fl oz add to 22 + 26 = 48 fl oz, and they differ by 4 fl oz, so 22 fl oz checks out.

Guess and check (tool 6): try smaller = 22, larger = 26, sum 48, difference 4; both conditions fit, confirming 22 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 4 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 3 answer: 22 fl oz

A total of 5050 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 66 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

50 fl oz of cooking oil is split between two bottles whose capacities differ by 6 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 50 fl oz when both are full.
  • The two capacities differ by 6 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 6 fl oz more than the smaller one.
  • The two amounts add up to 50 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 6 fl oz of the bigger bottle, the two bottles would be equal. Remove that 6 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 6 fl oz more than the smaller. Take those 6 fl oz away from the total of 50 fl oz, leaving 50 - 6 = 44 fl oz to share as two equal smaller bottles.
50−6=44 fl oz50 - 6 = 44\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 44 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 44 divided by 2 = 22 fl oz.
44÷2=22 fl oz44 \div 2 = 22\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 22 fl oz

Review

Smaller bottle 22 fl oz and larger bottle 22 + 6 = 28 fl oz add to 22 + 28 = 50 fl oz, and they differ by 6 fl oz, so 22 fl oz checks out.

Guess and check (tool 6): try smaller = 22, larger = 28, sum 50, difference 6; both conditions fit, confirming 22 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 6 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 4 answer: 34 fl oz

A total of 8080 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 1212 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

80 fl oz of cooking oil is split between two bottles whose capacities differ by 12 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 80 fl oz when both are full.
  • The two capacities differ by 12 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 12 fl oz more than the smaller one.
  • The two amounts add up to 80 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 12 fl oz of the bigger bottle, the two bottles would be equal. Remove that 12 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 12 fl oz more than the smaller. Take those 12 fl oz away from the total of 80 fl oz, leaving 80 - 12 = 68 fl oz to share as two equal smaller bottles.
80−12=68 fl oz80 - 12 = 68\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 68 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 68 divided by 2 = 34 fl oz.
68÷2=34 fl oz68 \div 2 = 34\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 34 fl oz

Review

Smaller bottle 34 fl oz and larger bottle 34 + 12 = 46 fl oz add to 34 + 46 = 80 fl oz, and they differ by 12 fl oz, so 34 fl oz checks out.

Guess and check (tool 6): try smaller = 34, larger = 46, sum 80, difference 12; both conditions fit, confirming 34 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 12 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 5 answer: 30 fl oz

A total of 6666 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 66 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

66 fl oz of cooking oil is split between two bottles whose capacities differ by 6 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 66 fl oz when both are full.
  • The two capacities differ by 6 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 6 fl oz more than the smaller one.
  • The two amounts add up to 66 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 6 fl oz of the bigger bottle, the two bottles would be equal. Remove that 6 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 6 fl oz more than the smaller. Take those 6 fl oz away from the total of 66 fl oz, leaving 66 - 6 = 60 fl oz to share as two equal smaller bottles.
66−6=60 fl oz66 - 6 = 60\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 60 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 60 divided by 2 = 30 fl oz.
60÷2=30 fl oz60 \div 2 = 30\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 30 fl oz

Review

Smaller bottle 30 fl oz and larger bottle 30 + 6 = 36 fl oz add to 30 + 36 = 66 fl oz, and they differ by 6 fl oz, so 30 fl oz checks out.

Guess and check (tool 6): try smaller = 30, larger = 36, sum 66, difference 6; both conditions fit, confirming 30 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 6 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 6 answer: 20 fl oz

A total of 5656 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 1616 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

56 fl oz of cooking oil is split between two bottles whose capacities differ by 16 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 56 fl oz when both are full.
  • The two capacities differ by 16 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 16 fl oz more than the smaller one.
  • The two amounts add up to 56 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 16 fl oz of the bigger bottle, the two bottles would be equal. Remove that 16 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 16 fl oz more than the smaller. Take those 16 fl oz away from the total of 56 fl oz, leaving 56 - 16 = 40 fl oz to share as two equal smaller bottles.
56−16=40 fl oz56 - 16 = 40\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 40 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 40 divided by 2 = 20 fl oz.
40÷2=20 fl oz40 \div 2 = 20\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 20 fl oz

Review

Smaller bottle 20 fl oz and larger bottle 20 + 16 = 36 fl oz add to 20 + 36 = 56 fl oz, and they differ by 16 fl oz, so 20 fl oz checks out.

Guess and check (tool 6): try smaller = 20, larger = 36, sum 56, difference 16; both conditions fit, confirming 20 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 16 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 7 answer: 40 fl oz

A total of 100100 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 2020 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

100 fl oz of cooking oil is split between two bottles whose capacities differ by 20 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 100 fl oz when both are full.
  • The two capacities differ by 20 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 20 fl oz more than the smaller one.
  • The two amounts add up to 100 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 20 fl oz of the bigger bottle, the two bottles would be equal. Remove that 20 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 20 fl oz more than the smaller. Take those 20 fl oz away from the total of 100 fl oz, leaving 100 - 20 = 80 fl oz to share as two equal smaller bottles.
100−20=80 fl oz100 - 20 = 80\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 80 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 80 divided by 2 = 40 fl oz.
80÷2=40 fl oz80 \div 2 = 40\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 40 fl oz

Review

Smaller bottle 40 fl oz and larger bottle 40 + 20 = 60 fl oz add to 40 + 60 = 100 fl oz, and they differ by 20 fl oz, so 40 fl oz checks out.

Guess and check (tool 6): try smaller = 40, larger = 60, sum 100, difference 20; both conditions fit, confirming 40 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 20 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 8 answer: 40 fl oz

A total of 9090 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 1010 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

90 fl oz of cooking oil is split between two bottles whose capacities differ by 10 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 90 fl oz when both are full.
  • The two capacities differ by 10 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 10 fl oz more than the smaller one.
  • The two amounts add up to 90 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 10 fl oz of the bigger bottle, the two bottles would be equal. Remove that 10 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 10 fl oz more than the smaller. Take those 10 fl oz away from the total of 90 fl oz, leaving 90 - 10 = 80 fl oz to share as two equal smaller bottles.
90−10=80 fl oz90 - 10 = 80\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 80 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 80 divided by 2 = 40 fl oz.
80÷2=40 fl oz80 \div 2 = 40\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 40 fl oz

Review

Smaller bottle 40 fl oz and larger bottle 40 + 10 = 50 fl oz add to 40 + 50 = 90 fl oz, and they differ by 10 fl oz, so 40 fl oz checks out.

Guess and check (tool 6): try smaller = 40, larger = 50, sum 90, difference 10; both conditions fit, confirming 40 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 10 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 9 answer: 28 fl oz

A total of 6464 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 88 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

64 fl oz of cooking oil is split between two bottles whose capacities differ by 8 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 64 fl oz when both are full.
  • The two capacities differ by 8 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 8 fl oz more than the smaller one.
  • The two amounts add up to 64 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 8 fl oz of the bigger bottle, the two bottles would be equal. Remove that 8 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 8 fl oz more than the smaller. Take those 8 fl oz away from the total of 64 fl oz, leaving 64 - 8 = 56 fl oz to share as two equal smaller bottles.
64−8=56 fl oz64 - 8 = 56\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 56 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 56 divided by 2 = 28 fl oz.
56÷2=28 fl oz56 \div 2 = 28\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 28 fl oz

Review

Smaller bottle 28 fl oz and larger bottle 28 + 8 = 36 fl oz add to 28 + 36 = 64 fl oz, and they differ by 8 fl oz, so 28 fl oz checks out.

Guess and check (tool 6): try smaller = 28, larger = 36, sum 64, difference 8; both conditions fit, confirming 28 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 8 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!
Variant 10 answer: 25 fl oz

A total of 6060 fluid ounces of cooking oil is poured into two bottles whose capacities differ by 1010 fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

60 fl oz of cooking oil is split between two bottles whose capacities differ by 10 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens
  • The two bottles together hold 60 fl oz when both are full.
  • The two capacities differ by 10 fl oz.
  • Both bottles are filled to the top.
Unknowns
  • The capacity (oil amount) of the smaller bottle in fluid ounces.
Constraints
  • The larger bottle holds exactly 10 fl oz more than the smaller one.
  • The two amounts add up to 60 fl oz.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

If we set aside the extra 10 fl oz of the bigger bottle, the two bottles would be equal. Remove that 10 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8
The larger bottle has 10 fl oz more than the smaller. Take those 10 fl oz away from the total of 60 fl oz, leaving 60 - 10 = 50 fl oz to share as two equal smaller bottles.
60−10=50 fl oz60 - 10 = 50\ \text{fl oz}
Setting aside the extra makes both bottles the same size, which is easier to share.
#11 Work Backwards 3.OA.A.3
The 50 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 50 divided by 2 = 25 fl oz.
50÷2=25 fl oz50 \div 2 = 25\ \text{fl oz}
Two equal parts means cut the leftover total in half.
Answer: 25 fl oz

Review

Smaller bottle 25 fl oz and larger bottle 25 + 10 = 35 fl oz add to 25 + 35 = 60 fl oz, and they differ by 10 fl oz, so 25 fl oz checks out.

Guess and check (tool 6): try smaller = 25, larger = 35, sum 60, difference 10; both conditions fit, confirming 25 fl oz.

Standards · min grade 3

  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 10 fl oz difference then halving to find the smaller bottle.
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.
💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!