← Subtracting two weighings cancels whatever they share · Track a Quantity Through Changes

Subtracting two weighings cancels whatever they share · 12 practice problems

6.NS.B.36.EE.B.76.RP.A.3

Generated variants — 12

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

Variant 1 easy answer: 0.3 kg

A container holding $3\ \text{L}$ of milk weighs $4.05\ \text{kg}$ . After $1.2\ \text{L}$ of milk is poured out, it weighs $2.55\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 3 L of milk weighs 4.05 kg, and 2.55 kg after 1.2 L is poured out. We want the container alone.

Givens

With 3 L of milk it weighs 4.05 kg.
After pouring 1.2 L out it weighs 2.55 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

4.05 - 2.55 = 1.5

1.2 L of milk weighs 1.5 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

1.5 \div 1.2 = 1.25

A litre of milk is 1.25 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 3 L of it.

1.25 \times 3 = 3.75

3.75 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

4.05 - 3.75 = 0.3

The container weighs 0.3 kg.

Answer: 0.3 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.3 kg of container plus 1.8 L of milk at 1.25 kg is 2.55 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 2 easy answer: 0.38 kg

A container holding $4\ \text{L}$ of milk weighs $5.18\ \text{kg}$ . After $1.5\ \text{L}$ of milk is poured out, it weighs $3.38\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 4 L of milk weighs 5.18 kg, and 3.38 kg after 1.5 L is poured out. We want the container alone.

Givens

With 4 L of milk it weighs 5.18 kg.
After pouring 1.5 L out it weighs 3.38 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

5.18 - 3.38 = 1.8

1.5 L of milk weighs 1.8 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

1.8 \div 1.5 = 1.2

A litre of milk is 1.2 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 4 L of it.

1.2 \times 4 = 4.8

4.8 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

5.18 - 4.8 = 0.38

The container weighs 0.38 kg.

Answer: 0.38 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.38 kg of container plus 2.5 L of milk at 1.2 kg is 3.38 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 3 easy answer: 0.45 kg

A container holding $5\ \text{L}$ of milk weighs $5.45\ \text{kg}$ . After $2\ \text{L}$ of milk is poured out, it weighs $3.45\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 5 L of milk weighs 5.45 kg, and 3.45 kg after 2 L is poured out. We want the container alone.

Givens

With 5 L of milk it weighs 5.45 kg.
After pouring 2 L out it weighs 3.45 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

5.45 - 3.45 = 2

2 L of milk weighs 2 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

2 \div 2 = 1

A litre of milk is 1 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 5 L of it.

1 \times 5 = 5

5 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

5.45 - 5 = 0.45

The container weighs 0.45 kg.

Answer: 0.45 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.45 kg of container plus 3 L of milk at 1 kg is 3.45 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 4 easy answer: 0.4 kg

A container holding $5\ \text{L}$ of milk weighs $6.4\ \text{kg}$ . After $2.5\ \text{L}$ of milk is poured out, it weighs $3.4\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 5 L of milk weighs 6.4 kg, and 3.4 kg after 2.5 L is poured out. We want the container alone.

Givens

With 5 L of milk it weighs 6.4 kg.
After pouring 2.5 L out it weighs 3.4 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

6.4 - 3.4 = 3

2.5 L of milk weighs 3 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

3 \div 2.5 = 1.2

A litre of milk is 1.2 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 5 L of it.

1.2 \times 5 = 6

6 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

6.4 - 6 = 0.4

The container weighs 0.4 kg.

Answer: 0.4 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.4 kg of container plus 2.5 L of milk at 1.2 kg is 3.4 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 5 medium answer: 0.57 kg

A container holding $6\ \text{L}$ of milk weighs $6.87\ \text{kg}$ . After $2.4\ \text{L}$ of milk is poured out, it weighs $4.35\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 6 L of milk weighs 6.87 kg, and 4.35 kg after 2.4 L is poured out. We want the container alone.

Givens

With 6 L of milk it weighs 6.87 kg.
After pouring 2.4 L out it weighs 4.35 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

6.87 - 4.35 = 2.52

2.4 L of milk weighs 2.52 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

2.52 \div 2.4 = 1.05

A litre of milk is 1.05 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 6 L of it.

1.05 \times 6 = 6.3

6.3 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

6.87 - 6.3 = 0.57

The container weighs 0.57 kg.

Answer: 0.57 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.57 kg of container plus 3.6 L of milk at 1.05 kg is 4.35 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 6 medium answer: 0.5 kg

A container holding $7\ \text{L}$ of milk weighs $7.5\ \text{kg}$ . After $3.5\ \text{L}$ of milk is poured out, it weighs $4\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 7 L of milk weighs 7.5 kg, and 4 kg after 3.5 L is poured out. We want the container alone.

Givens

With 7 L of milk it weighs 7.5 kg.
After pouring 3.5 L out it weighs 4 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

7.5 - 4 = 3.5

3.5 L of milk weighs 3.5 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

3.5 \div 3.5 = 1

A litre of milk is 1 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 7 L of it.

1 \times 7 = 7

7 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

7.5 - 7 = 0.5

The container weighs 0.5 kg.

Answer: 0.5 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.5 kg of container plus 3.5 L of milk at 1 kg is 4 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 7 medium answer: 0.62 kg

A container holding $8\ \text{L}$ of milk weighs $9.42\ \text{kg}$ . After $2.5\ \text{L}$ of milk is poured out, it weighs $6.67\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 8 L of milk weighs 9.42 kg, and 6.67 kg after 2.5 L is poured out. We want the container alone.

Givens

With 8 L of milk it weighs 9.42 kg.
After pouring 2.5 L out it weighs 6.67 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

9.42 - 6.67 = 2.75

2.5 L of milk weighs 2.75 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

2.75 \div 2.5 = 1.1

A litre of milk is 1.1 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 8 L of it.

1.1 \times 8 = 8.8

8.8 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

9.42 - 8.8 = 0.62

The container weighs 0.62 kg.

Answer: 0.62 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.62 kg of container plus 5.5 L of milk at 1.1 kg is 6.67 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 8 medium answer: 0.75 kg

A container holding $10\ \text{L}$ of milk weighs $10.25\ \text{kg}$ . After $4\ \text{L}$ of milk is poured out, it weighs $6.45\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 10 L of milk weighs 10.25 kg, and 6.45 kg after 4 L is poured out. We want the container alone.

Givens

With 10 L of milk it weighs 10.25 kg.
After pouring 4 L out it weighs 6.45 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

10.25 - 6.45 = 3.8

4 L of milk weighs 3.8 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

3.8 \div 4 = 0.95

A litre of milk is 0.95 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 10 L of it.

0.95 \times 10 = 9.5

9.5 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

10.25 - 9.5 = 0.75

The container weighs 0.75 kg.

Answer: 0.75 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.75 kg of container plus 6 L of milk at 0.95 kg is 6.45 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 9 hard answer: 0.66 kg

A container holding $9\ \text{L}$ of milk weighs $10.56\ \text{kg}$ . After $3\ \text{L}$ of milk is poured out, it weighs $7.26\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 9 L of milk weighs 10.56 kg, and 7.26 kg after 3 L is poured out. We want the container alone.

Givens

With 9 L of milk it weighs 10.56 kg.
After pouring 3 L out it weighs 7.26 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

10.56 - 7.26 = 3.3

3 L of milk weighs 3.3 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

3.3 \div 3 = 1.1

A litre of milk is 1.1 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 9 L of it.

1.1 \times 9 = 9.9

9.9 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

10.56 - 9.9 = 0.66

The container weighs 0.66 kg.

Answer: 0.66 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.66 kg of container plus 6 L of milk at 1.1 kg is 7.26 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 10 hard answer: 0.84 kg

A container holding $12\ \text{L}$ of milk weighs $13.44\ \text{kg}$ . After $5\ \text{L}$ of milk is poured out, it weighs $8.19\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 12 L of milk weighs 13.44 kg, and 8.19 kg after 5 L is poured out. We want the container alone.

Givens

With 12 L of milk it weighs 13.44 kg.
After pouring 5 L out it weighs 8.19 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

13.44 - 8.19 = 5.25

5 L of milk weighs 5.25 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

5.25 \div 5 = 1.05

A litre of milk is 1.05 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 12 L of it.

1.05 \times 12 = 12.6

12.6 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

13.44 - 12.6 = 0.84

The container weighs 0.84 kg.

Answer: 0.84 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.84 kg of container plus 7 L of milk at 1.05 kg is 8.19 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 11 hard answer: 0.9 kg

A container holding $15\ \text{L}$ of milk weighs $15.9\ \text{kg}$ . After $7.5\ \text{L}$ of milk is poured out, it weighs $8.4\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 15 L of milk weighs 15.9 kg, and 8.4 kg after 7.5 L is poured out. We want the container alone.

Givens

With 15 L of milk it weighs 15.9 kg.
After pouring 7.5 L out it weighs 8.4 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

15.9 - 8.4 = 7.5

7.5 L of milk weighs 7.5 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

7.5 \div 7.5 = 1

A litre of milk is 1 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 15 L of it.

1 \times 15 = 15

15 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

15.9 - 15 = 0.9

The container weighs 0.9 kg.

Answer: 0.9 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 0.9 kg of container plus 7.5 L of milk at 1 kg is 8.4 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.

Variant 12 hard answer: 1.1 kg

A container holding $20\ \text{L}$ of milk weighs $20.1\ \text{kg}$ . After $8\ \text{L}$ of milk is poured out, it weighs $12.5\ \text{kg}$ . How many kilograms does the empty container weigh?

Show solution

1 · Understandwhat's really being asked

A container with 20 L of milk weighs 20.1 kg, and 12.5 kg after 8 L is poured out. We want the container alone.

Givens

With 20 L of milk it weighs 20.1 kg.
After pouring 8 L out it weighs 12.5 kg.

Unknowns

The weight of the empty container.

Constraints

The container itself never changes weight.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #8 Analyze the Units#7 Identify Subproblems

The container's weight is buried in both readings, so neither reading alone says anything about it. Subtract them and it cancels, leaving pure milk -- which is where a weight per litre comes from.

3 · Execute4 carry out the plan

1Subtract the two weighings

#16 Count the Complement 6.NS.B.3

The container is in both, so the difference is milk and nothing else.

20.1 - 12.5 = 7.6

8 L of milk weighs 7.6 kg.

2Find one litre

#8 Analyze the Units 6.RP.A.3

Divide that weight by the litres poured.

7.6 \div 8 = 0.95

A litre of milk is 0.95 kg.

3Weigh the milk that was there at first

#7 Identify Subproblems 6.RP.A.3

All 20 L of it.

0.95 \times 20 = 19

19 kg of milk at the start.

4Take the milk off the first weighing

#7 Identify Subproblems 6.EE.B.7

What is left is the container.

20.1 - 19 = 1.1

The container weighs 1.1 kg.

Answer: 1.1 kg

4 · Reviewdoes it hold up?

The second weighing checks out too: 1.1 kg of container plus 12 L of milk at 0.95 kg is 12.5 kg.

Another way: Using the second weighing instead of the first gives the same container weight, which is a way of checking the rate without repeating the subtraction.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Adding and subtracting the decimal weights.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the container once the milk is known.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Getting and using the weight of one litre.

💡Takeaway. If something unknown is in both measurements, subtract them and it disappears.