Practice · Use weight differences to find the empty container · Sensim Math

Variant 1 easy

A box holding 3 identical bottles of juice weighs $2.25 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.90 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 3 identical juice bottles inside weighs 2.25 lb. Take out one bottle and the box now weighs 1.90 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 3 bottles = 2.25 lb.
Box + 2 bottles = 1.90 lb.
All 3 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 2.25 lb to 1.90 lb. That drop is one bottle: 2.25 - 1.90 = 0.35 lb.

2.25 - 1.90 = 0.35\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 2 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 2 bottles remain in the box. 2 bottles weigh 2 x 0.35 = 0.70 lb.

2 \times 0.35 = 0.70\ \text{lb}

2 equal bottles is just the one-bottle weight added 2 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.90 lb weighing is the box plus those 2 bottles. Take the bottles away: 1.90 - 0.70 = 1.20 lb is the empty box.

1.90 - 0.70 = 1.20\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 1.20 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 1.20 + 3 bottles (3 x 0.35 = 1.05) = 2.25 lb, exactly as given. The empty box (1.20 lb) is lighter than the box with juice (2.25 lb), which makes sense.

Another way: Use the full weighing directly: box = 2.25 - 3 x 0.35 = 2.25 - 1.05 = 1.20 lb - same answer by Work Backwards from the 3-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 2.25 - 1.90 and multiplying 2 x 0.35.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 2 medium

A box holding 4 identical bottles of juice weighs $1.98 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.76 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 4 identical juice bottles inside weighs 1.98 lb. Take out one bottle and the box now weighs 1.76 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 4 bottles = 1.98 lb.
Box + 3 bottles = 1.76 lb.
All 4 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 1.98 lb to 1.76 lb. That drop is one bottle: 1.98 - 1.76 = 0.22 lb.

1.98 - 1.76 = 0.22\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 3 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 3 bottles remain in the box. 3 bottles weigh 3 x 0.22 = 0.66 lb.

3 \times 0.22 = 0.66\ \text{lb}

3 equal bottles is just the one-bottle weight added 3 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.76 lb weighing is the box plus those 3 bottles. Take the bottles away: 1.76 - 0.66 = 1.10 lb is the empty box.

1.76 - 0.66 = 1.10\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 1.10 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 1.10 + 4 bottles (4 x 0.22 = 0.88) = 1.98 lb, exactly as given. The empty box (1.10 lb) is lighter than the box with juice (1.98 lb), which makes sense.

Another way: Use the full weighing directly: box = 1.98 - 4 x 0.22 = 1.98 - 0.88 = 1.10 lb - same answer by Work Backwards from the 4-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 1.98 - 1.76 and multiplying 3 x 0.22.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 3 easy

A box holding 4 identical bottles of juice weighs $1.50 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.25 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 4 identical juice bottles inside weighs 1.50 lb. Take out one bottle and the box now weighs 1.25 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 4 bottles = 1.50 lb.
Box + 3 bottles = 1.25 lb.
All 4 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 1.50 lb to 1.25 lb. That drop is one bottle: 1.50 - 1.25 = 0.25 lb.

1.50 - 1.25 = 0.25\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 3 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 3 bottles remain in the box. 3 bottles weigh 3 x 0.25 = 0.75 lb.

3 \times 0.25 = 0.75\ \text{lb}

3 equal bottles is just the one-bottle weight added 3 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.25 lb weighing is the box plus those 3 bottles. Take the bottles away: 1.25 - 0.75 = 0.50 lb is the empty box.

1.25 - 0.75 = 0.50\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.50 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.50 + 4 bottles (4 x 0.25 = 1.00) = 1.50 lb, exactly as given. The empty box (0.50 lb) is lighter than the box with juice (1.50 lb), which makes sense.

Another way: Use the full weighing directly: box = 1.50 - 4 x 0.25 = 1.50 - 1.00 = 0.50 lb - same answer by Work Backwards from the 4-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 1.50 - 1.25 and multiplying 3 x 0.25.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 4 easy

A box holding 4 identical bottles of juice weighs $2.40 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.95 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 4 identical juice bottles inside weighs 2.40 lb. Take out one bottle and the box now weighs 1.95 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 4 bottles = 2.40 lb.
Box + 3 bottles = 1.95 lb.
All 4 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 2.40 lb to 1.95 lb. That drop is one bottle: 2.40 - 1.95 = 0.45 lb.

2.40 - 1.95 = 0.45\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 3 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 3 bottles remain in the box. 3 bottles weigh 3 x 0.45 = 1.35 lb.

3 \times 0.45 = 1.35\ \text{lb}

3 equal bottles is just the one-bottle weight added 3 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.95 lb weighing is the box plus those 3 bottles. Take the bottles away: 1.95 - 1.35 = 0.60 lb is the empty box.

1.95 - 1.35 = 0.60\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.60 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.60 + 4 bottles (4 x 0.45 = 1.80) = 2.40 lb, exactly as given. The empty box (0.60 lb) is lighter than the box with juice (2.40 lb), which makes sense.

Another way: Use the full weighing directly: box = 2.40 - 4 x 0.45 = 2.40 - 1.80 = 0.60 lb - same answer by Work Backwards from the 4-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 2.40 - 1.95 and multiplying 3 x 0.45.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 5 medium

A box holding 5 identical bottles of juice weighs $3.25 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $2.75 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 5 identical juice bottles inside weighs 3.25 lb. Take out one bottle and the box now weighs 2.75 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 5 bottles = 3.25 lb.
Box + 4 bottles = 2.75 lb.
All 5 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 3.25 lb to 2.75 lb. That drop is one bottle: 3.25 - 2.75 = 0.50 lb.

3.25 - 2.75 = 0.50\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 4 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 4 bottles remain in the box. 4 bottles weigh 4 x 0.50 = 2.00 lb.

4 \times 0.50 = 2.00\ \text{lb}

4 equal bottles is just the one-bottle weight added 4 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 2.75 lb weighing is the box plus those 4 bottles. Take the bottles away: 2.75 - 2.00 = 0.75 lb is the empty box.

2.75 - 2.00 = 0.75\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.75 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.75 + 5 bottles (5 x 0.50 = 2.50) = 3.25 lb, exactly as given. The empty box (0.75 lb) is lighter than the box with juice (3.25 lb), which makes sense.

Another way: Use the full weighing directly: box = 3.25 - 5 x 0.50 = 3.25 - 2.50 = 0.75 lb - same answer by Work Backwards from the 5-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 3.25 - 2.75 and multiplying 4 x 0.50.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 6 medium

A box holding 5 identical bottles of juice weighs $1.96 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.66 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 5 identical juice bottles inside weighs 1.96 lb. Take out one bottle and the box now weighs 1.66 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 5 bottles = 1.96 lb.
Box + 4 bottles = 1.66 lb.
All 5 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 1.96 lb to 1.66 lb. That drop is one bottle: 1.96 - 1.66 = 0.30 lb.

1.96 - 1.66 = 0.30\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 4 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 4 bottles remain in the box. 4 bottles weigh 4 x 0.30 = 1.20 lb.

4 \times 0.30 = 1.20\ \text{lb}

4 equal bottles is just the one-bottle weight added 4 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.66 lb weighing is the box plus those 4 bottles. Take the bottles away: 1.66 - 1.20 = 0.46 lb is the empty box.

1.66 - 1.20 = 0.46\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.46 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.46 + 5 bottles (5 x 0.30 = 1.50) = 1.96 lb, exactly as given. The empty box (0.46 lb) is lighter than the box with juice (1.96 lb), which makes sense.

Another way: Use the full weighing directly: box = 1.96 - 5 x 0.30 = 1.96 - 1.50 = 0.46 lb - same answer by Work Backwards from the 5-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 1.96 - 1.66 and multiplying 4 x 0.30.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 7 medium

A box holding 5 identical bottles of juice weighs $1.35 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.15 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 5 identical juice bottles inside weighs 1.35 lb. Take out one bottle and the box now weighs 1.15 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 5 bottles = 1.35 lb.
Box + 4 bottles = 1.15 lb.
All 5 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 1.35 lb to 1.15 lb. That drop is one bottle: 1.35 - 1.15 = 0.20 lb.

1.35 - 1.15 = 0.20\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 4 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 4 bottles remain in the box. 4 bottles weigh 4 x 0.20 = 0.80 lb.

4 \times 0.20 = 0.80\ \text{lb}

4 equal bottles is just the one-bottle weight added 4 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.15 lb weighing is the box plus those 4 bottles. Take the bottles away: 1.15 - 0.80 = 0.35 lb is the empty box.

1.15 - 0.80 = 0.35\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.35 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.35 + 5 bottles (5 x 0.20 = 1.00) = 1.35 lb, exactly as given. The empty box (0.35 lb) is lighter than the box with juice (1.35 lb), which makes sense.

Another way: Use the full weighing directly: box = 1.35 - 5 x 0.20 = 1.35 - 1.00 = 0.35 lb - same answer by Work Backwards from the 5-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 1.35 - 1.15 and multiplying 4 x 0.20.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 8 hard

A box holding 6 identical bottles of juice weighs $2.38 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $2.05 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 6 identical juice bottles inside weighs 2.38 lb. Take out one bottle and the box now weighs 2.05 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 6 bottles = 2.38 lb.
Box + 5 bottles = 2.05 lb.
All 6 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 2.38 lb to 2.05 lb. That drop is one bottle: 2.38 - 2.05 = 0.33 lb.

2.38 - 2.05 = 0.33\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 5 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 5 bottles remain in the box. 5 bottles weigh 5 x 0.33 = 1.65 lb.

5 \times 0.33 = 1.65\ \text{lb}

5 equal bottles is just the one-bottle weight added 5 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 2.05 lb weighing is the box plus those 5 bottles. Take the bottles away: 2.05 - 1.65 = 0.40 lb is the empty box.

2.05 - 1.65 = 0.40\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.40 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.40 + 6 bottles (6 x 0.33 = 1.98) = 2.38 lb, exactly as given. The empty box (0.40 lb) is lighter than the box with juice (2.38 lb), which makes sense.

Another way: Use the full weighing directly: box = 2.38 - 6 x 0.33 = 2.38 - 1.98 = 0.40 lb - same answer by Work Backwards from the 6-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 2.38 - 2.05 and multiplying 5 x 0.33.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 9 hard

A box holding 6 identical bottles of juice weighs $3.20 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $2.80 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 6 identical juice bottles inside weighs 3.20 lb. Take out one bottle and the box now weighs 2.80 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 6 bottles = 3.20 lb.
Box + 5 bottles = 2.80 lb.
All 6 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 3.20 lb to 2.80 lb. That drop is one bottle: 3.20 - 2.80 = 0.40 lb.

3.20 - 2.80 = 0.40\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 5 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 5 bottles remain in the box. 5 bottles weigh 5 x 0.40 = 2.00 lb.

5 \times 0.40 = 2.00\ \text{lb}

5 equal bottles is just the one-bottle weight added 5 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 2.80 lb weighing is the box plus those 5 bottles. Take the bottles away: 2.80 - 2.00 = 0.80 lb is the empty box.

2.80 - 2.00 = 0.80\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.80 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.80 + 6 bottles (6 x 0.40 = 2.40) = 3.20 lb, exactly as given. The empty box (0.80 lb) is lighter than the box with juice (3.20 lb), which makes sense.

Another way: Use the full weighing directly: box = 3.20 - 6 x 0.40 = 3.20 - 2.40 = 0.80 lb - same answer by Work Backwards from the 6-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 3.20 - 2.80 and multiplying 5 x 0.40.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Variant 10 hard

A box holding 7 identical bottles of juice weighs $1.95 \text{ lb}$ . After taking 1 bottle of juice out of the box, the box weighs $1.80 \text{ lb}$ . How many pounds does the empty box weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A box with 7 identical juice bottles inside weighs 1.95 lb. Take out one bottle and the box now weighs 1.80 lb. All bottles weigh the same. Find the weight of the empty box.

Givens

Box + 7 bottles = 1.95 lb.
Box + 6 bottles = 1.80 lb.
All 7 bottles weigh the same.

Unknowns

The weight of the empty box, in pounds.

Constraints

Each bottle has the same positive weight.
The box has a fixed positive weight.

2 · Planchoose the strategy

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

The only difference between the two weighings is one bottle, so the drop in weight is exactly one bottle. Once one bottle is known, we work backwards from a weighing to peel off the bottles and leave the empty box.

3 · Execute3 carry out the plan

1Find one bottle's weight from the change

#8 Analyze the Units 5.NBT.B.7

Removing exactly one bottle made the weight fall from 1.95 lb to 1.80 lb. That drop is one bottle: 1.95 - 1.80 = 0.15 lb.

1.95 - 1.80 = 0.15\ \text{lb}

The weight that disappeared equals what you took out - one bottle.

2Find the weight of the 6 remaining bottles

#11 Work Backwards 5.NBT.B.7

After removing one bottle, 6 bottles remain in the box. 6 bottles weigh 6 x 0.15 = 0.90 lb.

6 \times 0.15 = 0.90\ \text{lb}

6 equal bottles is just the one-bottle weight added 6 times.

3Subtract the bottles to leave the empty box

#11 Work Backwards 4.MD.A.2

The 1.80 lb weighing is the box plus those 6 bottles. Take the bottles away: 1.80 - 0.90 = 0.90 lb is the empty box.

1.80 - 0.90 = 0.90\ \text{lb}

Removing the bottle weight from the total leaves only the box.

Answer: 0.90 lb

4 · Reviewdoes it hold up?

Check the first weighing: box 0.90 + 7 bottles (7 x 0.15 = 1.05) = 1.95 lb, exactly as given. The empty box (0.90 lb) is lighter than the box with juice (1.95 lb), which makes sense.

Another way: Use the full weighing directly: box = 1.95 - 7 x 0.15 = 1.95 - 1.05 = 0.90 lb - same answer by Work Backwards from the 7-bottle weighing.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 1.95 - 1.80 and multiplying 6 x 0.15.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about the measured weights to isolate the empty box.

💡Takeaway. The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!

Use weight differences to find the empty container · 10 practice problems

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