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← 4-2 · Use weight differences to find the empty container · Track a Quantity Through Changes

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

5.NBT.B.74.MD.A.2

Generated variants — 10

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

Variant 1 answer: 1.10 lb

A box holding 4 identical bottles of juice weighs 1.98 lb1.98 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.76 lb1.76 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb1.98 - 1.76 = 0.22\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
3×0.22=0.66 lb3 \times 0.22 = 0.66\ \text{lb}
3 equal bottles is just the one-bottle weight added 3 times.
#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 lb1.76 - 0.66 = 1.10\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 1.10 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 2 answer: 0.90 lb

A box holding 7 identical bottles of juice weighs 1.95 lb1.95 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.80 lb1.80 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb1.95 - 1.80 = 0.15\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
6×0.15=0.90 lb6 \times 0.15 = 0.90\ \text{lb}
6 equal bottles is just the one-bottle weight added 6 times.
#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 lb1.80 - 0.90 = 0.90\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.90 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 3 answer: 1.20 lb

A box holding 3 identical bottles of juice weighs 2.25 lb2.25 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.90 lb1.90 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb2.25 - 1.90 = 0.35\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
2×0.35=0.70 lb2 \times 0.35 = 0.70\ \text{lb}
2 equal bottles is just the one-bottle weight added 2 times.
#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 lb1.90 - 0.70 = 1.20\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 1.20 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 4 answer: 0.40 lb

A box holding 6 identical bottles of juice weighs 2.38 lb2.38 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 2.05 lb2.05 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb2.38 - 2.05 = 0.33\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
5×0.33=1.65 lb5 \times 0.33 = 1.65\ \text{lb}
5 equal bottles is just the one-bottle weight added 5 times.
#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 lb2.05 - 1.65 = 0.40\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.40 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 5 answer: 0.50 lb

A box holding 4 identical bottles of juice weighs 1.50 lb1.50 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.25 lb1.25 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb1.50 - 1.25 = 0.25\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
3×0.25=0.75 lb3 \times 0.25 = 0.75\ \text{lb}
3 equal bottles is just the one-bottle weight added 3 times.
#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 lb1.25 - 0.75 = 0.50\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.50 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 6 answer: 0.75 lb

A box holding 5 identical bottles of juice weighs 3.25 lb3.25 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 2.75 lb2.75 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb3.25 - 2.75 = 0.50\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
4×0.50=2.00 lb4 \times 0.50 = 2.00\ \text{lb}
4 equal bottles is just the one-bottle weight added 4 times.
#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 lb2.75 - 2.00 = 0.75\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.75 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 7 answer: 0.46 lb

A box holding 5 identical bottles of juice weighs 1.96 lb1.96 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.66 lb1.66 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb1.96 - 1.66 = 0.30\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
4×0.30=1.20 lb4 \times 0.30 = 1.20\ \text{lb}
4 equal bottles is just the one-bottle weight added 4 times.
#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 lb1.66 - 1.20 = 0.46\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.46 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 8 answer: 0.60 lb

A box holding 4 identical bottles of juice weighs 2.40 lb2.40 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.95 lb1.95 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb2.40 - 1.95 = 0.45\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
3×0.45=1.35 lb3 \times 0.45 = 1.35\ \text{lb}
3 equal bottles is just the one-bottle weight added 3 times.
#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 lb1.95 - 1.35 = 0.60\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.60 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 9 answer: 0.35 lb

A box holding 5 identical bottles of juice weighs 1.35 lb1.35 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 1.15 lb1.15 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb1.35 - 1.15 = 0.20\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
4×0.20=0.80 lb4 \times 0.20 = 0.80\ \text{lb}
4 equal bottles is just the one-bottle weight added 4 times.
#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 lb1.15 - 0.80 = 0.35\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.35 lb

Review

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.

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.

Standards · min 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.
💡 The weight that vanished when you removed one bottle IS one bottle - then just subtract the bottles away!
Variant 10 answer: 0.80 lb

A box holding 6 identical bottles of juice weighs 3.20 lb3.20 \text{ lb}. After taking 1 bottle of juice out of the box, the box weighs 2.80 lb2.80 \text{ lb}. How many pounds does the empty box weigh?

Show solution

Understand

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.

Plan

#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.

Execute

#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 lb3.20 - 2.80 = 0.40\ \text{lb}
The weight that disappeared equals what you took out - one bottle.
#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.
5×0.40=2.00 lb5 \times 0.40 = 2.00\ \text{lb}
5 equal bottles is just the one-bottle weight added 5 times.
#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 lb2.80 - 2.00 = 0.80\ \text{lb}
Removing the bottle weight from the total leaves only the box.
Answer: 0.80 lb

Review

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.

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.

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