← Find empty container weight by subtracting fractions · Track a Quantity Through Changes

Find empty container weight by subtracting fractions · 12 practice problems

5.NF.A.15.NF.A.2

Generated variants — 12

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

Variant 1 easy answer: 35\frac{3}{5} kg

A bottle that is completely full of water weighs 33 kg. After someone drinks half of the water in the bottle, it weighs 1451\dfrac{4}{5} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 33 kg. After half the water is drunk it weighs 1451\dfrac{4}{5} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 33 kg.
  • Bottle with half the water gone: 1451\dfrac{4}{5} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
3−145=1153 - 1\dfrac{4}{5} = 1\dfrac{1}{5}
Half the water weighs 1151\dfrac{1}{5} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
115×2=2251\dfrac{1}{5} \times 2 = 2\dfrac{2}{5}
A full bottle holds 2252\dfrac{2}{5} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
3−225=353 - 2\dfrac{2}{5} = \dfrac{3}{5}
The empty bottle weighs 35\dfrac{3}{5} kg.
Answer: 35\frac{3}{5} kg
4 · Reviewdoes it hold up?

Check both weighings: 35\dfrac{3}{5} + 2252\dfrac{2}{5} = 33, and 35\dfrac{3}{5} + 1151\dfrac{1}{5} = 1451\dfrac{4}{5}. Both match.

Another way: Work from the half-drunk weighing instead: 1451\dfrac{4}{5} - 1151\dfrac{1}{5} = 35\dfrac{3}{5}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 2 easy answer: 34\frac{3}{4} kg

A bottle that is completely full of water weighs 4144\dfrac{1}{4} kg. After someone drinks half of the water in the bottle, it weighs 2122\dfrac{1}{2} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 4144\dfrac{1}{4} kg. After half the water is drunk it weighs 2122\dfrac{1}{2} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 4144\dfrac{1}{4} kg.
  • Bottle with half the water gone: 2122\dfrac{1}{2} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
414−212=1344\dfrac{1}{4} - 2\dfrac{1}{2} = 1\dfrac{3}{4}
Half the water weighs 1341\dfrac{3}{4} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
134×2=3121\dfrac{3}{4} \times 2 = 3\dfrac{1}{2}
A full bottle holds 3123\dfrac{1}{2} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
414−312=344\dfrac{1}{4} - 3\dfrac{1}{2} = \dfrac{3}{4}
The empty bottle weighs 34\dfrac{3}{4} kg.
Answer: 34\frac{3}{4} kg
4 · Reviewdoes it hold up?

Check both weighings: 34\dfrac{3}{4} + 3123\dfrac{1}{2} = 4144\dfrac{1}{4}, and 34\dfrac{3}{4} + 1341\dfrac{3}{4} = 2122\dfrac{1}{2}. Both match.

Another way: Work from the half-drunk weighing instead: 2122\dfrac{1}{2} - 1341\dfrac{3}{4} = 34\dfrac{3}{4}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 3 easy answer: 56\frac{5}{6} kg

A bottle that is completely full of water weighs 4164\dfrac{1}{6} kg. After someone drinks half of the water in the bottle, it weighs 2122\dfrac{1}{2} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 4164\dfrac{1}{6} kg. After half the water is drunk it weighs 2122\dfrac{1}{2} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 4164\dfrac{1}{6} kg.
  • Bottle with half the water gone: 2122\dfrac{1}{2} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
416−212=1234\dfrac{1}{6} - 2\dfrac{1}{2} = 1\dfrac{2}{3}
Half the water weighs 1231\dfrac{2}{3} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
123×2=3131\dfrac{2}{3} \times 2 = 3\dfrac{1}{3}
A full bottle holds 3133\dfrac{1}{3} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
416−313=564\dfrac{1}{6} - 3\dfrac{1}{3} = \dfrac{5}{6}
The empty bottle weighs 56\dfrac{5}{6} kg.
Answer: 56\frac{5}{6} kg
4 · Reviewdoes it hold up?

Check both weighings: 56\dfrac{5}{6} + 3133\dfrac{1}{3} = 4164\dfrac{1}{6}, and 56\dfrac{5}{6} + 1231\dfrac{2}{3} = 2122\dfrac{1}{2}. Both match.

Another way: Work from the half-drunk weighing instead: 2122\dfrac{1}{2} - 1231\dfrac{2}{3} = 56\dfrac{5}{6}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 4 easy answer: 12\frac{1}{2} kg

A bottle that is completely full of water weighs 3343\dfrac{3}{4} kg. After someone drinks half of the water in the bottle, it weighs 2182\dfrac{1}{8} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 3343\dfrac{3}{4} kg. After half the water is drunk it weighs 2182\dfrac{1}{8} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 3343\dfrac{3}{4} kg.
  • Bottle with half the water gone: 2182\dfrac{1}{8} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
334−218=1583\dfrac{3}{4} - 2\dfrac{1}{8} = 1\dfrac{5}{8}
Half the water weighs 1581\dfrac{5}{8} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
158×2=3141\dfrac{5}{8} \times 2 = 3\dfrac{1}{4}
A full bottle holds 3143\dfrac{1}{4} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
334−314=123\dfrac{3}{4} - 3\dfrac{1}{4} = \dfrac{1}{2}
The empty bottle weighs 12\dfrac{1}{2} kg.
Answer: 12\frac{1}{2} kg
4 · Reviewdoes it hold up?

Check both weighings: 12\dfrac{1}{2} + 3143\dfrac{1}{4} = 3343\dfrac{3}{4}, and 12\dfrac{1}{2} + 1581\dfrac{5}{8} = 2182\dfrac{1}{8}. Both match.

Another way: Work from the half-drunk weighing instead: 2182\dfrac{1}{8} - 1581\dfrac{5}{8} = 12\dfrac{1}{2}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 5 medium answer: 710\frac{7}{10} kg

A bottle that is completely full of water weighs 23102\dfrac{3}{10} kg. After someone drinks half of the water in the bottle, it weighs 1121\dfrac{1}{2} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 23102\dfrac{3}{10} kg. After half the water is drunk it weighs 1121\dfrac{1}{2} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 23102\dfrac{3}{10} kg.
  • Bottle with half the water gone: 1121\dfrac{1}{2} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
2310−112=452\dfrac{3}{10} - 1\dfrac{1}{2} = \dfrac{4}{5}
Half the water weighs 45\dfrac{4}{5} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
45×2=135\dfrac{4}{5} \times 2 = 1\dfrac{3}{5}
A full bottle holds 1351\dfrac{3}{5} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
2310−135=7102\dfrac{3}{10} - 1\dfrac{3}{5} = \dfrac{7}{10}
The empty bottle weighs 710\dfrac{7}{10} kg.
Answer: 710\frac{7}{10} kg
4 · Reviewdoes it hold up?

Check both weighings: 710\dfrac{7}{10} + 1351\dfrac{3}{5} = 23102\dfrac{3}{10}, and 710\dfrac{7}{10} + 45\dfrac{4}{5} = 1121\dfrac{1}{2}. Both match.

Another way: Work from the half-drunk weighing instead: 1121\dfrac{1}{2} - 45\dfrac{4}{5} = 710\dfrac{7}{10}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 6 medium answer: 512\frac{5}{12} kg

A bottle that is completely full of water weighs 2342\dfrac{3}{4} kg. After someone drinks half of the water in the bottle, it weighs 17121\dfrac{7}{12} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 2342\dfrac{3}{4} kg. After half the water is drunk it weighs 17121\dfrac{7}{12} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 2342\dfrac{3}{4} kg.
  • Bottle with half the water gone: 17121\dfrac{7}{12} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
234−1712=1162\dfrac{3}{4} - 1\dfrac{7}{12} = 1\dfrac{1}{6}
Half the water weighs 1161\dfrac{1}{6} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
116×2=2131\dfrac{1}{6} \times 2 = 2\dfrac{1}{3}
A full bottle holds 2132\dfrac{1}{3} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
234−213=5122\dfrac{3}{4} - 2\dfrac{1}{3} = \dfrac{5}{12}
The empty bottle weighs 512\dfrac{5}{12} kg.
Answer: 512\frac{5}{12} kg
4 · Reviewdoes it hold up?

Check both weighings: 512\dfrac{5}{12} + 2132\dfrac{1}{3} = 2342\dfrac{3}{4}, and 512\dfrac{5}{12} + 1161\dfrac{1}{6} = 17121\dfrac{7}{12}. Both match.

Another way: Work from the half-drunk weighing instead: 17121\dfrac{7}{12} - 1161\dfrac{1}{6} = 512\dfrac{5}{12}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 7 hard answer: 13\frac{1}{3} kg

A bottle that is completely full of water weighs 31123\dfrac{1}{12} kg. After someone drinks half of the water in the bottle, it weighs 117241\dfrac{17}{24} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 31123\dfrac{1}{12} kg. After half the water is drunk it weighs 117241\dfrac{17}{24} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 31123\dfrac{1}{12} kg.
  • Bottle with half the water gone: 117241\dfrac{17}{24} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
3112−11724=1383\dfrac{1}{12} - 1\dfrac{17}{24} = 1\dfrac{3}{8}
Half the water weighs 1381\dfrac{3}{8} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
138×2=2341\dfrac{3}{8} \times 2 = 2\dfrac{3}{4}
A full bottle holds 2342\dfrac{3}{4} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
3112−234=133\dfrac{1}{12} - 2\dfrac{3}{4} = \dfrac{1}{3}
The empty bottle weighs 13\dfrac{1}{3} kg.
Answer: 13\frac{1}{3} kg
4 · Reviewdoes it hold up?

Check both weighings: 13\dfrac{1}{3} + 2342\dfrac{3}{4} = 31123\dfrac{1}{12}, and 13\dfrac{1}{3} + 1381\dfrac{3}{8} = 117241\dfrac{17}{24}. Both match.

Another way: Work from the half-drunk weighing instead: 117241\dfrac{17}{24} - 1381\dfrac{3}{8} = 13\dfrac{1}{3}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 8 medium answer: 23\frac{2}{3} kg

A bottle that is completely full of water weighs 211122\dfrac{11}{12} kg. After someone drinks half of the water in the bottle, it weighs 119241\dfrac{19}{24} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 211122\dfrac{11}{12} kg. After half the water is drunk it weighs 119241\dfrac{19}{24} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 211122\dfrac{11}{12} kg.
  • Bottle with half the water gone: 119241\dfrac{19}{24} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
21112−11924=1182\dfrac{11}{12} - 1\dfrac{19}{24} = 1\dfrac{1}{8}
Half the water weighs 1181\dfrac{1}{8} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
118×2=2141\dfrac{1}{8} \times 2 = 2\dfrac{1}{4}
A full bottle holds 2142\dfrac{1}{4} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
21112−214=232\dfrac{11}{12} - 2\dfrac{1}{4} = \dfrac{2}{3}
The empty bottle weighs 23\dfrac{2}{3} kg.
Answer: 23\frac{2}{3} kg
4 · Reviewdoes it hold up?

Check both weighings: 23\dfrac{2}{3} + 2142\dfrac{1}{4} = 211122\dfrac{11}{12}, and 23\dfrac{2}{3} + 1181\dfrac{1}{8} = 119241\dfrac{19}{24}. Both match.

Another way: Work from the half-drunk weighing instead: 119241\dfrac{19}{24} - 1181\dfrac{1}{8} = 23\dfrac{2}{3}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 9 medium answer: 58\frac{5}{8} kg

A bottle that is completely full of water weighs 47244\dfrac{7}{24} kg. After someone drinks half of the water in the bottle, it weighs 211242\dfrac{11}{24} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 47244\dfrac{7}{24} kg. After half the water is drunk it weighs 211242\dfrac{11}{24} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 47244\dfrac{7}{24} kg.
  • Bottle with half the water gone: 211242\dfrac{11}{24} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
4724−21124=1564\dfrac{7}{24} - 2\dfrac{11}{24} = 1\dfrac{5}{6}
Half the water weighs 1561\dfrac{5}{6} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
156×2=3231\dfrac{5}{6} \times 2 = 3\dfrac{2}{3}
A full bottle holds 3233\dfrac{2}{3} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
4724−323=584\dfrac{7}{24} - 3\dfrac{2}{3} = \dfrac{5}{8}
The empty bottle weighs 58\dfrac{5}{8} kg.
Answer: 58\frac{5}{8} kg
4 · Reviewdoes it hold up?

Check both weighings: 58\dfrac{5}{8} + 3233\dfrac{2}{3} = 47244\dfrac{7}{24}, and 58\dfrac{5}{8} + 1561\dfrac{5}{6} = 211242\dfrac{11}{24}. Both match.

Another way: Work from the half-drunk weighing instead: 211242\dfrac{11}{24} - 1561\dfrac{5}{6} = 58\dfrac{5}{8}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 10 hard answer: 27\frac{2}{7} kg

A bottle that is completely full of water weighs 411144\dfrac{11}{14} kg. After someone drinks half of the water in the bottle, it weighs 215282\dfrac{15}{28} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 411144\dfrac{11}{14} kg. After half the water is drunk it weighs 215282\dfrac{15}{28} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 411144\dfrac{11}{14} kg.
  • Bottle with half the water gone: 215282\dfrac{15}{28} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
41114−21528=2144\dfrac{11}{14} - 2\dfrac{15}{28} = 2\dfrac{1}{4}
Half the water weighs 2142\dfrac{1}{4} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
214×2=4122\dfrac{1}{4} \times 2 = 4\dfrac{1}{2}
A full bottle holds 4124\dfrac{1}{2} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
41114−412=274\dfrac{11}{14} - 4\dfrac{1}{2} = \dfrac{2}{7}
The empty bottle weighs 27\dfrac{2}{7} kg.
Answer: 27\frac{2}{7} kg
4 · Reviewdoes it hold up?

Check both weighings: 27\dfrac{2}{7} + 4124\dfrac{1}{2} = 411144\dfrac{11}{14}, and 27\dfrac{2}{7} + 2142\dfrac{1}{4} = 215282\dfrac{15}{28}. Both match.

Another way: Work from the half-drunk weighing instead: 215282\dfrac{15}{28} - 2142\dfrac{1}{4} = 27\dfrac{2}{7}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 11 hard answer: 49\frac{4}{9} kg

A bottle that is completely full of water weighs 217182\dfrac{17}{18} kg. After someone drinks half of the water in the bottle, it weighs 125361\dfrac{25}{36} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 217182\dfrac{17}{18} kg. After half the water is drunk it weighs 125361\dfrac{25}{36} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 217182\dfrac{17}{18} kg.
  • Bottle with half the water gone: 125361\dfrac{25}{36} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
21718−12536=1142\dfrac{17}{18} - 1\dfrac{25}{36} = 1\dfrac{1}{4}
Half the water weighs 1141\dfrac{1}{4} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
114×2=2121\dfrac{1}{4} \times 2 = 2\dfrac{1}{2}
A full bottle holds 2122\dfrac{1}{2} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
21718−212=492\dfrac{17}{18} - 2\dfrac{1}{2} = \dfrac{4}{9}
The empty bottle weighs 49\dfrac{4}{9} kg.
Answer: 49\frac{4}{9} kg
4 · Reviewdoes it hold up?

Check both weighings: 49\dfrac{4}{9} + 2122\dfrac{1}{2} = 217182\dfrac{17}{18}, and 49\dfrac{4}{9} + 1141\dfrac{1}{4} = 125361\dfrac{25}{36}. Both match.

Another way: Work from the half-drunk weighing instead: 125361\dfrac{25}{36} - 1141\dfrac{1}{4} = 49\dfrac{4}{9}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.
Variant 12 hard answer: 78\frac{7}{8} kg

A bottle that is completely full of water weighs 23402\dfrac{3}{40} kg. After someone drinks half of the water in the bottle, it weighs 119401\dfrac{19}{40} kg. How many kilograms does the empty bottle weigh?

Show solution
1 · Understandwhat's really being asked

A full bottle weighs 23402\dfrac{3}{40} kg. After half the water is drunk it weighs 119401\dfrac{19}{40} kg. We need the weight of the bottle with no water in it.

Givens
  • Full bottle: 23402\dfrac{3}{40} kg.
  • Bottle with half the water gone: 119401\dfrac{19}{40} kg.
Unknowns
  • The weight of the empty bottle.
Constraints
  • The bottle itself weighs the same in both weighings; only water left.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Only water left between the two weighings, so their difference is water and nothing else. That gives half the water, then all of it, and the bottle is what remains.

3 · Execute3 carry out the plan

1Find the weight of half the water

#11 Work Backwards 5.NF.A.1
The drop between the two weighings is exactly the water that was drunk -- the bottle cancels out.
2340−11940=352\dfrac{3}{40} - 1\dfrac{19}{40} = \dfrac{3}{5}
Half the water weighs 35\dfrac{3}{5} kg.

2Find the weight of all the water

#7 Identify Subproblems 5.NF.A.2
If half weighs that much, the whole is twice as much.
35×2=115\dfrac{3}{5} \times 2 = 1\dfrac{1}{5}
A full bottle holds 1151\dfrac{1}{5} kg of water.

3Subtract the water from the full bottle

#11 Work Backwards 5.NF.A.2
Taking all the water off the full weight leaves the bottle.
2340−115=782\dfrac{3}{40} - 1\dfrac{1}{5} = \dfrac{7}{8}
The empty bottle weighs 78\dfrac{7}{8} kg.
Answer: 78\frac{7}{8} kg
4 · Reviewdoes it hold up?

Check both weighings: 78\dfrac{7}{8} + 1151\dfrac{1}{5} = 23402\dfrac{3}{40}, and 78\dfrac{7}{8} + 35\dfrac{3}{5} = 119401\dfrac{19}{40}. Both match.

Another way: Work from the half-drunk weighing instead: 119401\dfrac{19}{40} - 35\dfrac{3}{5} = 78\dfrac{7}{8}, one step shorter.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting the two weighings over a common denominator.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Doubling to the full water weight and removing it from the total.
💡Takeaway. When the same container sits in both weighings, subtracting them cancels it -- what is left is only what changed.