Practice · Find empty container weight by subtracting fractions · Sensim Math

Variant 1 easy

A bottle that is completely full of water weighs $3$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{4}{5}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $3$ kg. After half the water is drunk it weighs $1\dfrac{4}{5}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $3$ kg.
Bottle with half the water gone: $1\dfrac{4}{5}$ 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 - 1\dfrac{4}{5} = 1\dfrac{1}{5}

Half the water weighs

1\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.

1\dfrac{1}{5} \times 2 = 2\dfrac{2}{5}

A full bottle holds

2\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 - 2\dfrac{2}{5} = \dfrac{3}{5}

The empty bottle weighs

\dfrac{3}{5}

kg.

Answer:

\frac{3}{5}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{3}{5}$ + $2\dfrac{2}{5}$ = $3$ , and $\dfrac{3}{5}$ + $1\dfrac{1}{5}$ = $1\dfrac{4}{5}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{4}{5}$ - $1\dfrac{1}{5}$ = $\dfrac{3}{5}$ , 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

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

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

Full bottle: $4\dfrac{1}{4}$ kg.
Bottle with half the water gone: $2\dfrac{1}{2}$ 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.

4\dfrac{1}{4} - 2\dfrac{1}{2} = 1\dfrac{3}{4}

Half the water weighs

1\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.

1\dfrac{3}{4} \times 2 = 3\dfrac{1}{2}

A full bottle holds

3\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.

4\dfrac{1}{4} - 3\dfrac{1}{2} = \dfrac{3}{4}

The empty bottle weighs

\dfrac{3}{4}

kg.

Answer:

\frac{3}{4}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{3}{4}$ + $3\dfrac{1}{2}$ = $4\dfrac{1}{4}$ , and $\dfrac{3}{4}$ + $1\dfrac{3}{4}$ = $2\dfrac{1}{2}$ . Both match.

Another way: Work from the half-drunk weighing instead: $2\dfrac{1}{2}$ - $1\dfrac{3}{4}$ = $\dfrac{3}{4}$ , 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

A bottle that is completely full of water weighs $4\dfrac{1}{6}$ kg. After someone drinks half of the water in the bottle, it weighs $2\dfrac{1}{2}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $4\dfrac{1}{6}$ kg. After half the water is drunk it weighs $2\dfrac{1}{2}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $4\dfrac{1}{6}$ kg.
Bottle with half the water gone: $2\dfrac{1}{2}$ 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.

4\dfrac{1}{6} - 2\dfrac{1}{2} = 1\dfrac{2}{3}

Half the water weighs

1\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.

1\dfrac{2}{3} \times 2 = 3\dfrac{1}{3}

A full bottle holds

3\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.

4\dfrac{1}{6} - 3\dfrac{1}{3} = \dfrac{5}{6}

The empty bottle weighs

\dfrac{5}{6}

kg.

Answer:

\frac{5}{6}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{5}{6}$ + $3\dfrac{1}{3}$ = $4\dfrac{1}{6}$ , and $\dfrac{5}{6}$ + $1\dfrac{2}{3}$ = $2\dfrac{1}{2}$ . Both match.

Another way: Work from the half-drunk weighing instead: $2\dfrac{1}{2}$ - $1\dfrac{2}{3}$ = $\dfrac{5}{6}$ , 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

A bottle that is completely full of water weighs $3\dfrac{3}{4}$ kg. After someone drinks half of the water in the bottle, it weighs $2\dfrac{1}{8}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $3\dfrac{3}{4}$ kg. After half the water is drunk it weighs $2\dfrac{1}{8}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $3\dfrac{3}{4}$ kg.
Bottle with half the water gone: $2\dfrac{1}{8}$ 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.

3\dfrac{3}{4} - 2\dfrac{1}{8} = 1\dfrac{5}{8}

Half the water weighs

1\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.

1\dfrac{5}{8} \times 2 = 3\dfrac{1}{4}

A full bottle holds

3\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.

3\dfrac{3}{4} - 3\dfrac{1}{4} = \dfrac{1}{2}

The empty bottle weighs

\dfrac{1}{2}

kg.

Answer:

\frac{1}{2}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{1}{2}$ + $3\dfrac{1}{4}$ = $3\dfrac{3}{4}$ , and $\dfrac{1}{2}$ + $1\dfrac{5}{8}$ = $2\dfrac{1}{8}$ . Both match.

Another way: Work from the half-drunk weighing instead: $2\dfrac{1}{8}$ - $1\dfrac{5}{8}$ = $\dfrac{1}{2}$ , 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

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

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

Full bottle: $2\dfrac{3}{10}$ kg.
Bottle with half the water gone: $1\dfrac{1}{2}$ 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.

2\dfrac{3}{10} - 1\dfrac{1}{2} = \dfrac{4}{5}

Half the water weighs

\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.

\dfrac{4}{5} \times 2 = 1\dfrac{3}{5}

A full bottle holds

1\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.

2\dfrac{3}{10} - 1\dfrac{3}{5} = \dfrac{7}{10}

The empty bottle weighs

\dfrac{7}{10}

kg.

Answer:

\frac{7}{10}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{7}{10}$ + $1\dfrac{3}{5}$ = $2\dfrac{3}{10}$ , and $\dfrac{7}{10}$ + $\dfrac{4}{5}$ = $1\dfrac{1}{2}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{1}{2}$ - $\dfrac{4}{5}$ = $\dfrac{7}{10}$ , 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

A bottle that is completely full of water weighs $2\dfrac{3}{4}$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{7}{12}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $2\dfrac{3}{4}$ kg. After half the water is drunk it weighs $1\dfrac{7}{12}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $2\dfrac{3}{4}$ kg.
Bottle with half the water gone: $1\dfrac{7}{12}$ 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.

2\dfrac{3}{4} - 1\dfrac{7}{12} = 1\dfrac{1}{6}

Half the water weighs

1\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.

1\dfrac{1}{6} \times 2 = 2\dfrac{1}{3}

A full bottle holds

2\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.

2\dfrac{3}{4} - 2\dfrac{1}{3} = \dfrac{5}{12}

The empty bottle weighs

\dfrac{5}{12}

kg.

Answer:

\frac{5}{12}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{5}{12}$ + $2\dfrac{1}{3}$ = $2\dfrac{3}{4}$ , and $\dfrac{5}{12}$ + $1\dfrac{1}{6}$ = $1\dfrac{7}{12}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{7}{12}$ - $1\dfrac{1}{6}$ = $\dfrac{5}{12}$ , 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

A bottle that is completely full of water weighs $3\dfrac{1}{12}$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{17}{24}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $3\dfrac{1}{12}$ kg. After half the water is drunk it weighs $1\dfrac{17}{24}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $3\dfrac{1}{12}$ kg.
Bottle with half the water gone: $1\dfrac{17}{24}$ 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.

3\dfrac{1}{12} - 1\dfrac{17}{24} = 1\dfrac{3}{8}

Half the water weighs

1\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.

1\dfrac{3}{8} \times 2 = 2\dfrac{3}{4}

A full bottle holds

2\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.

3\dfrac{1}{12} - 2\dfrac{3}{4} = \dfrac{1}{3}

The empty bottle weighs

\dfrac{1}{3}

kg.

Answer:

\frac{1}{3}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{1}{3}$ + $2\dfrac{3}{4}$ = $3\dfrac{1}{12}$ , and $\dfrac{1}{3}$ + $1\dfrac{3}{8}$ = $1\dfrac{17}{24}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{17}{24}$ - $1\dfrac{3}{8}$ = $\dfrac{1}{3}$ , 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

A bottle that is completely full of water weighs $2\dfrac{11}{12}$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{19}{24}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $2\dfrac{11}{12}$ kg. After half the water is drunk it weighs $1\dfrac{19}{24}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $2\dfrac{11}{12}$ kg.
Bottle with half the water gone: $1\dfrac{19}{24}$ 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.

2\dfrac{11}{12} - 1\dfrac{19}{24} = 1\dfrac{1}{8}

Half the water weighs

1\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.

1\dfrac{1}{8} \times 2 = 2\dfrac{1}{4}

A full bottle holds

2\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.

2\dfrac{11}{12} - 2\dfrac{1}{4} = \dfrac{2}{3}

The empty bottle weighs

\dfrac{2}{3}

kg.

Answer:

\frac{2}{3}

kg

4 · Reviewdoes it hold up?

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

Another way: Work from the half-drunk weighing instead: $1\dfrac{19}{24}$ - $1\dfrac{1}{8}$ = $\dfrac{2}{3}$ , 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

A bottle that is completely full of water weighs $4\dfrac{7}{24}$ kg. After someone drinks half of the water in the bottle, it weighs $2\dfrac{11}{24}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $4\dfrac{7}{24}$ kg. After half the water is drunk it weighs $2\dfrac{11}{24}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $4\dfrac{7}{24}$ kg.
Bottle with half the water gone: $2\dfrac{11}{24}$ 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.

4\dfrac{7}{24} - 2\dfrac{11}{24} = 1\dfrac{5}{6}

Half the water weighs

1\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.

1\dfrac{5}{6} \times 2 = 3\dfrac{2}{3}

A full bottle holds

3\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.

4\dfrac{7}{24} - 3\dfrac{2}{3} = \dfrac{5}{8}

The empty bottle weighs

\dfrac{5}{8}

kg.

Answer:

\frac{5}{8}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{5}{8}$ + $3\dfrac{2}{3}$ = $4\dfrac{7}{24}$ , and $\dfrac{5}{8}$ + $1\dfrac{5}{6}$ = $2\dfrac{11}{24}$ . Both match.

Another way: Work from the half-drunk weighing instead: $2\dfrac{11}{24}$ - $1\dfrac{5}{6}$ = $\dfrac{5}{8}$ , 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

A bottle that is completely full of water weighs $4\dfrac{11}{14}$ kg. After someone drinks half of the water in the bottle, it weighs $2\dfrac{15}{28}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $4\dfrac{11}{14}$ kg. After half the water is drunk it weighs $2\dfrac{15}{28}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $4\dfrac{11}{14}$ kg.
Bottle with half the water gone: $2\dfrac{15}{28}$ 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.

4\dfrac{11}{14} - 2\dfrac{15}{28} = 2\dfrac{1}{4}

Half the water weighs

2\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.

2\dfrac{1}{4} \times 2 = 4\dfrac{1}{2}

A full bottle holds

4\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.

4\dfrac{11}{14} - 4\dfrac{1}{2} = \dfrac{2}{7}

The empty bottle weighs

\dfrac{2}{7}

kg.

Answer:

\frac{2}{7}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{2}{7}$ + $4\dfrac{1}{2}$ = $4\dfrac{11}{14}$ , and $\dfrac{2}{7}$ + $2\dfrac{1}{4}$ = $2\dfrac{15}{28}$ . Both match.

Another way: Work from the half-drunk weighing instead: $2\dfrac{15}{28}$ - $2\dfrac{1}{4}$ = $\dfrac{2}{7}$ , 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

A bottle that is completely full of water weighs $2\dfrac{17}{18}$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{25}{36}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $2\dfrac{17}{18}$ kg. After half the water is drunk it weighs $1\dfrac{25}{36}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $2\dfrac{17}{18}$ kg.
Bottle with half the water gone: $1\dfrac{25}{36}$ 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.

2\dfrac{17}{18} - 1\dfrac{25}{36} = 1\dfrac{1}{4}

Half the water weighs

1\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.

1\dfrac{1}{4} \times 2 = 2\dfrac{1}{2}

A full bottle holds

2\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.

2\dfrac{17}{18} - 2\dfrac{1}{2} = \dfrac{4}{9}

The empty bottle weighs

\dfrac{4}{9}

kg.

Answer:

\frac{4}{9}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{4}{9}$ + $2\dfrac{1}{2}$ = $2\dfrac{17}{18}$ , and $\dfrac{4}{9}$ + $1\dfrac{1}{4}$ = $1\dfrac{25}{36}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{25}{36}$ - $1\dfrac{1}{4}$ = $\dfrac{4}{9}$ , 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

A bottle that is completely full of water weighs $2\dfrac{3}{40}$ kg. After someone drinks half of the water in the bottle, it weighs $1\dfrac{19}{40}$ kg. How many kilograms does the empty bottle weigh?

Your answer

Show solution

1 · Understandwhat's really being asked

A full bottle weighs $2\dfrac{3}{40}$ kg. After half the water is drunk it weighs $1\dfrac{19}{40}$ kg. We need the weight of the bottle with no water in it.

Givens

Full bottle: $2\dfrac{3}{40}$ kg.
Bottle with half the water gone: $1\dfrac{19}{40}$ 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.

2\dfrac{3}{40} - 1\dfrac{19}{40} = \dfrac{3}{5}

Half the water weighs

\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.

\dfrac{3}{5} \times 2 = 1\dfrac{1}{5}

A full bottle holds

1\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.

2\dfrac{3}{40} - 1\dfrac{1}{5} = \dfrac{7}{8}

The empty bottle weighs

\dfrac{7}{8}

kg.

Answer:

\frac{7}{8}

kg

4 · Reviewdoes it hold up?

Check both weighings: $\dfrac{7}{8}$ + $1\dfrac{1}{5}$ = $2\dfrac{3}{40}$ , and $\dfrac{7}{8}$ + $\dfrac{3}{5}$ = $1\dfrac{19}{40}$ . Both match.

Another way: Work from the half-drunk weighing instead: $1\dfrac{19}{40}$ - $\dfrac{3}{5}$ = $\dfrac{7}{8}$ , 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.

Find empty container weight by subtracting fractions · 12 practice problems

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