← Fraction of a quantity by multiplying · Part-Whole Fraction Reasoning

Fraction of a quantity by multiplying · 12 practice problems

5.NF.B.45.NF.B.64.OA.A.3

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Generated variants — 12

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

Variant 1 easy

There are $24$ apples. The number of pears is greater than the number of apples by $\dfrac{3}{4}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 24 apples. There are more pears than apples, and the extra amount is 3 over 4 of the 24 apples. We need the total number of pears.

Givens

There are 24 apples.
The pears exceed the apples by $\frac{3}{4}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 24 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 24 apples' worth, and then some more on top.

\text{pears} = 24 + \frac{3}{4} \times 24

The word 'by' marks the extra, not the whole.

2Find $\frac{3}{4}$ of 24

#9 Solve an Easier Related Problem 5.NF.B.4

Split 24 into 4 equal parts of 6, then take 3 of them.

24 \div 4 = 6,\quad 6 \times 3 = 18

The extra is 18.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 24 apples' worth plus the 18 extra.

24 + 18 = 42

There are 42 pears.

Answer: 42 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 7 over 4 of 24, which is 42 -- the two routes agree.

Another way: Reading '3 over 4 more' as one-and-3-over-4 times as many gives $24 \times \frac{7}{4} = 42$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{3}{4}$ of 24.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 2 easy

There are $28$ apples. The number of pears is greater than the number of apples by $\dfrac{3}{7}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 28 apples. There are more pears than apples, and the extra amount is 3 over 7 of the 28 apples. We need the total number of pears.

Givens

There are 28 apples.
The pears exceed the apples by $\frac{3}{7}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 28 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 28 apples' worth, and then some more on top.

\text{pears} = 28 + \frac{3}{7} \times 28

The word 'by' marks the extra, not the whole.

2Find $\frac{3}{7}$ of 28

#9 Solve an Easier Related Problem 5.NF.B.4

Split 28 into 7 equal parts of 4, then take 3 of them.

28 \div 7 = 4,\quad 4 \times 3 = 12

The extra is 12.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 28 apples' worth plus the 12 extra.

28 + 12 = 40

There are 40 pears.

Answer: 40 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 10 over 7 of 28, which is 40 -- the two routes agree.

Another way: Reading '3 over 7 more' as one-and-3-over-7 times as many gives $28 \times \frac{10}{7} = 40$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{3}{7}$ of 28.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 3 easy

There are $30$ apples. The number of pears is greater than the number of apples by $\dfrac{5}{6}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 30 apples. There are more pears than apples, and the extra amount is 5 over 6 of the 30 apples. We need the total number of pears.

Givens

There are 30 apples.
The pears exceed the apples by $\frac{5}{6}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 30 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 30 apples' worth, and then some more on top.

\text{pears} = 30 + \frac{5}{6} \times 30

The word 'by' marks the extra, not the whole.

2Find $\frac{5}{6}$ of 30

#9 Solve an Easier Related Problem 5.NF.B.4

Split 30 into 6 equal parts of 5, then take 5 of them.

30 \div 6 = 5,\quad 5 \times 5 = 25

The extra is 25.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 30 apples' worth plus the 25 extra.

30 + 25 = 55

There are 55 pears.

Answer: 55 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 11 over 6 of 30, which is 55 -- the two routes agree.

Another way: Reading '5 over 6 more' as one-and-5-over-6 times as many gives $30 \times \frac{11}{6} = 55$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{5}{6}$ of 30.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 4 easy

There are $33$ apples. The number of pears is greater than the number of apples by $\dfrac{8}{11}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 33 apples. There are more pears than apples, and the extra amount is 8 over 11 of the 33 apples. We need the total number of pears.

Givens

There are 33 apples.
The pears exceed the apples by $\frac{8}{11}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 33 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 33 apples' worth, and then some more on top.

\text{pears} = 33 + \frac{8}{11} \times 33

The word 'by' marks the extra, not the whole.

2Find $\frac{8}{11}$ of 33

#9 Solve an Easier Related Problem 5.NF.B.4

Split 33 into 11 equal parts of 3, then take 8 of them.

33 \div 11 = 3,\quad 3 \times 8 = 24

The extra is 24.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 33 apples' worth plus the 24 extra.

33 + 24 = 57

There are 57 pears.

Answer: 57 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 19 over 11 of 33, which is 57 -- the two routes agree.

Another way: Reading '8 over 11 more' as one-and-8-over-11 times as many gives $33 \times \frac{19}{11} = 57$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{8}{11}$ of 33.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 5 medium

There are $36$ apples. The number of pears is greater than the number of apples by $\dfrac{7}{9}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 36 apples. There are more pears than apples, and the extra amount is 7 over 9 of the 36 apples. We need the total number of pears.

Givens

There are 36 apples.
The pears exceed the apples by $\frac{7}{9}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 36 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 36 apples' worth, and then some more on top.

\text{pears} = 36 + \frac{7}{9} \times 36

The word 'by' marks the extra, not the whole.

2Find $\frac{7}{9}$ of 36

#9 Solve an Easier Related Problem 5.NF.B.4

Split 36 into 9 equal parts of 4, then take 7 of them.

36 \div 9 = 4,\quad 4 \times 7 = 28

The extra is 28.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 36 apples' worth plus the 28 extra.

36 + 28 = 64

There are 64 pears.

Answer: 64 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 16 over 9 of 36, which is 64 -- the two routes agree.

Another way: Reading '7 over 9 more' as one-and-7-over-9 times as many gives $36 \times \frac{16}{9} = 64$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{7}{9}$ of 36.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 6 medium

There are $40$ apples. The number of pears is greater than the number of apples by $\dfrac{7}{10}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 40 apples. There are more pears than apples, and the extra amount is 7 over 10 of the 40 apples. We need the total number of pears.

Givens

There are 40 apples.
The pears exceed the apples by $\frac{7}{10}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 40 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 40 apples' worth, and then some more on top.

\text{pears} = 40 + \frac{7}{10} \times 40

The word 'by' marks the extra, not the whole.

2Find $\frac{7}{10}$ of 40

#9 Solve an Easier Related Problem 5.NF.B.4

Split 40 into 10 equal parts of 4, then take 7 of them.

40 \div 10 = 4,\quad 4 \times 7 = 28

The extra is 28.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 40 apples' worth plus the 28 extra.

40 + 28 = 68

There are 68 pears.

Answer: 68 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 17 over 10 of 40, which is 68 -- the two routes agree.

Another way: Reading '7 over 10 more' as one-and-7-over-10 times as many gives $40 \times \frac{17}{10} = 68$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{7}{10}$ of 40.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 7 medium

There are $45$ apples. The number of pears is greater than the number of apples by $\dfrac{2}{5}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 45 apples. There are more pears than apples, and the extra amount is 2 over 5 of the 45 apples. We need the total number of pears.

Givens

There are 45 apples.
The pears exceed the apples by $\frac{2}{5}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 45 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 45 apples' worth, and then some more on top.

\text{pears} = 45 + \frac{2}{5} \times 45

The word 'by' marks the extra, not the whole.

2Find $\frac{2}{5}$ of 45

#9 Solve an Easier Related Problem 5.NF.B.4

Split 45 into 5 equal parts of 9, then take 2 of them.

45 \div 5 = 9,\quad 9 \times 2 = 18

The extra is 18.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 45 apples' worth plus the 18 extra.

45 + 18 = 63

There are 63 pears.

Answer: 63 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 7 over 5 of 45, which is 63 -- the two routes agree.

Another way: Reading '2 over 5 more' as one-and-2-over-5 times as many gives $45 \times \frac{7}{5} = 63$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{2}{5}$ of 45.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 8 medium

There are $48$ apples. The number of pears is greater than the number of apples by $\dfrac{5}{6}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 48 apples. There are more pears than apples, and the extra amount is 5 over 6 of the 48 apples. We need the total number of pears.

Givens

There are 48 apples.
The pears exceed the apples by $\frac{5}{6}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 48 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 48 apples' worth, and then some more on top.

\text{pears} = 48 + \frac{5}{6} \times 48

The word 'by' marks the extra, not the whole.

2Find $\frac{5}{6}$ of 48

#9 Solve an Easier Related Problem 5.NF.B.4

Split 48 into 6 equal parts of 8, then take 5 of them.

48 \div 6 = 8,\quad 8 \times 5 = 40

The extra is 40.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 48 apples' worth plus the 40 extra.

48 + 40 = 88

There are 88 pears.

Answer: 88 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 11 over 6 of 48, which is 88 -- the two routes agree.

Another way: Reading '5 over 6 more' as one-and-5-over-6 times as many gives $48 \times \frac{11}{6} = 88$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{5}{6}$ of 48.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 9 hard

There are $54$ apples. The number of pears is greater than the number of apples by $\dfrac{5}{9}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 54 apples. There are more pears than apples, and the extra amount is 5 over 9 of the 54 apples. We need the total number of pears.

Givens

There are 54 apples.
The pears exceed the apples by $\frac{5}{9}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 54 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 54 apples' worth, and then some more on top.

\text{pears} = 54 + \frac{5}{9} \times 54

The word 'by' marks the extra, not the whole.

2Find $\frac{5}{9}$ of 54

#9 Solve an Easier Related Problem 5.NF.B.4

Split 54 into 9 equal parts of 6, then take 5 of them.

54 \div 9 = 6,\quad 6 \times 5 = 30

The extra is 30.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 54 apples' worth plus the 30 extra.

54 + 30 = 84

There are 84 pears.

Answer: 84 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 14 over 9 of 54, which is 84 -- the two routes agree.

Another way: Reading '5 over 9 more' as one-and-5-over-9 times as many gives $54 \times \frac{14}{9} = 84$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{5}{9}$ of 54.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 10 hard

There are $60$ apples. The number of pears is greater than the number of apples by $\dfrac{4}{5}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 60 apples. There are more pears than apples, and the extra amount is 4 over 5 of the 60 apples. We need the total number of pears.

Givens

There are 60 apples.
The pears exceed the apples by $\frac{4}{5}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 60 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 60 apples' worth, and then some more on top.

\text{pears} = 60 + \frac{4}{5} \times 60

The word 'by' marks the extra, not the whole.

2Find $\frac{4}{5}$ of 60

#9 Solve an Easier Related Problem 5.NF.B.4

Split 60 into 5 equal parts of 12, then take 4 of them.

60 \div 5 = 12,\quad 12 \times 4 = 48

The extra is 48.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 60 apples' worth plus the 48 extra.

60 + 48 = 108

There are 108 pears.

Answer: 108 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 9 over 5 of 60, which is 108 -- the two routes agree.

Another way: Reading '4 over 5 more' as one-and-4-over-5 times as many gives $60 \times \frac{9}{5} = 108$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{4}{5}$ of 60.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 11 hard

There are $63$ apples. The number of pears is greater than the number of apples by $\dfrac{4}{7}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 63 apples. There are more pears than apples, and the extra amount is 4 over 7 of the 63 apples. We need the total number of pears.

Givens

There are 63 apples.
The pears exceed the apples by $\frac{4}{7}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 63 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 63 apples' worth, and then some more on top.

\text{pears} = 63 + \frac{4}{7} \times 63

The word 'by' marks the extra, not the whole.

2Find $\frac{4}{7}$ of 63

#9 Solve an Easier Related Problem 5.NF.B.4

Split 63 into 7 equal parts of 9, then take 4 of them.

63 \div 7 = 9,\quad 9 \times 4 = 36

The extra is 36.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 63 apples' worth plus the 36 extra.

63 + 36 = 99

There are 99 pears.

Answer: 99 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 11 over 7 of 63, which is 99 -- the two routes agree.

Another way: Reading '4 over 7 more' as one-and-4-over-7 times as many gives $63 \times \frac{11}{7} = 99$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{4}{7}$ of 63.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Variant 12 hard

There are $72$ apples. The number of pears is greater than the number of apples by $\dfrac{5}{8}$ of the number of apples.

How many pears are there in all?

Your answer

Show solution

1 · Understandwhat's really being asked

There are 72 apples. There are more pears than apples, and the extra amount is 5 over 8 of the 72 apples. We need the total number of pears.

Givens

There are 72 apples.
The pears exceed the apples by $\frac{5}{8}$ of the apples.

Unknowns

How many pears there are.

Constraints

The fraction is taken of the 72 apples, not of the pears.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The pears are the apples plus an extra part. Work out that extra part on its own -- a fraction of a number we already know -- and then add it on.

3 · Execute3 carry out the plan

1See the total as a base plus an extra part

#7 Identify Subproblems 4.OA.A.3

'Greater than the apples by ...' means the pears are all 72 apples' worth, and then some more on top.

\text{pears} = 72 + \frac{5}{8} \times 72

The word 'by' marks the extra, not the whole.

2Find $\frac{5}{8}$ of 72

#9 Solve an Easier Related Problem 5.NF.B.4

Split 72 into 8 equal parts of 9, then take 5 of them.

72 \div 8 = 9,\quad 9 \times 5 = 45

The extra is 45.

3Add the extra onto the base

#7 Identify Subproblems 5.NF.B.6

The pears are the 72 apples' worth plus the 45 extra.

72 + 45 = 117

There are 117 pears.

Answer: 117 pears

4 · Reviewdoes it hold up?

Adding the extra 25 percent-style share onto the base is the same as taking 13 over 8 of 72, which is 117 -- the two routes agree.

Another way: Reading '5 over 8 more' as one-and-5-over-8 times as many gives $72 \times \frac{13}{8} = 117$ in a single step.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Taking $\frac{5}{8}$ of 72.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Combining the base and the extra share into the total.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Reading 'greater than by' as an addition onto the base.

💡Takeaway. 'More by a fraction of it' means the fraction is of the thing you started with -- so find that part, then add it on.

Where next?

← Back to the problem: Fraction of a quantity by multiplying