← 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

Generated variants — 12

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

Variant 1 easy answer: 42 pears

There are 2424 apples. The number of pears is greater than the number of apples by 34\dfrac{3}{4} of the number of apples.

How many pears are there in all?

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 34\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.
pears=24+34×24\text{pears} = 24 + \frac{3}{4} \times 24
The word 'by' marks the extra, not the whole.

2Find 34\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÷4=6,6×3=1824 \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=4224 + 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 24×74=4224 \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 answer: 40 pears

There are 2828 apples. The number of pears is greater than the number of apples by 37\dfrac{3}{7} of the number of apples.

How many pears are there in all?

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 37\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.
pears=28+37×28\text{pears} = 28 + \frac{3}{7} \times 28
The word 'by' marks the extra, not the whole.

2Find 37\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÷7=4,4×3=1228 \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=4028 + 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 28×107=4028 \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 answer: 55 pears

There are 3030 apples. The number of pears is greater than the number of apples by 56\dfrac{5}{6} of the number of apples.

How many pears are there in all?

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 56\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.
pears=30+56×30\text{pears} = 30 + \frac{5}{6} \times 30
The word 'by' marks the extra, not the whole.

2Find 56\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÷6=5,5×5=2530 \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=5530 + 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 30×116=5530 \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 answer: 57 pears

There are 3333 apples. The number of pears is greater than the number of apples by 811\dfrac{8}{11} of the number of apples.

How many pears are there in all?

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 811\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.
pears=33+811×33\text{pears} = 33 + \frac{8}{11} \times 33
The word 'by' marks the extra, not the whole.

2Find 811\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÷11=3,3×8=2433 \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=5733 + 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 33×1911=5733 \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 answer: 64 pears

There are 3636 apples. The number of pears is greater than the number of apples by 79\dfrac{7}{9} of the number of apples.

How many pears are there in all?

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 79\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.
pears=36+79×36\text{pears} = 36 + \frac{7}{9} \times 36
The word 'by' marks the extra, not the whole.

2Find 79\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÷9=4,4×7=2836 \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=6436 + 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 36×169=6436 \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 answer: 68 pears

There are 4040 apples. The number of pears is greater than the number of apples by 710\dfrac{7}{10} of the number of apples.

How many pears are there in all?

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 710\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.
pears=40+710×40\text{pears} = 40 + \frac{7}{10} \times 40
The word 'by' marks the extra, not the whole.

2Find 710\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÷10=4,4×7=2840 \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=6840 + 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 40×1710=6840 \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 answer: 63 pears

There are 4545 apples. The number of pears is greater than the number of apples by 25\dfrac{2}{5} of the number of apples.

How many pears are there in all?

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 25\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.
pears=45+25×45\text{pears} = 45 + \frac{2}{5} \times 45
The word 'by' marks the extra, not the whole.

2Find 25\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÷5=9,9×2=1845 \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=6345 + 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 45×75=6345 \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 answer: 88 pears

There are 4848 apples. The number of pears is greater than the number of apples by 56\dfrac{5}{6} of the number of apples.

How many pears are there in all?

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 56\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.
pears=48+56×48\text{pears} = 48 + \frac{5}{6} \times 48
The word 'by' marks the extra, not the whole.

2Find 56\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÷6=8,8×5=4048 \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=8848 + 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 48×116=8848 \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 answer: 84 pears

There are 5454 apples. The number of pears is greater than the number of apples by 59\dfrac{5}{9} of the number of apples.

How many pears are there in all?

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 59\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.
pears=54+59×54\text{pears} = 54 + \frac{5}{9} \times 54
The word 'by' marks the extra, not the whole.

2Find 59\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÷9=6,6×5=3054 \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=8454 + 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 54×149=8454 \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 answer: 108 pears

There are 6060 apples. The number of pears is greater than the number of apples by 45\dfrac{4}{5} of the number of apples.

How many pears are there in all?

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 45\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.
pears=60+45×60\text{pears} = 60 + \frac{4}{5} \times 60
The word 'by' marks the extra, not the whole.

2Find 45\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÷5=12,12×4=4860 \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=10860 + 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 60×95=10860 \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 answer: 99 pears

There are 6363 apples. The number of pears is greater than the number of apples by 47\dfrac{4}{7} of the number of apples.

How many pears are there in all?

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 47\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.
pears=63+47×63\text{pears} = 63 + \frac{4}{7} \times 63
The word 'by' marks the extra, not the whole.

2Find 47\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÷7=9,9×4=3663 \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=9963 + 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 63×117=9963 \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 answer: 117 pears

There are 7272 apples. The number of pears is greater than the number of apples by 58\dfrac{5}{8} of the number of apples.

How many pears are there in all?

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 58\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.
pears=72+58×72\text{pears} = 72 + \frac{5}{8} \times 72
The word 'by' marks the extra, not the whole.

2Find 58\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÷8=9,9×5=4572 \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=11772 + 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 72×138=11772 \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.