# Fractions Of Numbers

Here we will learn about fractions of numbers and how to solve problems involving fractions of numbers.

Students first learn about fractions of numbers in fourth grade in their work with number and operations – fractions. They will extend this understanding as they progress through 5th and 6th grade.

## What are fractions of numbers?

Fractions of numbers are calculated when we multiply a fractional number by the whole number. The word “of” means to multiply.

For example, let’s look at how to use a visual model and the algorithm to calculate fractions of numbers.

## Common Core Standards

How does this relate to 4th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.B.4.a)
Understand a fraction \cfrac{a}{b} as a multiple of \cfrac{1}{b}. For example, use a visual fraction model to represent \cfrac{5}{4} as the product 5 \times (\cfrac{1}{4}), recording the conclusion by the equation \cfrac{5}{4} = 5 \times (\cfrac{1}{4}).

• Grade 4 – Number and Operations – Fractions (4.NF.B.4.b)
Understand a multiple of \cfrac{a}{b} as a multiple of \cfrac{1}{b}, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 \times (\cfrac{2}{5}) as 6 \times (\cfrac{1}{5}), recognizing this product as \cfrac{6}{5}. (In general, n \times (\cfrac{a}{b}) = \cfrac{(n \, \times \, a)}{b}.)

## How to calculate fractions of numbers

In order to find a fraction of a number using a model.

1. Draw one fraction horizontally and the other vertically.
2. Connect the fractions all the way across with 2 different colors.
3. Count up the shaded, overlapping parts.
4. If possible, simplify or convert to a mixed number.

In order to find a fraction of a number using the algorithm

1. Convert to a multiplication statement.
2. Convert the whole number to an improper fraction.
3. Multiply the numerators together and the denominators together.
4. If possible, simplify or convert to a whole number or mixed number.

## Fractions of numbers examples

### Example 1: unit fraction of a number using a model

\cfrac{1}{4} \, of 7

1. Draw one fraction horizontally and the other vertically.

7 or \, \cfrac{7}{1} \, is drawn vertically and \, \cfrac{1}{4} \, is drawn horizontally which is 7 wholes divided into 4 equal pieces.

2Connect the fractions all the way across with 2 different colors.

3Count up the shaded, overlapping parts.

7 groups of \, \cfrac{1}{4} \, represents the overlap shaded pieces which is

\cfrac{1}{4}+\cfrac{1}{4}+\cfrac{1}{4}+\cfrac{1}{4}+\cfrac{1}{4}+\cfrac{1}{4}+\cfrac{1}{4}=\cfrac{7}{4}

4If possible, simplify or convert to a mixed number.

There is no common factor between 7 and 4 which means \, \cfrac{7}{4} \, is in its simplest form.

\cfrac{7}{4} \, as a mixed number is 1\cfrac{3}{4}

\cfrac{1}{4} \, of 7=1\cfrac{3}{4}

### Example 2: unit fraction of a number

\cfrac{1}{2} \, of 16

\cfrac{1}{2} \times 16

16 as an improper fraction is \, \cfrac{16}{1}

\cfrac{1}{2} \times \cfrac{16}{1}=\cfrac{1 \, \times \, 16}{2 \, \times \, 1}

\cfrac{16}{2} \, has a common factor of 2

\cfrac{16\div 2}{2\div 2}=\cfrac{8}{1} = 8

### Example 3: proper fraction of a number using a model

\cfrac{3}{10} \, of 5

5 or \cfrac{5}{1} draw vertically and \cfrac{1}{10} drawn horizontally which is 5 wholes divided into 10 equal pieces

5 groups of \cfrac{3}{10} \, represent the overlap shaded pieces which is

\cfrac{3}{10}+\cfrac{3}{10}+\cfrac{3}{10}+\cfrac{3}{10}+\cfrac{3}{10}=\cfrac{15}{10}

The common factor between \cfrac{15}{10} \, is 5.

\cfrac{15 \, \div \, 5}{10 \, \div \, 5}=\cfrac{3}{2}=1\cfrac{1}{2}

### Example 4: proper fractions of numbers

Find \, \cfrac{2}{7} \, of 28

\cfrac{2}{7} \times 28

\cfrac{2}{7}\times \cfrac{28}{1}

\cfrac{2}{7}\times \cfrac{28}{1} = \cfrac{2 \, \times \, 28}{7 \, \times \, 1}=\cfrac{56}{7}

The common factor between \, \cfrac{56}{7} \, is 7.

\cfrac{56 \, \div \, 7}{7 \, \div \, 7}=\cfrac{8}{1}=8

\cfrac{2}{7} \, of 28 is 8

### Example 5: proper fraction of a number

What is \, \cfrac{3}{4} \, of 20?

\cfrac{3}{4}\times 20

\cfrac{3}{4}\times \cfrac{20}{1}

\cfrac{3}{4}\times \cfrac{20}{1}= \cfrac{3\times 20}{4\times1}= \cfrac{60}{4}

The common factor between 4 and 60 is 4.

\cfrac{60 \, \div \, 4}{4 \, \div \, 4}=\cfrac{15}{1}=15

\cfrac{3}{4} \, of 20 is 15

### Example 6: word problem

Jenny went hiking and only walked two thirds of the 24 mile hiking trail. How many miles did she hike?

\cfrac{2}{3} \, of 24 is \, \cfrac{2}{3} \times 24

\cfrac{2}{3} \times\cfrac{24}{1}

\cfrac{2}{3} \times\cfrac{24}{1} = \cfrac{2 \, \times \, 24}{3 \, \times \, 1} =\cfrac{48}{3}

The common factor between 48 and 3 is 3.

\cfrac{48 \, \div \, 3}{3 \, \div \, 3}=\cfrac{16}{1}=16

Two thirds of 24 is 16, so she hiked 16 miles

### Example 7: word problem

Dylan only filled \, \cfrac{2}{5} \, of his 20 -ounce water bottle. How much water is there in the bottle?

\cfrac{2}{5}\times 20

\cfrac{2}{5}\times \cfrac{20}{1}

\cfrac{2}{5}\times \cfrac{20}{1}=\cfrac{2 \, \times \, 20}{5 \, \times \, 1}=\cfrac{40}{5}

The common factor between 40 and 5 is 5.

\cfrac{40 \, \div \, 5}{5 \, \div \, 5}= \cfrac{8}{1}= 8

\cfrac{2}{5} \, of 20 is 8, so there are 8 ounces of water in the bottle.

### Teaching tips for a fraction of a number

• Use visual models so students can develop conceptual understanding.

• Explore patterns so that students can make sense of the simple steps involved in multiplying a fraction by a whole number.

• Use a number line to show student

• Although practice worksheets have their place, reinforcing skills with visual models and hands-on activities is more effective for students to formulate deep understanding.

### Easy mistakes to make

• Confusing the meaning of the word “of”
Thinking “of” means division instead of multiplication.
For example, thinking \, \cfrac{1}{2} \, of 10 means \, \cfrac{1}{2} \div 10

• Cross multiplying
If students have been introduced to cross multiplying, they may get confused and use it when multiplying two fractions.
For example,

• Confusing multiplying fractions and dividing fractions
For example,

• Drawing a fraction model incorrectly
For example, when drawing a model to represent \, \cfrac{1}{2} \, of 4, not dividing the model into the correct number of equal parts.

### Practice fractions of numbers questions

1. Find \, \cfrac{1}{3} \, of 9

27

6

3

4.5

To use a model to find the answer:

9 or \, \cfrac{9}{1} \, is vertical and \, \cfrac{1}{3} \, is horizontal which is 9 divided into 3 equal pieces.

The overlap shaded region represents 9 groups of \, \cfrac{1}{3} \, or

\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}+\cfrac{1}{3}=\cfrac{9}{3}

\cfrac{9}{3} = 3

\cfrac{1}{3} \, of 9 is 3

2. Find \, \cfrac{1}{2} \, of 18

9

9.5

36

6

\cfrac{1}{2} \, of 18 is \, \cfrac{1}{2} \times 18

18 as an improper fraction is \, \cfrac{18}{1}

So, \cfrac{1}{2}\times\cfrac{18}{1}=\cfrac{18}{2}

The common factor of 18 and 2 is 2

\cfrac{18 \, \div \, 2}{2 \, \div \, 2}=\cfrac{9}{1}=9

\cfrac{1}{2} \, of 18 is 9

3. Find \, \cfrac{3}{4} \, of 8

12

6

\cfrac{32}{3}

\cfrac{3}{24}

To use a model to find the answer:

\cfrac{8}{1} \, is horizontal and \, \cfrac{3}{4} \, is vertical.

The overlap shaded region represents 8 groups of \, \cfrac{3}{4} \, or

\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}+\cfrac{3}{4}=\cfrac{24}{4}

\cfrac{24}{4}=6

4. Find \, \cfrac{5}{8} \, of 56

36

35

40

28

\cfrac{5}{8} \, of 56 is \, \cfrac{5}{8}\times 56

56 as an improper fraction is \, \cfrac{56}{1}

So, \, \cfrac{5}{8}\times\cfrac{56}{1}=\cfrac{5 \, \times \, 56}{8 \, \times \, 1}= \cfrac{280}{8}

The common factor of 280 and 8 is 8.

\cfrac{280 \, \div \, 8}{8 \, \div \, 8}=\cfrac{35}{1}=35

\cfrac{5}{8} \, of 56 is 35

5. Lucas ran \, \cfrac{1}{5} \, of a 15 mile running path. How far did he run?

7.5 miles

4 miles

10 miles

3 miles

\cfrac{1}{5} \, of 15 is \, \cfrac{1}{5}\times 15

15 as an improper fraction is \, \cfrac{15}{1}

So, \, \cfrac{1}{5}\times\cfrac{15}{1}=\cfrac{1 \, \times \, 15}{5 \, \times \, 1}= \cfrac{15}{5}

The common factor between 15 and 5 is 5.

\cfrac{15 \, \div \, 5}{5 \, \div \, 5}=\cfrac{3}{1}=3

\cfrac{1}{5} \, of 15 is 3

6. Maddie reads four-fifths of the 65 pages of her book. How many pages did she read?

52 pages

13 pages

48 pages

54 pages

\cfrac{4}{5} \, of 65 is \, \cfrac{4}{5}\times 65

65 as an improper fraction is \, \cfrac{65}{1}

\cfrac{4}{5}\times \cfrac{65}{1}=\cfrac{4 \, \times \, 65}{5 \, \times \, 1}=\cfrac{260}{5}

The common factor between 260 and 5 is 5.

\cfrac{260 \, \div \, 5}{5 \, \div \, 5}=\cfrac{52}{1} =52

\cfrac{4}{5} \, of 65 is 52 pages

## Fractions of numbers FAQs

Is finding a fraction of a number the same as when you multiply fractions?

Yes, the word “of” means to multiply. So finding the fraction of a number is the same as when you multiply fractions.

Is 1 always the denominator of the fraction of a whole number?

Yes, in order to make a whole number a fraction, place it over 1. So the denominator of the fraction is always 1.

Is a mixed number the same as a mixed fraction?

Mixed fractions and mixed numbers mean the same thing.

Do you have to get a common denominator when multiplying fractions?

You can get a common denominator to multiply fractions, but it isn’t necessary for multiplication.

What are the different types of fractions?

There are proper fractions where the numerator (top number) is smaller than the denominator (bottom number). There are unit fractions where the numerator (top number) is 1 and the denominator (bottom number) is a whole number. There are improper fractions where the numerator (top number) is greater than the denominator (bottom number). There are mixed numbers or mixed fractions that are made up of a whole number and a proper fraction.

Are decimal numbers and fractional numbers the same?

They are similar because they can have a whole part and a fractional part.

## The next lessons are

• Decimals
• Place value
• Multipy
• Improper fractions and mixed numbers

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