# 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.

### What are fractions of numbers? ## Common Core State 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

Convert to a multiplication statement.

Convert the whole number to an improper fraction.

Multiply the numerators together and the denominators together.

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

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

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

Draw one fraction horizontally and the other vertically.

Connect the fractions all the way across with \bf{2} different colors.

Count up the shaded, overlapping parts.

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

### Example 4: proper fractions of numbers

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

Convert to a multiplication statement.

Convert the whole number to an improper fraction.

Multiply the numerators together and the denominators together.

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

### Example 5: proper fraction of a number

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

Convert to a multiplication statement.

Convert the whole number to an improper fraction.

Multiply the numerators together and the denominators together.

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

### Example 6: word problem

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

Convert to a multiplication statement.

Convert the whole number to an improper fraction.

Multiply the numerators together and the denominators together.

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

### Example 7: word problem

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

Convert to a multiplication statement.

Convert the whole number to an improper fraction.

Multiply the numerators together and the denominators together.

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

### 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 provide a visual representation of fractions of numbers and help students understand the concept as points on the number line.

• 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.

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