What is a Fraction?
A fraction is used to represent a whole number that is divided into equal parts. The general form of a fraction is \(\frac{a}{b}\). Since a fraction is the same as division, \(\frac{a}{b}\) is the same as \(a\div b\). \(\frac{1}{2},\frac{2}{3},\) and \(\frac{4}{5}\) are some examples of fractions.
We can perform math operations on fractions just like we do with whole numbers.
Using Models to Multiply a Fraction by a Whole Number
When we multiply a fraction by a whole number, we are essentially finding the fraction of that whole number.
For example, \(\frac{2}{3}\times 6\) is the same as \(\frac{2}{3}\) of 6.
In both cases, the result is 4. So, we can find the result either by multiplying \(\frac{2}{3}\times 6\), or by finding the value of \(\frac{2}{3}\) of 6.
We can also use a tape diagram model to represent the product of a fraction and a whole number.
Step 1: Since we are finding \(\frac{2}{3}\) of 6, we divide 6 into 3 equal parts.
Step 2: As we want to find \(\frac{2}{3}\) of 6, we select 2 of the 3 parts.
In this diagram, we can see that 6 wholes are divided into 3 equal parts. Each part is the same as 2 wholes. We are selecting 2 out of 3 parts.
So, \(\frac{2}{3}\times 6\) or \(\frac{2}{3}\) of 6 is 4.
Using Models to Multiply Fractions
Multiplying a fraction with another fraction is similar to the steps involved in the multiplication of a fraction and a whole number. Multiplication of a fraction with another fraction essentially means that we need to find the fraction of another fraction.
For example, \(\frac{1}{2}\times \frac{1}{3}\) is the same as \(\frac{1}{2}\) of \(\frac{1}{3}\). We can use a tape diagram or an area model to represent this visually.
Using a Tape Diagram to Multiply Fractions
Suppose you want to find \(\frac{1}{3}\times \frac{1}{4}\).
Step 1: Model \(\frac{1}{4}\). Divide 1 whole into 4 equal parts.
Step 2: To find \(\frac{1}{3}\) of \(\frac{1}{4}\), divide each \(\frac{1}{4}\) of the whole into 3 equal parts.
Since 12 parts make 1 whole, 1 part represents \(\frac{1}{12}\).
So, \(\frac{1}{3}\times \frac{1}{4}\) or \(\frac{1}{3}\) of \(\frac{1}{4}=\frac{1}{12}\)
Using an Area model to Multiply Fractions
Suppose you want to find \(\frac{1}{3}\times \frac{1}{4}\) using an area model.
To find the product of \(\frac{1}{3}\) and \(\frac{1}{4}\), the area model should have 3 rows and 4 columns.
Step 1: Shade 1 out of 3 rows blue to represent \(\frac{1}{3}\).
Step 2: Shade 1 out of 4 columns red to represent \(\frac{1}{4}\).
The purple overlap shows the product of the two fractions.
1 out of 12 parts is purple.
So, \(\frac{1}{3}\times \frac{1}{4}\) or \(\frac{1}{3}\) of \(\frac{1}{4}=\frac{1}{12}\)
Direct Multiplication of Fractions
Even though we now have a visual idea of the multiplication of fractions by using models, we can also multiply two fractions directly.
To multiply two fractions, we can simply multiply the numerators and the denominators of the two fractions to get the product.
For example, \(\frac{1}{3}\times \frac{1}{4}=\frac{1\times 1}{3\times 4}\)
\(=\frac{1}{12}\)
We got the same result when we used models, and when we performed the direct multiplication.
Solved Examples
Example 1: Find \(\frac{3}{5}\) of 10.
Solution:
\(\frac{3}{5}\) of 10 is the same as \(\frac{3}{5}\times 10\).
We can use a tape diagram to find the product.
Step 1: Since we are finding \(\frac{3}{5}\) of 10, we divide 10 into 5 equal parts.
Step 2: As we want to find \(\frac{3}{5}\) of 10, we select 3 of the 5 parts.
In this diagram, we can see that 10 wholes are divided into 5 equal parts. Each part is the same as 2 wholes. We are selecting 3 out of 5 parts.
So, \(\frac{3}{5}\times 10\) or \(\frac{3}{5}\) of 10 is 6.
Example 2: You bought 12 oranges. You used \(\frac{1}{3}\) of those oranges to make orange juice. How many oranges did you use?
Solution:
Since you used \(\frac{1}{3}\) of 12 oranges, we need to find the value of \(\frac{1}{3}\times 12\).
Step 1: Since we are finding \(\frac{1}{3}\) of 12, we divide 12 into 3 equal parts.
Step 2: As we want to find \(\frac{1}{3}\) of 12, we select 1 of the 3 parts.
We can see that 12 wholes are divided into 3 equal parts. Each part is the same as 4 wholes. We are selecting 1 out of the 3 parts.
So, \(\frac{1}{3}\times 12\) or \(\frac{1}{3}\) of 12 is 4.
Therefore, you used 4 out of 12 oranges to make orange juice.
Example 3: Use a model to find \(\frac{1}{4}\) of \(\frac{3}{6}\).
Solution:
\(\frac{1}{4}\) of \(\frac{3}{6}\) is the same as \(\frac{1}{4}\times \frac{3}{6}\). We can use an area model to find the product.
To find the product of \(\frac{1}{4}\) and \(\frac{3}{6}\), the area model should have 4 rows and 6 columns.
Step 1: Shade 1 out of 4 rows blue to represent \(\frac{1}{4}\).
Step 2: Shade 3 out of 6 columns in red to represent \(\frac{3}{6}\).
The purple overlap shows the product of the two fractions.
3 out of 24 parts are purple.
So, \(\frac{1}{4}\times \frac{3}{6}\) or \(\frac{1}{4}\) of \(\frac{3}{6}=\frac{3}{24}\)
Example 4: There are 60 students in grade 8. Out of these students, \(\frac{2}{3}\) of them have opted for science. Out of the students who have opted for science, \(\frac{1}{2}\) of them are girls. How many girls have opted for science?
Solution:
First, we need to express the number of girls who opted for science as a fraction of the total number of students.
Fraction of students who opted for science \(=\frac{2}{3}\)
Fraction of science students who are girls \(=\frac{1}{2}\)
Fraction of girls who have opted for science out of the total number of students\(=\frac{1}{2}\) of \(\frac{2}{3}\) or \(\frac{1}{2}\times\frac{2}{3}\)
\(\frac{1}{2}{\times}\frac{2}{3}=\frac{1\times2}{2\times3}\)
\(=\frac{1}{3}\)
So, \(\frac{1}{3}\) of the total number of students who opted for science are girls.
Number of girl students who opted for science = Fraction of girls who opted for science × Total number of students
\(=\frac{1}{3}\)of 60 \(=\frac{1}{3}\times 60\)
\(=\frac{1\times 60}{3}\)
\(=\frac{60}{3}\)
\(=20\)
Therefore, there were 20 girls who opted for science.
Using models to multiply fractions gives us a visual image of what the multiplication process means. This helps us better understand the multiplication process while multiplying fractions.
The multiplication operation with fractions can be expressed in words by replacing the multiplication symbol with “of”. So, the product of two fractions is a fraction of the other fraction.