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by Ron Kurtus (updated 18 January 2022)
Multiplying fractions simply consists of multiplying the numerators (top numbers) by each other and the denominators (bottom numbers) by each other. If you want to multiply a whole number by a fraction, you must first convert the whole number to a fraction and then proceed. When multiplying fractions and mixed numbers, you must first convert the mixed number to a fraction and then proceed.
Questions you may have include:
- How do you multiply two fractions?
- How do you multiply a fraction and whole number?
- How do you multiply a fraction and a mixed number?
This lesson will answer those questions.
Multiply two fractions
It is easy to multiple two fractions together. You just multiply the top numbers (numerators) by each other and the bottom numbers (denominators) by each other.
Consider the multiplying the fractions:
2/7 × 3/5
Multiply top numbers or numerators (2 and 3) together and bottom numbers or denominators (7 and 5) together:
(2 × 3) / (7 × 5) = 6/35
You can apply this technique to multiply a series of fractions.
Multiply whole number with fraction
You can multiply a whole number and a fraction by considering the whole number as a form of fraction. For example, the number 5 can be considered the fraction 5/1 (5 divided by 1 equals 5). Thus, you follow the same technique as in multiply two fractions together.
Consider the multiplying a whole number by a fraction:
5 × 3/7
Change the whole number to a fraction.
5/1 × 3/7
Multiply numerators together and denominators together.
(5 × 3)/(1 × 7) = 15/7
Note that 15/7 is considered an improper fraction. The proper form for your fraction answer is that the numerator is smaller than the denominator.
Divide 15 by 7 to reduce your answer a mixed number.
15 ÷ 7 = 2 1/7
Consider the multiplying a fraction by a whole number:
1/2 × 9
Change whole number to fraction:
1/2 × 9/1
Multiply numerators together and denominators together:
(1 × 9)/(2 × 1) = 9/2
Reduce improper fraction to a mixed number:
9 ÷ 2 = 4 1/2
Multiply with mixed numbers
When you multiply a fraction times a mixed number, you must first convert the mixed number into a fraction and then perform the multiplication.
First convert mixed number
A mixed number is the sum of a whole number and a fraction: 5 1/2 = 5 + 1/2.
You can convert a mixed number into a fraction by first converting the whole number into a fraction over 1. For example: 5 = 5/1.
In order to complete the addition of 5/1 + 1/2, both of the denominators must be the same. In other words, in this case, you want to have both denominators equal to 2.
You can achieve this by multiplying 5/1 × 2/2 = 10/2. (Note that 2/2 = 1, so you are just multiplying by 1.)
5 1/2 = 10/2 + 1/2 = 11/2
Instead of going through those steps, you can shortcut the operation for 5 1/2 by simply multiplying the 5 by the denominator of the fraction (2) and then adding to the numerator of the fraction (1). Thus, (5 × 2) + 1 = 11 and 5 1/2 = 11/2.
To convert 7 2/3 to a mixed number, change the whole number 7 to a fraction over 1.
7/1 + 2/3
Multiply the whole number fraction by 3/3:
7/1 × 3/3 = 21/3
Add the fractions:
21/3 + 2/3 = 23/3
Once you get good at it, you can use the shortcut method of multiplying the whole number by the denominator of the fraction and adding to the numerator:
7 2/3 = (7 × 3 + 2)/3 =
You always do multiplication before addition within the parentheses:
(21 + 3)/3 = 23/3
Multiply fraction and mixed number
It doesn't matter whether you multiply the fraction times the mixed number or the mixed number times the fraction.
3/5 × 6 2/7 =
Change the mixed number to a fraction:
3/5 × (6 × 7 + 2)/7 =
3/5 × (42 + 2)/7 =
Multiply the fractions:
3/5 × 44/7 =
(3 × 44)/(5 × 7)
Since 132 is greater than 35, you need to reduce the fraction to a proper fraction or mixed number. Thus, by dividing 132 ÷ 35 = 3 21/35.
Multiply two mixed numbers
A similar process is followed when you multiply two mixed numbers.
2 1/2 × 3 2/3 =
Change both mixed numbers to fractions:
(2 × 2 + 1)/ = 5/2
(3 × 3 + 2)/ = 11/3
5/2 × 11/3 =
Multiplying fractions consists of multiplying the numerators by each other and the denominators by each other. In multiplying a whole number by a fraction, you must first convert the whole number to a fraction and then proceed. When multiplying fractions and mixed numbers, you must first convert the mixed number to a fraction and then proceed.
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