Summary
How to Multiply Whole Numbers by Fractions
- Change the whole number into a fraction.
- Multiply the numerators.
- Multiply the denominators.
- Reduce, if needed.
- Convert to a mixed number, if desired.
If you are multiplying fractions with really big numbers in the numerator or denominator, it can be helpful to cross-reduce before you multiply.
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Examples
Multiply: \[5\times\frac{2}{3}\]
How can you write 5 as a fraction?
The simplest way to write 5 as a fraction is \(\frac{5}{1}\).
What is the product when you multiply the numerators? \[\frac{\green{5}}{1}\times\frac{\green{2}}{3}=\frac{\green{10}}{}\]
What is the product when you multiply the denominators? \[\frac{5}{\green{1}}\times\frac{2}{\green{3}}=\frac{10}{\green{3}}\]
Can the answer be reduced?
No, 10 and 3 don’t have any common factors so \(\frac{10}{3}\) can’t be reduced any further.
Can the answer be converted to a mixed number?
Yes, \(\frac{10}{3}\) is an improper fraction, so it can be converted to \(3\frac{1}{3}\).
Click here to see how to convert improper fractions to mixed numbers.
When you multiply… \[5\times\frac{2}{3}\]
The answer as an improper fraction is… \[\frac{10}{3}\]
And the answer as a mixed number is… \[3\frac{1}{3}\]
Multiply: \[2\times\frac{3}{8}\]
How can you write 2 as a fraction?
The simplest way to write 2 as a fraction is \(\frac{2}{1}\).
What is the product when you multiply the numerators? \[\frac{\yellow{2}}{1}\times\frac{\yellow{3}}{8}=\frac{\yellow{6}}{}\]
What is the product when you multiply the denominators? \[\frac{2}{\yellow{1}}\times\frac{3}{\yellow{8}}=\frac{6}{\yellow{8}}\]
Can the answer be reduced?
Yes, 6 and 8 are both divisible by 2 so \(\frac{6}{8}\) can be reduced: \[\frac{6}{8}=\frac{3}{4}\]
Click here to see how to reduce fractions.
Can the answer be converted to a mixed number?
No, \(\frac{3}{4}\) isn’t an improper fraction, so it can’t be converted to a mixed number.
When you multiply… \[2\times\frac{3}{8}\]
The reduced answer is… \[\frac{3}{4}\]
Multiply: \[6\times\frac{4}{9}\]
How can you write 6 as a fraction?
The simplest way to write 6 as a fraction is \(\frac{6}{1}\).
What is the product when you multiply the numerators? \[\frac{\blue{6}}{1}\times\frac{\blue{4}}{9}=\frac{\blue{24}}{}\]
What is the product when you multiply the denominators? \[\frac{6}{\blue{1}}\times\frac{4}{\blue{9}}=\frac{24}{\blue{9}}\]
Can the answer be reduced?
Yes, 24 and 9 are both divisible by 3 so \(\frac{24}{9}\) can be reduced: \[\frac{24}{9}=\frac{8}{3}\]
Click here to see how to reduce fractions.
Can the answer be converted to a mixed number?
Yes, \(\frac{8}{3}\) is an improper fraction, so it can be converted to \(2\frac{2}{3}\).
Click here to see how to convert improper fractions to mixed numbers.
When you multiply… \[6\times\frac{4}{9}\]
The answer as an improper fraction is… \[\frac{8}{3}\]
And the answer as a mixed number is… \[2\frac{2}{3}\]
Why It Works
When you multiply whole numbers by fractions, the whole number represents the original amount and the fraction represents the number of copies you are making of that group.
Multiplying by a fraction means that you have to make a partial copy instead of multiple copies. That means that your answer will be smaller than the original amount.
For example, if you were multiplying \(5\times\frac{2}{3}\) the whole number means that the original amount is 5 wholes.
Multiplying the 5 wholes by \(\frac{2}{3}\) means that you are going to make a partial copy of each whole and only keep two-thirds of each whole.
When you make \(\frac{2}{3}\) of a copy of the 5 wholes, the wholes get split up into thirds.
This is why we multiply the denominators when we multiply whole numbers by fractions.
\[\frac{5}{\red{1}}\times\frac{2}{\red{3}}=\frac{}{\red{3}}\]
When you keep 2 of the thirds for each of the 5 wholes, you end up with a total of 10 thirds. This is why we multiply the numerators.
\[\frac{\red{5}}{1}\times\frac{\red{2}}{3}=\frac{\red{10}}{3}\]
If you write the final answer as a mixed number, you get…
\[5\times\frac{2}{3}=3\frac{1}{3}\]
Notice that the final answer is smaller than the original whole number. This is because \(\frac{2}{3}\) is smaller than 1. We only made a partial copy of the whole number so the result is smaller than what we started with.
If you multiply a whole number by an improper fraction, the answer will be bigger than the original whole number because improper fractions are bigger than 1.
Improper fractions are equivalent to mixed numbers and multiplying by mixed numbers means that you are making a certain number of whole copies and then a partial copy of the original amount.
