How to Multiply Mixed Numbers?
In math, while mixed numbers are multiplied with each other, we may encounter two types of scenarios:
- Multiplying a mixed number with another mixed number
- Multiplying a mixed number with a fraction.
Let us discuss each of these two methods in detail along with a few examples.
Multiplying Two or More Mixed Numbers
Here are the steps to follow when multiplying two or more mixed numbers:
Step 1: Write the given mixed numbers as improper fractions.
Step 2: Multiply one numerator with the other numerator and the denominators with each other.
Step 3: The product of the numerators and the denominators in the previous step form the numerator and the denominator of the product.
Step 4: Simplify or convert the resultant fraction, into a mixed or a whole number if necessary.
Take a look at the following example.
Multiplying a Mixed Number with Fraction
There are two methods to multiply a mixed number with a fraction.
Method 1:
In order to multiply a mixed number with a fraction, convert the mixed number into an improper fraction and then follow the steps 2 to 4 as explained in the previous method to obtain the product. Here’s an example: \(\frac{2}{5}\times 3\frac{1}{3}\).
Method 2:
In this method, we will use the Distributive Property to multiply. Let’s see how it is done with the help of an example \(\frac{5}{6}\times 4\frac{9}{11}\)
Step 1: Write the mixed number as the sum of a whole number and a fraction, that is \(4 + \frac{9}{11}\).
Now the new equation is, \(\frac{5}{6}\times (4 + \frac{9}{11})\)
Step 2: Apply the Distributive Property, that is,
\((\frac{5}{6}\times 4) + (\frac{5}{6}\times \frac{9}{11})\)
Step 3: Divide the common factors from numerator and denominator, if applicable.
Step 4: Now, multiply the fractions in parentheses as usual, \((\frac{5~\times ~2}{3~\times ~1}) + (\frac{5~\times ~3}{2~\times ~11})=\frac{10}{3}+\frac{15}{22}\)
Step 5: Add both the fractions to get the result, \(\frac{10}{3}+\frac{15}{22}=\frac{265}{66}\)
So, \(\frac{5}{6}\times 4\frac{9}{11}=\frac{265}{66}\) and is also \(4\frac{1}{66}\) as a mixed number.
[Note: While multiplying a whole number with a fraction, always write the whole number as \(\frac{\text{whole number}}{1}\)]
Solved Examples
Example 1:
Find the product of \(8\frac{5}{10} \times 9\frac{1}{15}\)
Solution:
We have to find the product: \(8\frac{5}{10} \times 9\frac{1}{15}\)
Let’s convert both the mixed numbers into improper fractions,
\(8\frac{5}{10}=\frac{85}{10}\) and \(9\frac{1}{15}=\frac{136}{15}\).
So, \(8\frac{5}{10} \times 9\frac{1}{15}\)
= \(\frac{85}{10} \times \frac{136}{15}\) [Rewrite as improper fractions]
= \(\frac{85\times 136}{10 \times 15}\) [Multiply numerators and denominators]
= \(\frac{17\times 68}{5 \times 3}\) [Simplify]
= \(\frac{1156}{15}\) [Simplify further]
Thus, \(8\frac{5}{10} \times 9\frac{1}{15} = \frac{1156}{15} \) which is expressed as \(77\frac{1}{15}\) as a mixed fraction.
Example 2:
Evaluate: \(7\frac{3}{2}\times \frac{10}{17}\).
Solution:
We have to find the product \(7\frac{3}{2}\times \frac{10}{17}\).
Let’s convert the mixed number 732 into an improper fraction, \(7\frac{3}{2} = \frac{17}{2}\).
Now,
\(\frac{17}{2}\times \frac{10}{17}\) [Rewrite as an improper fraction]
= \(\frac{17 \times 10}{2 \times 17}\) [Multiply]
= \(\frac{10}{2}\) [Divide both numerator and denominator by 17]
= 5 [Simplify]
Therefore, \(7\frac{3}{2}\times \frac{10}{17}=5\).
Example 3:
Peter reads an article that says an orange contains 80 milligrams of vitamin C. He wonders how much vitamin C he consumes as he eats \(2\frac{3}{4}\) oranges everyday? Help Peter find the answer.
Solution:
It is mentioned that an orange contains 80 mg of vitamin C. We can write this as a fraction = \(.
Number of oranges consumed by Peter everyday = \(\)2\frac{3}{4}=\frac{11}{4}\)
To find the total amount of vitamin C consumed we multiply the quantity of vitamin C in one orange with the number of oranges.
\(\Rightarrow \frac{80}{1}\times \frac{11}{4}\)
\(\Rightarrow \frac{80\times 11}{1 \times 4}\) [Multiply]
\(\Rightarrow 20 \times 11\) [Simplify by dividing both numerator and denominator by 4]
\(\Rightarrow 220\) [Simplify further]
Hence, Peter consumes 220 mg of vitamin C everyday.
When we have to multiply a mixed number with a whole number, write the whole number as a fraction by keeping the denominator as 1 to make it an improper fraction.
If any number is multiplied with zero, the result will also be 0. Hence, the product of any mixed number and zero is also 0.
As we know, while multiplying mixed numbers we multiply the numerator of the first fraction with the numerator of the second fraction. Then, the equation given is 7 x a = 21. Hence, the value of a is 3.