What are Whole numbers and Mixed numbers?
A set of numbers that includes all the natural numbers and “0” are known as whole numbers. The set of whole numbers also belongs to a set of numbers called real numbers. Whole numbers include counting numerals as well, called natural numbers. 0, 1, 2, 3, and so on are examples of whole numbers.
A mixed number is a combination of a whole number and a proper fraction. It usually denotes a number that appears between two whole numbers. A whole number, a numerator, and a denominator are combined to create a mixed number. The numerator and denominator both represent parts of the divisions that exist between two mixed numbers on a number line.
Examine the image above. It depicts a fraction \(2~\frac{1}{5}\) which when converted into an improper fraction is \(\frac{11}{5}\), which cannot be simplified further.
This mixed fraction is further than the whole number 2 but less than the whole number 3, that is, it lies between whole numbers 2 and 3. As a result, \(2~\frac{1}{5}\) is a mixed number.
How to find the product of a Whole number and a Mixed number?
We can multiply a whole number and a mixed number in two ways:
- Converting mixed number to fraction
- Applying the distributive property
Let’s discuss each of these methods in detail.
Converting a Mixed number to a Fraction
Let us find the product of \(1~\frac{1}{2}\) and 5 by converting the mixed number \(1~\frac{1}{2}\) into an improper fraction. By definition, an improper fraction is a fraction in which the value of the numerator is greater than or equal to the value of the denominator.
So, \(5\times~1~\frac{1}{2}\)
\(=~5\times~\left ( \frac{1}{1}+\frac{1}{2} \right )\)
\(=~5\times~\left ( \frac{2+1}{2} \right )\) [Add]
\(=~5\times~ \frac{3}{2} \) [Simplify]
\(=~ \frac{15}{2}\) [Multiply]
Applying the Distributive property
To multiply a mixed number and a whole number, express the mixed number as the sum of the whole number part and the fractional part. The whole number is multiplied by each part of the mixed number. Finally, these products are added together to get the final product.
Let’s solve \(5\times~1~\frac{1}{2}\) using the distributive property.
We can express the mixed number as the sum of the whole number part and fraction part, that is , \(~1~\frac{1}{2}~=~1+\frac{1}{2}\).
So the multiplication equation now becomes,
\(=~5\times~\left ( 1+\frac{1}{2} \right )\)
\(=~5\times~1+5\times~\frac{1}{2}\) [Apply distributive property]
\(=~5+\frac{5}{2}\) [Multiply]
\(=~\frac{5}{1}+\frac{5}{2}\)
\(=~\frac{10+5}{2}\) [Add]
\(=~\frac{15}{2}\) [Simplify]
So the product \(1~\frac{1}{2}\times~5\) is equal to \(\frac{15}{2}\).
Solved Examples
Example 1: Let’s solve \(4~\frac{5}{6}\times~7\) using the distributive property and by simplifying the mixed fractions.
Solution:
By using the distributive property
\(4~\frac{5}{6}\times~7\)
First, the mixed fractions are expressed as a sum of the whole number and the fraction
\(=\left ( 4+\frac{5}{6} \right )\times~7\)
\(=~(4\times~7)+\left ( \frac{5}{6}~\times~7 \right )\) [Apply distributive property]
\(=~(28)+\left ( \frac{35}{6} \right )\) [Multiply]
\(=~\left ( \frac{28}{1} \right )+\left ( \frac{35}{6} \right )\)
\(=~\left ( \frac{168+35}{6} \right )\) [Add]
\(=~\left ( \frac{203}{6} \right )\) [Simplify]
By simplifying the mixed fractions:
\(4\frac{5}{6}\times~7\)
\(=~\frac{6\times~4+5}{6}\times~7\) [Simplify mixed fraction into improper fraction]
\(=~\frac{24+5}{6}\times~7\) [Simplify further]
\(=~\frac{29}{6}\times~7\) [Add]
\(=~\frac{29}{6}\times~\frac{7}{1}\)
\(=~\frac{203}{6}\) [Multiply]
The result obtained is \(\frac{203}{6}\) using both the methods.
Example 2: Find \( 3~\frac{1}{6}~\times~2\) by using the distributive property.
Solution:
By using the distributive property,
\(3~\frac{1}{6}\times~2\)
First, the mixed fraction is expressed as a sum of the whole number and the fraction
\(=~\left ( 3+\frac{1}{6} \right )\times2\)
\(=~(3\times~2)+\left ( \frac{1}{6}\times~2 \right )\) [Apply distributive property]
\(=~(6)+\left ( \frac{2}{6} \right )\) [Multiply]
\(=~\left ( \frac{6}{1} \right )+\left ( \frac{2}{6} \right )\)
\(=~\left ( \frac{36+2}{6} \right ) \) [Add]
\(=~\left ( \frac{38}{6} \right ) \) [Add]
By simplifying the mixed fractions:
\(3~\frac{1}{6}\times2\)
\(=~\frac{6\times~3+1}{6}\times2\) [Simplify mixed fraction into improper fraction]
\(=~\frac{18+1}{6}\times2\) [Simplify further]
\(=~\frac{19}{6}\times2\) [Add]
\(=~\frac{19}{6}\times~\frac{2}{1}\)
\(=~\frac{38}{6}\) [Multiply]
The result obtained is \(\frac{38}{6}\) using both the methods.
Example 3: One group of children requires four and a half pieces of cake. There are a total of 12 groups. What is the total quantity of cake pieces required?
Solution:
To find out the total number of cake pieces that are required, multiply \( 4~\frac{1}{2}\) with 12.
\( =~4~\frac{1}{2}~\times~12\)
First, the mixed fraction is expressed as a sum of the whole number and the fraction
\(=~\left ( \frac{1}{2}+4 \right )\times~12\)
\(=~\left ( \frac{1}{2}\times~12 \right )+(4\times~12)\) [Apply distributive property]
\(=~(6)+(48)\) [Multiply]
\(=~54\) [Add]
Hence, we need a total of 54 pieces of cake for 12 groups of children.
Example 4: Three and a quarter portions of a plate of rice is required to feed a large family. If three large families are staying in one house, how many portions of rice plates are required?
Solution:
To find the total portions of the plates of rice, we need to multiply three by three and a quarter.
\(3\times~3\frac{1}{2}\)
The mixed fractions are expressed as a sum of the whole number and the fraction,
\(3\times~\left ( 3+\frac{1}{2} \right )\)
\(\left ( 3\times~3+\frac{1}{2}\times~3 \right )\) [Apply distributive property]
\( \left ( 9+\frac{3}{2} \right )\) [Multiply]
\( =~\frac{18+3}{2}\) [Add]
\( =~\frac{21}{2}\) [Simplify]
So, we need a total of \( \frac{21}{2}\) plates of rice to satisfy the entire family staying in the house.
We can multiply a whole number and a mixed number by expressing the mixed number as the sum of the whole number and fraction part. Then we apply the distributive property by multiplying the whole number with the addends of the mixed number and add these products to get the final product.
The two ways to multiply a mixed number with a whole number are:
- Using the distributive property
- Expressing the mixed number as an improper fraction
Two mixed numbers can be multiplied by expressing the mixed fractions as improper fractions and then multiplying the corresponding numerators and denominators.