About Division of Mixed Numbers
What are Fractions?
Fractions represent parts of an entire item, or a ‘whole’. Let us say we have a pizza which is divided into 6 equal parts.
How do we represent two slices out of 6. To represent such parts of a whole we use fractions. Two slices out of 6 can also be represented as \( \frac{2}{6} \) and this is a fraction. Fractions are generally written in the format of \( \frac{\square }{\square } \) where the number above the horizontal line is called the numerator, and the number below is called the denominator. The numerator is the number of equal parts chosen by us and the denominator represents the number of equal parts present in the whole.
What are Mixed Numbers?
A mix of whole numbers and fractions is called mixed numbers. The mixed number \( 2\frac{1}{6} \) will represent two whole pizzas and one equal part out of six.
What if we just have one slice of pizza left and we wish to divide it into three equal parts. In that case we will have to divide \( \frac{1}{6} \) by 3. Division of fractions can be performed by multiplying the reciprocal of the divisor to the dividend. But how do we divide mixed numbers? Here are some situations which would require us to divide mixed numbers.
How many two thirds are there in two and two thirds?
How many three fourths are there in three and three fourths?
These types of situations require us to divide mixed numbers.
Steps to divide Mixed Numbers
Dividing mix numbers involves two of the following steps.
Step 1: Write each mixed number as a proper fraction.
Step 2: Divide as would in case of proper fractions.
Let’s look at an example to understand this better.
Example 1 : Find \( 2\frac{2}{3}\div \frac{2}{3} \) :
Solution :
The first step would be to write the mix number \( 2\frac{2}{3} \) as an improper fraction:
\( 2\frac{2}{3}=\frac{(3\times 2)+2}{3} \)
\( 2\frac{2}{3}=\frac{8}{3} \)
The second step would be to divide as we do in the case of proper fractions.
\( 2\frac{2}{3}\div \frac{2}{3}=\frac{8}{3}\div \frac{2}{3} \)
\( =\frac{8}{3}\times \frac{3}{2} \) Take reciprocal of \( \frac{2}{3} \)
\( =\frac{8\times 3}{3\times 2} \) Multiply fractions
\( =\frac{26}{4} \) or 4 Simplify.
Example 2 : Find \( 4\frac{3}{4}\div 5\frac{2}{5} \) :
Solution :
The first step would be to write the mixed numbers \( 4\frac{3}{4} \) and \( 5\frac{2}{5} \) as an improper fractions:
\( 4\frac{3}{4}=\frac{(4\times 4)+3}{4} \) and \( 5\frac{2}{5}=\frac{(5\times 5)+2}{5} \)
\( 4\frac{3}{4}=\frac{19}{4} \) and \( 5\frac{2}{5}=\frac{27}{5} \)
The second step would be to divide as we do in the case of proper fractions:
\( 4\frac{3}{4}\div 5\frac{2}{5}=\frac{19}{4}\div \frac{27}{5} \)
\( =\frac{19}{4}\times \frac{5}{27} \) Take reciprocal of \( \frac{27}{5} \)
\( =\frac{19\times 5}{4\times 27} \) Multiply fractions
\( =\frac{95}{108} \) Simplify.
Example 3: Find \( 9\div 4\frac{5}{6} \)
Solution :
First step, write the mixed number \( 4\frac{5}{6} \) as an improper fraction:
\( 4\frac{5}{6}=\frac{(4\times 6)+5}{6} \)
\( 4\frac{5}{6}=\frac{29}{6} \)
Second step, divide as we do in the case of proper fractions:
\( 9\div 4\frac{5}{6}=9\div \frac{29}{6} \)
\( =9\times \frac{6}{29} \) Take reciprocal of \( \frac{29}{6} \)
\( =\frac{9\times 6}{29} \) Multiply fractions
\( =\frac{54}{29} \) Simplify.
Real Life Modeling Questions
Example 1:
In a wedding ceremony 5 identical cakes are ordered and every cake is divided into ten equal slices. If \( 3\frac{7}{10} \) of the total cakes are eaten then how many slices remain? Denote the number of slices left as a mixed number. If the leftover slices are divided among 5 charity kitchens then find the fraction of cake each kitchen would receive.
Solution:
There are 10 equal slices of five identical cakes, and there are 5 cakes (or 50 identical cake slices), out of which \( 3\frac{7}{10} \) is consumed. This can be shown through the following diagram:
From the figure we can clearly see that 1 whole cake(10 slices) and 3 slices out of 10 are left. So, the total number of slices left are :
10 + 3 = 13
In the fraction form, 13 slices are 10 slices and 3 slices. 10 slices represent one whole and three slices out of 10.
Therefore, 13 slices in fraction form is 1 and \( \frac{3}{10} \) or \( 1\frac{3}{10} \) .
Now, this fraction of the cake(\( 1\frac{3}{10} \)) has to be divided amongst 5 charity kitchens.
This can be represented as :
\( 1\frac{3}{10}\div 5 \)
\( =\frac{13}{10}\div 5 \) Changing mixed fraction to improper fraction
\( =\frac{13}{10}\times \frac{1}{5} \)
\( =\frac{13}{50} \)
This shows each charity kitchen will receive \( \frac{13}{50} \) of the remaining cake.
Example 2 :
Richard has \( 20\frac{1}{5} \) pounds of clay which is used for modeling. If he needs to divide this clay between 15 of his friends, then find the fraction of clay each of Richard’s friends would get.
Solution :
To find the required quantity of clay we need to find:
\( 20\frac{1}{5}\div 15 \)
\( =\frac{101}{5}\div 15 \)
\( =\frac{101}{5}\times \frac{1}{15} \)
\( =\frac{101}{75} \)
\( =1\frac{26}{75} \)
So each of Richard’s friends would receive \( 1\frac{26}{75} \) pounds of clay.
Fractions having their numerators greater than their denominators are called improper fractions.
No, proper fractions cannot be converted into mixed fractions. However, improper fractions can be converted to mixed fractions and vice versa.