Step-by-Step Guide: How to Divide Mixed Fractions
Let's start with an example You and your friend ate a whole pizza, and a quarter of a Let's start with an example You and your friend ate a whole pizza and a quarter of a pizza. As a mixed fraction, we can write this as 1 and $\dfrac{1}{4}$. A fraction consists of an integer, and a real fraction is called a mixed number. Example $6 \dfrac{4}{5}, 9 \dfrac{2}{3}$. A mixed fraction division is a division operation performed between two mixed fractions. In this article, you will learn how to divide mixed fractions with the help of examples.
The pictorial representation of the Mixed fraction
Dividing Mixed Fractions
A mixed fraction division is a division operation performed between two mixed fractions. It is similar to the multiplication operation in that it takes the reciprocal of the second fraction.
Dividing Mixed Fractions with Like Denominators
As you know, the term "same denominator" means that the denominators are the same. To understand the procedure with an example, consider mixed numbers with the same denominator.
Let's look at the following key points to help you divide mixed numbers.
The mixed fraction $a \dfrac{b}{c}$ can also be written $a+\left(\dfrac{b}{c}\right)$ and vice versa.
To convert a mixed number to an improper fraction, multiply the integer by the denominator and add the result to the numerator of the proper fraction. The denominator remains the same. For example, to convert $3 \dfrac{1}{6}$ to an improper function, we multiply 6 and 3, i.e., $3 \times 6=18$, and the result is added to 1, i.e., $18+$ $1=19$. Thus the improper fraction is $\dfrac{19}{6}$
To convert an improper fraction to a mixed fraction, divide the numerator of the improper fraction by the denominator. The quotient becomes the integer part, the remainder becomes the numerator of the appropriate fraction, and the denominator remains unchanged. For example, to convert $\dfrac{21}{4}$ to a mixed number, we first divide 21 by 4 and get the quotient as 5 and the remainder as 1 . Thus, the mixed fraction is $5 \dfrac{1}{4}$
Now, let's see an example of how to divide mixed fraction numbers with like denominators.
Example: $3 \dfrac{1}{8}$ by $2 \dfrac{3}{8}$
Ans: $3 \dfrac{1}{8} \div 2 \dfrac{3}{8}$
Convert the mixed fraction to improper faction i.e.,
$3 \dfrac{1}{8}=\dfrac{25}{8}$ and $2 \dfrac{3}{8}=\dfrac{19}{8}$
We will multiply the reciprocal of the second fraction with the first fraction
$\dfrac{25}{8} \times \dfrac{8}{19}$
The numerators and denominators of these fractions are multiplied separately
$\dfrac{25 \times 8}{8 \times 19}$
If there are common divisors in the numerator and denominator, cancel them out. Here we remove 8.
Now we get $\dfrac{25}{19}$
Now convert it to the mixed fraction i.e, $\dfrac{25}{19}=1 \dfrac{6}{19}$
Dividing Mixed Fractions with Unlike Denominators
Dividing mixed numbers with different denominators is defined as dividing mixed numbers with different denominators. The steps to perform splitting are the same as described in the previous section. Let's look at an example to understand the division of mixed numbers with unequal denominators.
Example : $2 \dfrac{2}{9}$ by $3 \dfrac{1}{7}$
Ans: Convert the mixed fraction to improper faction i.e, $2 \dfrac{2}{9}=\dfrac{20}{9}$ and $3 \dfrac{1}{7}=\dfrac{22}{7}$
We will now divide $\dfrac{20}{9}$ by $\dfrac{22}{7}$
$\dfrac{20}{9} \div \dfrac{22}{7}$
$\dfrac{20}{9} \times \dfrac{7}{22}$
$\dfrac{10 \times 7}{9 \times 11}$ (by cancelling 2 from the numerator and denominator)
$\dfrac{70}{99}$
Dividing Mixed Fractions by Whole Numbers
Integer can be written in fractions with denominators of 1. For example, 9 can be written as $\dfrac{9}{1}$. To divide a mixed number by an integer, convert the mixed number to an improper fraction, represent the integer as a fraction, and then perform the procedure for dividing a mixed number as described in the previous section. To understand, go through the example below.
Example 1: $4 \dfrac{1}{5}$ by 7
Ans. $4 \dfrac{1}{5} \div 7$
Now convert $4 \dfrac{1}{5}$ to improper function $\dfrac{21}{5}$ and the whole number 7 can be written as $\dfrac{7}{1}$
We will now divide $\dfrac{21}{5}$ by $\dfrac{7}{1}$
$=\dfrac{21}{5} \div \dfrac{7}{1}$
$=\dfrac{21}{5} \times \dfrac{1}{7}$
$=\dfrac{21 \times 1}{5 \times 7} \text { (Here we cancelled out } 21 \text { from the } 7)$
$=\dfrac{3}{5}$
Dividing Mixed Fractions with Fractions
Dividing a mixed number by a fraction is the division between a mixed number and a proper fraction. The steps to share them are the same as explained before. Convert the mixed fraction to an improper fraction and divide by the given fraction according to the division procedure. To understand, go through the example below.
Example $2: 5 \dfrac{2}{3}$ by the fraction $\dfrac{2}{9}$
Ans. We will convert the mixed fraction into improper fraction $\dfrac{17}{3}$
$\dfrac{17}{3} \div \dfrac{2}{9}$
$\dfrac{17}{3} \times \dfrac{9}{2}$
$\dfrac{17 \times 9}{3 \times 2} \text { (Here we cancelled out } 9 \text { by } 3 \text { ) }$
$\dfrac{17 \times 9}{2}=\dfrac{51}{2}=25 \dfrac{1}{2}$
Practice Questions
Q1. $\dfrac{3}{15} \div 2 \dfrac{6}{15}$
Ans. $\dfrac{1}{4}$
Q2. $\dfrac{2}{5} \div 3 \dfrac{4}{7}$
Ans. $\dfrac{14}{125}$
Q3. $6 \dfrac{1}{2} \div 2 \dfrac{1}{4}$
Ans. $\dfrac{26}{9}$
Q4. $8 \dfrac{2}{5} \div 4 \dfrac{2}{7}$
Ans. $1 \dfrac{24}{25}$
Summary
In this article, we learned that a mixed fraction consists of a whole number with a proper fraction. We also learned how to divide mixed fractions when both fractions have the same and different denominators. We also learned how to divide mixed fractions with whole numbers with the help of examples and when mixed fractions with the Improper fraction with the help of examples. There are some practice questions related to examples. Practice them then you will easily solve the mixed fractions division.
FAQs on Division of Mixed Fractions Made Simple
1. What are the main steps to divide mixed fractions?
To divide mixed fractions, you should follow a clear three-step process:
- Convert: Change all mixed fractions into improper fractions. To do this, multiply the whole number by the denominator and add the numerator. This result becomes the new numerator, and the denominator stays the same.
- Invert and Multiply: Keep the first fraction as it is, change the division sign to a multiplication sign, and take the reciprocal (or flip) of the second fraction. This is often called the 'Keep-Change-Flip' method.
- Simplify: Multiply the numerators together and the denominators together. If the resulting fraction is improper, convert it back to a mixed number and simplify to its lowest terms if possible.
2. Why must mixed fractions be converted to improper fractions before division?
Mixed fractions must be converted to improper fractions because the standard rules of fraction division and multiplication are designed to work with a single numerator and a single denominator. A mixed number, like $2 rac{1}{3}$, represents a whole number plus a fractional part. This format makes it mathematically complex to directly apply the 'multiply by the reciprocal' rule. By converting it to an improper fraction (like $ rac{7}{3}$), you express the entire value as one coherent fraction, allowing for straightforward and accurate calculation.
3. how do you divide a mixed fraction by a whole number?
To divide a mixed fraction by a whole number, you first need to express both quantities in fractional form. Follow these steps:
- First, convert the mixed fraction into an improper fraction.
- Next, convert the whole number into a fraction by placing it over a denominator of 1. For example, the whole number 5 becomes $ rac{5}{1}$.
- Finally, apply the standard division rule: multiply the first fraction by the reciprocal of the second fraction.
4. how is the division of mixed fractions different from their multiplication?
The initial step for both operations is identical: you must convert all mixed fractions into improper fractions. The key difference appears in the next step. For multiplication, you simply multiply the numerators and denominators directly. however, for division, you must perform an extra action: find the reciprocal of the second fraction (the divisor) and then change the operation to multiplication.
5. If the answer after dividing is an improper fraction, how is it converted back to a mixed fraction?
To convert an improper fraction back into a mixed fraction, you perform a simple division. Divide the numerator by the denominator. The result of this division gives you all the parts you need:
- The quotient becomes the whole number part of the mixed fraction.
- The remainder becomes the new numerator.
- The denominator remains unchanged.
For example, to convert $ rac{11}{4}$, divide 11 by 4. The quotient is 2 and the remainder is 3, so the mixed fraction is $2 rac{3}{4}$.
6. Does dividing by a mixed fraction always make the original number smaller?
Yes, this is correct. A mixed fraction always has a value greater than 1. In mathematics, dividing any positive number by a value greater than 1 will always result in a smaller number. For instance, dividing 10 by $2 rac{1}{2}$ (or $ rac{5}{2}$) gives 4, which is smaller than 10. This is a fundamental concept in division that helps in estimating the outcome of a calculation.
