How to Find the Reciprocal and Divide Fractions with Step by Step Examples
Reciprocal and Division of fractions are different from each other. When two terms, such that the second is the inverse of first, gives a product of 1 upon multiplication, then the fractions are called reciprocal or multiplicative inverse of each other. It can be achieved by interchanging the numerator and denominator of the fraction. If a term is x/y, then its reciprocal will be y/x. A fraction is a numerical quantity that represents a part of the whole number.
Dividing fractions requires inverting the divisor (reciprocal of the divisor) and then follow the steps of multiplication. The term x/y, when divided by a non-zero fraction p/q, then the expression looks like:
x/y ÷ p/q = x/y ÷ q/p
Division of fractions involves multiple steps.
Parts of Fraction
The parts of a fraction are:
Numerator: The numerator is the number which is on the top of the line. It shows how many equal parts of the whole or collection is taken.
Denominator: The number below the line is the denominator. It shows the total divisible number of equal parts or the total number of equal parts which they are in a collection.
Types of Fraction
There are three different types of fractions:
Proper Fraction: A fraction in which the numerator and denominator are positive, and the numerator is smaller than the denominator; those fractions are called proper fractions.
Example: 4/5, 1/6, 7/9, 3/7, etc.
Improper Fraction: A fraction in which the numerator is greater than the denominator, those fractions are called improper fractions.
Example: 5/2, 7/3, 8/5, 5,3, etc.
Mixed Fraction: When a whole number and a proper fraction is combined, it is known as a mixed fraction.
Example: 4²/₃, 3¹/₂, etc.
Reciprocal of Fractions
Interchanging or swapping the numerator or denominator with each other gives us the reciprocal of the fraction. Like for example, the reciprocal of 1/2 is 2/1, or that of 4 is 1/4.
In order to obtain a reciprocal from a mixed fraction, it must be converted to an improper fraction, and then the numerator and denominator must be swapped. For example, to find the reciprocal of 4²/₃, it is first converted into an improper fraction.
4²/₃ =14/3[improper fraction]
14/3 = 3/14
Therefore, the reciprocal of 4²/₃ is 3/14.
The product of a fraction and its reciprocal is always 1, as it is nothing but its inverse.
Division of Fractions
Division of fractions involves certain rules and follows multiple steps. To perform Division, we have to multiply the first fraction with the reciprocal of the second. Division involves some steps to be followed:
Step 1: Changing the division sign (÷) to the multiplication sign (×).
Step 2: If we change the sign to multiplication, we also have to write the reciprocal of the second term or fraction.
Step 3: Multiplying the fractions and simplifying the result.
Here, is an example of Division of fractions:
15³/₇ ÷ 1²³/₄₉
First, we have to change the mixed to an improper fraction.
= 108/ 7 ÷ 72/49.
Step 1: Changing the sign to multiplication from division and writing the reciprocal of the second term [72/49 = 49/72]
= 108/7 × 49/72
Step 2: Multiplying the first with the reciprocal of the second fraction.
= (108 × 49)/ (7 × 72)
= (3 × 7)/ (1 × 2)
= 21/2
Step 3: Getting the simplified result of the expression.
Division of fractions is the multiplication of fractions by just changing the second fraction to its reciprocal.
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Solved Example:
1. 5/9 ÷ 2/3
[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]
= 5/9 × 3/2
[Step II: Multiplying the first fraction by the reciprocal of the second: (3 × 5)/(2 × 9)]
= 5/6
[Step III: It is the simplified expression, hence no further simplifications].
2. 4/9 ÷ 2/3
[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]
= 4/9 × 3/2
[Step II: Multiplying the first fraction by the reciprocal of the second]
= (4 × 3)/ (9 × 2)
= (2 × 1)/(3 × 1)
= 2/3
[Step III: It is the simplified expression, hence no further simplifications]
FAQs on Reciprocal and Division of Fractions Made Simple
1. What is the reciprocal of a fraction?
The reciprocal of a fraction is found by swapping its numerator and denominator. For a fraction a/b (where a ≠ 0), its reciprocal is b/a.
- Example: The reciprocal of 3/5 is 5/3.
- For a whole number like 4, write it as 4/1; its reciprocal is 1/4.
- Zero has no reciprocal because division by 0 is undefined.
2. How do you divide fractions step by step?
To divide fractions, multiply the first fraction by the reciprocal of the second fraction. Follow these steps:
- Step 1: Keep the first fraction the same.
- Step 2: Change the division sign to multiplication.
- Step 3: Flip (take the reciprocal of) the second fraction.
- Step 4: Multiply numerators and denominators.
3. Why do we multiply by the reciprocal when dividing fractions?
We multiply by the reciprocal because division is the same as multiplying by a number’s multiplicative inverse. For any non-zero fraction a/b, multiplying by its reciprocal b/a gives 1.
- Example: 3/4 × 4/3 = 12/12 = 1.
- So dividing by 4/3 is the same as multiplying by 3/4.
4. What is the formula for dividing fractions?
The formula for dividing fractions is (a/b) ÷ (c/d) = (a/b) × (d/c), where c/d ≠ 0. After multiplying:
- Multiply numerators: a × d
- Multiply denominators: b × c
- Simplify the result if possible
5. How do you find the reciprocal of a whole number or mixed number?
To find the reciprocal of a whole or mixed number, first write it as an improper fraction, then flip it.
- Whole number example: 7 = 7/1, so reciprocal is 1/7.
- Mixed number example: 1 2/3 = 5/3, so reciprocal is 3/5.
6. What happens when you divide a fraction by a whole number?
Dividing a fraction by a whole number means multiplying the fraction by the reciprocal of the whole number. Write the whole number as a fraction over 1.
- Example: 3/4 ÷ 2 = 3/4 × 1/2 = 3/8.
7. Can you give an example of dividing mixed fractions?
To divide mixed fractions, convert them to improper fractions, then multiply by the reciprocal.
- Example: 1 1/2 ÷ 3/4
- Convert: 1 1/2 = 3/2
- Apply rule: 3/2 × 4/3 = 12/6 = 2
8. What is the difference between reciprocal and inverse?
In fraction maths, reciprocal and multiplicative inverse mean the same thing: a number that multiplies with the original number to give 1.
- Example: The reciprocal (inverse) of 5/8 is 8/5.
9. What are common mistakes when dividing fractions?
A common mistake in dividing fractions is forgetting to flip the second fraction before multiplying. Watch out for these errors:
- Not taking the reciprocal of the divisor.
- Flipping the wrong fraction.
- Forgetting to simplify the final answer.
- Trying to divide numerators and denominators directly.
10. Can a fraction have a reciprocal of zero?
No, a fraction cannot have a reciprocal of zero because no number multiplied by zero equals 1. If a fraction has numerator 0 (like 0/5), its value is 0, and zero has no reciprocal.
- Example: There is no number x such that 0 × x = 1.
