Reciprocal of Fractions

Fractions created by swapping the numerator and denominator of the given fraction are known as Reciprocal Fractions. For example, fraction b/a has the reciprocal fraction a/b. The characteristic of reciprocal fractions is that they always result in 1 when multiplied together.

In this article, we will learn about, Reciprocal Fraction Definition, What are Fractions? Reciprocal Function Graph, Reciprocal Mixed Fraction, Adding Reciprocal Fractions, Subtracting Reciprocal Fractions, Reciprocal Fractions Algebra, How to Find Reciprocal Fraction, etc and others.

What are Reciprocal Fractions?

In mathematics, reciprocals are similar to “opposite fractions.” The reciprocal of an integer or fraction is what you get when you turn it upside down. For instance, the reciprocal of 2 is 1/2, 3/4 has a reciprocal of 4/3, and so on.

Product of reciprocal fraction with its fraction always results in 1. This is explained as the reciprocal of 2 is 1/2 and 2 × 1/2 = 1.

Define Reciprocal Fractions

When numerator and denominator of a given fraction are switched, a new fraction is created which is known as a reciprocal fraction.

Before moving further we must know about, Fractions.

Learn, Reciprocal

What are Fractions?

Fractions are a way of representing a portion of a whole or ratio of two whole numbers. It is made when two numbers are divided by a horizontal or diagonal line, where the top number is called Numerator and the bottom number is called Denominator.

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Parts of Fractions

There are two parts of fractions,

• Numerator
• Denominator

Numerator: Numerator is the number above the fraction line that indicates how many components of the whole you have or are contemplating. It denotes the number of pieces being discussed or used. For example, in the fraction 3/5, 3 is numerator, indicating that you are evaluating 3 parts out of a total of 5.

Denominator: Number below the fraction line is the denominator, and it represents the total number of equal pieces that make up the whole. It denotes the total number of pieces in the unit or set as a whole. In the above example, denominator in 3/5 is 5, indicating that the whole has been divided into 5 equal pieces.

The numerator and denominator work together to convey the value of a fraction as well as its relationship to the whole, allowing us to express portions or divisions of a whole in simply accessible and quantitative terms.

Types of Fraction

Proper Fractions, Improper fractions, and Mixed Fractions are three types of fractions. Let us define each of them.

Proper Fraction: Proper fractions have a numerator (the top number) that is less than the denominator (the bottom number). When represented as decimals, they indicate portions of a whole and are always less than one. For example, 3/4​ or 5/8​ are proper fractions.

Improper Fraction: Improper fractions, on the other hand, have a numerator that is equal to or higher than the denominator. They signify a greater-than-one value and may be converted to mixed numbers. For example, 8/5​ or 7/4​ are proper fractions.

Mixed Fraction: A mixed number is a whole number plus a proper fraction. They are written with a whole number first, then a fraction. For example, [Tex]3\tfrac{1}{2}[/Tex]​ or [Tex]1\tfrac{3}{4}[/Tex]​ are mixed fractions.

Each fraction type serves a particular purpose in describing quantities, making it easier to understand and interact with different portions of a numbers.

Reciprocal Function Graph

There are several types of reciprocal functions. One of them takes the form k/x. Here, ‘k’ is a real number, and ‘x’ cannot be zero. Let us now build a graph of the function f(x) = 1/x using various x and y values.

x	-2	-1	1/2	1	2
y = 1/x	-0.5	-1	2	1	0.5

For a reciprocal function f(x) = 1/x, ‘x’ can never be 0. The graph shows that they never contact the x-axis or y-axis. The y-axis is called the vertical asymptote because the curve approaches it but never touches it. In addition, the x-axis is the horizontal asymptote since the curve never meets it.

Reciprocal Mixed Fraction

Reciprocal of mixed fraction is found using the following steps:

Step 1: First convert the mixed number to an improper fraction. After converting of a mixed fraction [Tex]a\tfrac{b}{c}[/Tex] to (ac+b)/c

Step 2: Then, find the reciprocal of the improper fraction. The reciprocal of a mixed fraction [Tex]a\tfrac{b}{c}[/Tex] is c/(ac + b)

This is explained by the example,

Example: Find reciprocal of mixed fraction: [Tex]3\tfrac{1}{4}[/Tex]

Step 1: First, convert to an improper fraction. (3 × 4) + 1 /4 = 13/4

Step 2: Find the reciprocal of 13/4 is 4/13.

So, the reciprocal of [Tex]3\tfrac{1}{4}[/Tex] is 4/13.

Adding Reciprocal Fractions

To add reciprocal fractions, use the following steps:

Step 1: First, determine the reciprocal of each fraction. The reciprocal of a fraction a/b is b/a.

Step 2: Once you get the reciprocals, add them like ordinary fractions.

Step 3: Simplify the resulting fraction if possible.

Now lets take a example for better understanding.

Example: Add reciprocal of 1/2 and 1/3

Step 1: First,

• Calculate reciprocal of 1/2 = 2/1
• Calculate reciprocal of 1/3 = 3/1

Step 2: Add 2/1 + 3/1 = 5/1

Step 3: Simplify fraction is 5/1, which equals only 5

Subtracting Reciprocal Fractions

To subtract reciprocal fractions, use the following steps:

Step 1: First, determine reciprocal of each fraction. The reciprocal of fraction a/b is b/a.

Step 2: Once you get reciprocals, subtract them like ordinary fractions.

Step 3: Simplify the resulting fraction if possible.

Now lets take a example for better understanding.

Example: Subtract reciproal of: 1/3 and 1/2

Step 1: First,

• Calculate Reciprocal of 1/3 = 3/1
• Calculate Reciprocal of 1/2 = 2/1

Step 2: Subtract reciprocals 3/1 – 2/1 = 1/1

Step 3: Simplify fraction 1/1, which equals only 1

Reciprocal Fractions Algebra

Reiprocal of a variable x is 1/x and is found by the example added below,

Example: Find reciprocal of 2/(x – 3)

Recprcal of 2(x – 3) = (x – 3)/2

How to Find a Reciprocal Fraction

To get a reciprocal of any fraction follow the following steps,

Step 1: Exchange the numerator (top number) for the denominator (bottom number).

Step 2: Reciprocal is the new fraction created by this swapping.

For example:

• Reciprocal of 3/5 is 5/3
• Reciprocal of 2/9 is 9/2

Keep in mind that a whole number is also a fraction with demoniator 1. So its reciprocal is also easily found, (for example, reciprocal of 5 is 1/5).

Reciprocal of a Negative Fraction

Rules for the reciprocal of a positive fraction and the reciprocal of a negative fraction are the same, i.e.

Step 1: Exchange the numerator (top number) for the denominator (bottom number).

Step 2: Reciprocal fraction is created by this swapping.

For example:

• Reciprocal of -4/8 is -8/4
• Reciprocal of -3/6 is -6/3

When working with negative fractions, you can place the negative sign on the denominator or the numerator.

Reciprocal of a Fraction with Exponents

Similar to a normal fraction, the reciprocal of a fraction with exponents may be determined.

Step 1: Swap Numerator and Denominator.

Step 2: Sign of exponent changes when you determine the reciprocal of a fraction.

For example, if you have a^m/bⁿ, its reciprocal is bⁿ/a^m​, where the exponents m and n change signs.

Example:

• Reciprocal of 3²/4³ is 4³/3²
• Reciprocal of x⁵/y⁴ is y⁴/x⁵

To get the reciprocal of a fraction using exponents, keep in mind that the most important is to flip the fraction and adjust the exponents’ signs.

Examples of Reciprocal Fractions

Various on reciprocal of fractions are,

Example 1: Find reciprocal of fraction 2/7.

Solution:

Reciprocal of a fraction is found by swapping the numerator and denominator

Given Fraction, 2/7

Reciprocal of 2/7 is 7/2

Example 2: If reciprocal of x is 8/12, calculate x + 6.

Solution:

Given,

Reciprocal of x is 8/12 i.e, x is 12/8. Now,

x + 6 = 8/12 + 6

⇒ 12/8 + 6/1

⇒ (12+48)/8

⇒ 60/8

As a result, x+6 has a value of 60/8.

Example 3: Determine reciprocal of (7/4)^-2?

Solution:

= (7/4)^-2

= (4/7)²

= (16/49)

Reciprocal of 16/49 is 49/16

Practice Questions on Reciprocal of Fraction

Some practice questions on reciprocal of fractions are,

Q1: Find Reciprocal of -11/5

Q2: Find Reciprocal of 1/5

Q3: Find Reciprocal of (1/3)²

Q4: Find Reciprocal of (x + 1)

Q5: Find Reciprocal of 1/(5x + 9)

Reciprocal Fractions Frequently Asked Questions

What is Reciprocal of a Fraction?

A fraction’s reciprocal is just the fraction generated by swapping the numerator and denominator. The denominator becomes the numerator, and the numerator becomes the denominator.

What is Reciprocal of Mixed Fraction?

To get the reciprocal of a mixed fraction, we must first convert it to an improper fraction. The reciprocal of the inappropriate fraction derived thus reflects the reciprocal of the mixed fraction.

What is Reciprocal of 0 (Zero)?

Reciprocal of 0 is undefined because every integer multiplied by zero is always zero, implying that no number can be multiplied by zero to produce 1 as result.

What is Negative Fraction Reciprocal?

The reciprocal of a negative fraction is calculated by switching the fraction’s numerator and denominator. The negative sign will remain with the reciprocal numerator. For example, -y/x is the reciprocal of -x/y.

What is Reciprocal of 8?

1/8 is the reciprocal of 8. 1 divided by the number equals the reciprocal. 1/8 is equal to 1 divided by 8.

What is Reciprocal of 5?

Reciprocal of 5 is 1/5

What is Reciprocal of 2 in Fractions?

Reciprocal of 2 is 1/2