Reciprocal of Fractions

The mathematical concept of reciprocal means the multiplicative inverse of a number. It is what you get when you reverse the positions of the numerator and the denominator. For example, the reciprocal of 2/7 is 7/2, 8/3 is 3/8, etc. 

 

When a fraction is multiplied by the reciprocal, it always results 1. Let us understand this by an example. Let’s say, 2, the reciprocal of 2 is ½ and then multiply 2 by its reciprocal, 2  1/2 = 1
 

A fraction represents a portion of a whole or the ratio of two whole numbers. It can be represented in the form of p/q, where p and q are integers and q≠0 is called a fraction. It is a horizontal line that separates the two integers, called the fractional bar. The top number is called the numerator and the bottom number is called the denominator. For example, a ratio of 5:10 can be written as

 

        5/10 =>  p/q => NumeratorDenominator 

A fraction has two parts:

 

• Numerator: Numerator means number above the fraction line. It tells us how many parts of the whole are being considered. It indicates the number of pieces being used. For instance, in a fraction 5/6, 5 is the numerator, meaning that you are working with 5 parts out of 6. 
  
   
• Denominator: This is the number below the fraction line. It shows how many total numbers of equal parts the whole is divided into. It indicates how many pieces are there in the entire unit or set. For instance, in a fraction 5/6, 6 is the denominator, meaning the whole is divided into 6 equal parts. 
   

There are three types of fractions. Let us understand more about fractions in detail.

 

• Proper Fraction: In a proper fraction, the numerator is always smaller than the denominator. Proper fractions are always smaller than one. For example, 5/8 and 3/4 are proper fractions.
  
   
• Improper Fractions: In an improper fraction, the numerator is equal to or greater than the denominator. Improper fractions are always greater than or equal to one when expressed as decimals. For example, 8/5 and 40/15 are improper fractions.
  
   
• Mixed Fractions:  A mixed fraction is made of a whole number and a proper fraction. The whole number will be written first and then followed by a fraction. For example, 4 3/6 and 8 5/9 are mixed fractions.

To find the reciprocal of a fraction, follow these simple steps.

 

Step 1: Understand which one is the numerator and which is the denominator. 

 

Step 2: Reverse the numerator and denominator. Thus, the numerator becomes the new denominator and the denominator becomes the new numerator.

 

Step 3: Write the new fraction, this is the reciprocal of the original fraction.

 

Now, let’s look how to find the reciprocal of different types of fractions.
 

Step 1: First, you have to convert the mixed number into an improper fraction. 

 

For example, 6 1/2  is a mixed fraction, it should be converted to improper fraction. For that, we follow these steps:
First, multiply the whole number by the denominator of the fraction.


Now, add the numerator to the result.


Write the sum as the new numerator, and denominator should be the same.


The answer is 13/2.

 

Step 2: Reverse the improper fraction now. 

 

        13/2 => 2/13

 

Step 3: The reciprocal of 6 1/2 is 2/13.
 

Step 1: For example, 3²/4³ is a fraction with exponents. 

 

Step 2: Reverse the numerator and denominator, that is, 4³/3². For reciprocal, powers of both numerator and denominator will also be swapped. 


Step 3: The reciprocal of 3²/4³ is 4³/3².

Step 1: Exchange the numerator for the denominator.  For example, – 3/2

 

            3/2 => 2/3

 

Step 2: The negative sign remains unchanged. So, the reciprocal of – 3/2 is – 2/3.
 

There are different types of reciprocal functions, one of which is expressed as f(x) = k/x, where k is a real number and x cannot be zero because division by zero is undefined. Now, let’s graph the function f(x) = 1/x by plotting x and y values.


In the reciprocal function f(x) = 1/x, x can never be 0 because division by zero is undefined. The graph reveals that it never touches the x-axis or y-axis. The y-axis is called the vertical asymptote and x-axis is the horizontal asymptote, as the hyperbola approaches it but never intersects it.
 

Now, let’s understand how to do operations on reciprocal fractions. 

 

Addition of Reciprocal Fractions

 

Step 1: Start by finding the reciprocal of each fraction. The reciprocal of a fraction a/b is b/a. For example, 3/4 and 5/6, its reciprocal will be 4/3 and 6/5.

 

Step 2: After the reciprocals, the fractions will be added. 
 

        4/3 + 6/5  

    Here, the fractions are with different denominator. So, we have to make fractions with similar denominators. Therefore, LCM of 3 and 5 is 15.

        4 x 5 / 3 x 5 = 20 /15 

6 x 3 / 5 x 3 = 18 /15

    Now, add them together
        
20 / 15 + 18 / 15 = 48 / 30

 

Step 3: Finally, simplify the resulting fraction if possible.

        48 / 30 => 8 / 5
 

• Cooking and Recipe: If a recipe calls for 2/3 cup of flour per serving, but you want to find how many servings you can make with 1 cup, you take the reciprocal (3/2) and multiply. By multiplying 1 cup by the reciprocal (3/2), you determine that you can make 1 1/2 servings.           1 x 3 / 2 = 3 / 2 = 1.5 servings
  
   
• Speed and Time: If a car travels at 4/5 of a mile per minute, its time per mile is the reciprocal (5/4 minutes per mile).
  
   
• Construction and Scaling: If a worker takes 2/3 of an hour to complete a task, the reciprocal (3/2) helps determine how many tasks they can complete per hour (1.5 tasks per hour). 
  
   
• Exchange Rates and Currency Conversion: If 1 dollar = 3/4 euros, then the reciprocal (4/3) tells you that 1 euro = 4/3 dollars.