What must be subtracted from the numerator and denominator of fraction 6 7 to give a fraction equal to 16 21?

In mathematics, a fraction is represented as a numerical value that defines a part of a whole. A fraction is a portion of any quantity taken from a whole, whereas the whole can be any number, a thing, or a certain value. For example, a circle is divided into six equal parts. If we have to express a selected part of the circle, then we can express it as 1/6, which means one in six equal parts. It can also be written as “one-sixth”, or “1 by 6”. A fraction has two parts, namely, a numerator, and a denominator. A numerator indicates the sections of the fraction and is the upper part of the fraction, whereas a denominator is the lower part of the fraction and indicates the total parts into which the fraction is divided.

Equivalent fractions

Equivalent fractions are defined as fractions that have the same value but have different numerators and denominators. For instance, 3/9, 4/12, 5/15, and 6/18 are equivalent fractions as both of their values are equal to 1/3. When all equivalent fractions are simplified, they are reduced to the same fraction. Equivalent fractions indicate the same portion of the whole. We can find an equivalent fraction for every fraction by multiplying or dividing both the numerator and the denominator with or by the same number.

How to determine Equivalent Fractions?

For every fraction, we can determine an equivalent fraction either by multiplying or dividing both the numerator and the denominator with or by the same number. That’s why when all equivalent fractions are simplified, they are reduced to the same fraction. Now, let’s study these two cases.

Case 1: Multiplying the numerator and the denominator by the same number

For every fraction, we can determine an equivalent fraction by multiplying both the numerator and the denominator with the same number. For instance, to determine the equivalent fraction of 2/3, multiply both the numerator and the denominator with the same number, i.e., 2. Thus, the equivalent fraction of 2/3 is 4/6. Similarly, we can find some other equivalent fractions by repeating the same process.

Equivalent fractions of 2/3

• Multiplying the numerator and denominator with 3, we get 2/3 × 3/3 = 6/9
• Multiplying the numerator and denominator with 4, we get 2/3 × 4/4 = 8/12
• Multiplying the numerator and denominator with 5, we get 2/3 × 5/5 = 10/15

Hence, we can conclude that, 2/3 = 4/6 = 8/12 = 10/15.

Case 2: Dividing the numerator and the denominator by the same number.

For every fraction, we can determine an equivalent fraction by dividing both the numerator and the denominator by the same number. For instance, to find the equivalent fraction of 45/300, first, we have to find their common factors. Here, 3 is a common factor of both 45 and 300. So, the equivalent fraction of 45/300 can be found by dividing its numerator and denominator by 3. Thus, the equivalent fraction of  45/300 is 15/100. 

Let us simplify the fraction further.

5 is a common factor of 15 and 100. So, 15/100 = (15÷5)/(100÷5) = 3/20

Hence, the equivalent fractions of 45/300 are 15/100 and 3/20. As there are no common factors for 3 and 20 other than 1, 3/20 is the simplified form of 45/300.

Equivalent Fractions Chart

From the given chart, we can observe that the equivalent fractions of 1/3 are 2/6, 4/12, 8/24,… Now, let us see the equivalent fractions of some unit fractions.

How to Find if two Fractions are Equivalent?

How can we find whether two fractions are equivalent or not? It is possible by the following methods:

1. Make the Denominators Equal
2. Determining the decimal form of both fractions
3. Cross Multiplication method

Making Denominators Equal

By making the denominators of the given fractions the same, we can find whether they are equivalent or not. 

Example: Determine whether 4/10 and 6/15 are equivalent fractions or not.

Solution:

The Least Common Multiple (LCM) of 10 and 15 is 30.

Multiply 4/10 by 3/3 and 6/15 by 2/2 to make their denominators equal to 30.

4/10 × 3/3 = 12/30

6/15 × 2/2 = 12/30

By making the denominators of the given fractions the same, we can find whether they are equivalent or not. Thus, by making the denominators the same, we can observe that 4/10 and 6/15 are equivalent to the same fraction, i.e., the given fractions are equivalent.

Note: If the fractions are NOT equivalent, we can check the greater or smaller fraction by looking at the numerator of both the resultant fractions. Hence, this method can also be used for comparing fractions.

Determining the Decimal form of both Fractions

By finding the decimals of the given fractions, we can check whether they are equivalent or not.

Example: Determine whether 3/4, 6/8, and 12/16 are equivalent fractions or not.

Solution:

3/4 = 0.25

6/8 = 0.25

12/16 = 0.25×

As the decimal values of the given fractions are the same, the given fractions are equivalent.

Cross Multiplication method

To identify whether two fractions are equivalent or not, cross multiply them. The fractions are equivalent if both the products are the same.

Example: Determine whether 3/6 and 4/8 are equivalent fractions or not.

Solution:

The given fractions are equivalent as both the products are 24.

Solved Example based on Equivalent Fractions

Example 1: Determine the value of “a” if 10/21 and a/16 are equivalent fractions.

Solution:

Given data,

 10/21 = a/16

⇒ a = (16 × 21)/10

⇒ a = 336/10

⇒ a =168/5

Hence, the value of “a” is 168/5.

Example 2: Determine the value of “x” if 5/8 and x/9 are equivalent fractions.

Solution: 

Given data,

5/8 = x/9

⇒ x = (5 × 9)/8

⇒ x = 45/8

Hence, the value of “x” is 45/8.

Example 3: Check whether 5/9 and 11/15 are equivalent fractions or not.

Solution:

Let us use the cross multiplication method to check whether 5/9 and 11/15 are equivalent fractions or not.

Now, by the cross multiplication of the given fractions we get 5 × 15 = 75, and 9 × 11 = 99. Here, both products are not equal, i.e., 75≠99. Hence, 5/9 and 11/15 are not equivalent fractions.

Example 4: What are the equivalent fractions of 7/8?

Solution: 

To find equivalent fractions of 7/8, multiply the numerator and denominator by the same numbers.

7/8 × (2/2) = 14/16

7/8 × (3/3) = 21/24

7/8 × (4/4) = 28/32

7/8 × (5/5) = 35/40

7/8 × (6/6) = 42/48 and so on.

Hence, the equivalent fractions of 7/8 are 14/16, 21/24, 28/32, 35/40, 42/48, and so on.

Example 5: What is its denominator, if the numerator of a fraction equivalent to 6/24 is 14?

Solution:

Let the unknown denominator be “x”.

Given that, 6/24 = 14/x

We know that, if two fractions are equivalent their products when they are cross-multiplied are equal.

Now, by the cross multiplication of the given fractions, we get 

6 × x = 24 × 14

x = (24 × 14)/6 = 56.

Hence, the value of x is 56.

FAQs on Equivalent Fractions

Question 1: What do we mean by equivalent fractions?

Answer:

Two or more fractions containing different numerators and denominators but their simplest form is same then they are called equivalent fractions. e.g. 2/10, 3/15 etc, all are simplified as 1/5 they all are equivalet fractions.

Question 2: What are the ways to determine whether a fraction is equivalent or not?

Answer:

To determine whether a fracion is equivalnt or not we need to simplify the given fractions into there simplest form and the simplest form is compared to get equivalent fractions.

Let us suppose 5/10 and 4/12 are two fractions. 5/10 can be further simplified to 1/2, and 4/12 is simplified as 1/3 and but 1/2 and 1/3 are not same so they not equivalent fractions.

Question 3: What are the equivalent fractions of 7/14?

Answer:

7/14 is simplified to ½. Hence, the equivalent fractions of 7/14 is 1/2

Question 4: Write some examples of equivalent fractions.

Answer:

Some examples of equivalent fractions are:
3/9 and 4/12 are simplified as 1/3
6/12 and 4/8 are simpified as 1/2

Question 5: What is the equivalent fraction of 1/7?

Answer:

Equivalent fraction of 1/7 are calculated by multiplying, the numerator and denominator by the same number.
1/7 = 1/7 × 2/2 = 2/14
1/7 = 1/7 × 3/3 = 3/21
1/7 = 1/7 × 4/4 = 4/28