Rational numbers between two rational numbers

We can find unlimited rational numbers between two rational numbers. A number between two rational numbers, can be a rational number or a whole number. A rational number is a real number that can be expressed as P/Q, where P and Q are any integers and Q ≠ 0.

When we speak about whole numbers, then between any two whole numbers, there are limited numbers only. For example, between 3 and 7, there are only three whole numbers, i.e. 4, 5 and 6. But there is no limit to finding the rational numbers between two rationals.

Also, read: Properties of Rational Numbers

How to Find Rational Numbers Between Two Rational Numbers?

To find the rational numbers between two rational numbers, we have to check with the following conditions:

  • When denominators are the same
  • When denominators are different

When Denominators are the same

As we already know, the arithmetic operations on rational numbers become easy when the denominators are the same. Hence, it also applies to finding the rational numbers between two rational numbers. Follow the steps:

Step 1: Check the values on the numerators of the rational numbers

Step 2: Find by how many values, the numerators differ from each other

Step 3: Since, the denominators are the same for the two rational numbers, therefore, we can write the rational numbers between the two given rationals, in the increasing order of numerator, if the difference between the two numerators is more.

Example: How many rational numbers are there between ⅗ and 9/5?

Solution: Given, ⅗ and 9/5 are two rational numbers.

Since, the denominators are the same, there we can compare the numerators here.

3 < 5

Therefore, the rational numbers between ⅗ and 9/5 are:

⅘, 5/5, 6/5, 7/5,8/5

Or

⅘, 1, 6/5, 7/5 and 8/5

Step 4: If the difference between two numerators is less, and we need to find more rational numbers, then multiply the numerator and denominator of the given rational numbers by multiples of 10.

Example: Find 10 rational numbers between 4/5 and 8/5.

Solution: As we can see, the denominators of given rational numbers are the same.

Now, on comparing numerators,

4 < 8

There are only three numbers between 4 and 8, i.e., 5, 6 and 7.

Hence, we will multiply both the rational numbers by 10/10.

⅘ x 10/10 = 40/50

8/5 x 10/10 = 80/50

Now again comparing the numerators, we can see,

80>40

And there are more than 10 numbers between 40 and 80. Hence, we can take any 10 rational numbers between 40/50 and 80/50.

Therefore, the ten rational numbers are 41/50, 42/50, 45/50, 50/50, 55/50, 60/50, 65/50, 70/50, 72/50, 75/50.

We can further write them in the simplest form:

41/50, 21/25, 9/10, 1, 11/10, 6/5, 13/10, 7/10, 36/25 and 3/2

When Denominators are Different

We have already discussed above to find the rational numbers, when the given two rational numbers have the same denominators. But what if the denominators are different? Here we apply the same rule, what we have applied while finding the sum and difference of two rational numbers.

Suppose we have two rational numbers with different denominators, then follow the below steps to find the rational numbers between them.

Step 1: Find the LCM of two rational numbers first.

Step 2: Multiply and divide the two rational numbers, by the value that results in the denominators equal to the obtained LCM.

Step 3: Once the denominators become the same, follow the same rules as we have discussed for the rational numbers with the same denominators.

Example: Find the rational numbers between ½ and ⅔.

Solution: The two given rational numbers are ½ and ⅔

LCM of denominators (2 and 3) = 6

Therefore, multiply and divide ½ and ⅔ by 3/3 and 2/2, respectively.

½ x (3/3) = 3/6

⅔ x (2/2) = 4/6

Now, the denominators are the same.

Numerators are 3 and 4. Hence, we cannot find any number in between them.

Thus, we have to multiply again the rational numbers 3/6 and 4/6 by 10/10, each.

3/6 x 10/10 = 30/60

4/6 x 10/10 = 40/60

Therefore, the rational numbers between ½ and ⅔ are:

31/60, 32/60, 33/60, 34/60, 35/60, 36/60, 37/60, 38/60 and 39/60.

Finding the Next Rational Numbers

To find the next rational numbers for a given pattern of rational numbers, we have to check the common factors of numerator and denominator.

Suppose the given pattern of rational numbers are: -⅓, -2/6, -3/9, -4/12,….

  • -⅓
  • -2/6 = -⅓ x (2/2)
  • -3/9 = -⅓ x (3/3)
  • -4/12 = -⅓ x (4/4)

Thus, we can see, each time, the rational number -⅓ is multiplied by increasing order of numbers both in numerator and denominator.

Hence, to find the next rational numbers, we can multiply -⅓ by 5/5, 6/6, and so on.

  • -⅓ x 5/5 = -5/15
  • -⅓ x 6/6 = -6/18
  • -⅓ x 7/7 = -7/21

Rational Numbers Related Articles

Problems and Solutions

Q.1: Find the rational number between 5/11 and 8/11.

Solution: Since, the rational numbers are having same denominators, therefore, we will compare the numerators here.

Numerator 5 < Numerator 8

The numbers between 5 and 8 are 6 and 7.

Therefore, the rational numbers between 5/11 and 8/11 are 6/11and 7/11.

Q.2: Find the rational numbers between ¼ and 1.

Solution. We can represent 1 as 1/1.

Now, the denominators of ¼ and 1/1 are different.

Thus, LCM of 1 and 4 = 4

Now multiplying and dividing 1/1 by 4/4, we get;

1/1 x 4/4 = 4/4

Therefore, the rational numbers between ¼ and 4/4 are 2/4 and ¾.

Practice Questions

  1. Find five rational numbers between ½ and 9/2.
  2. Find ten rational numbers between 1/11 and ⅜.
  3. Find any five rational numbers between -3 and -1.
  4. Write the next four rational numbers in the given pattern: -⅓, -2/6, -3/9, _, _,_,_.

Frequently Asked Questions – FAQs

Is there any rational number between two whole numbers?

Yes, we can find rational numbers between two whole numbers.

Are there only limited rational numbers between two rational numbers?

No, there are unlimited rational numbers between two rationals.

Can we find the rational numbers between two rational numbers if the denominators are different?

Yes, we can find rational numbers, even if the denominators of the two rational numbers are different. We have to make the denominators the same by finding the LCM.

How many rational numbers are there between 2 and 3?

We can find unlimited rational numbers between 2 and 3, by multiplying and dividing them with the multiples of 10. For example, multiplying and dividing 2 and 3 by 10/10, each, we get 20/10 and 30/10. Therefore, the rational numbers between them are:
21/10, 22/10, 23/10, 24/10, 25/10, 26/10, 27/10, 28/10, 29/10.

Find six rational numbers between 3 and 4.

Multiplying and dividing 3 and 4 by 10/10, each we get;
30/10 and 40/10
Thus, the six rational numbers between 3 and 4 are 31/10, 32/10, 33/10, 34/10, 35/10 and 36/10.

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