# CLASS-7INSERTING RATIONAL NUMBERS BETWEEN TWO NUMBERS

INSERTING RATIONAL NUMBERS BETWEEN TWO NUMBERS -

1. Infinite number of rational numbers exist between any two distinct rational numbers. We know that a rational number is a number which can be written in the form of p/q where p and q are integers and q≠0.

2. To find the rational numbers between two rational numbers with different denominators, the denominators should be equated. Equating

the denominators can be done by finding their LCM.

ª However, there is no integer between two consecutive integers.

For example, between -7 and 7 there are 7 - (-7) - 1 = 7 + 7 - 1 = 14 – 1 = 13 integers.

The integers are -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 and 6 but there is no integer between 2 and 3 since they are consecutive integers.

Thus, we find that between two given integers there may or may not lie any integer.

How to insert many rational numbers between two rational numbers?

ª Inserting rational numbers between rational numbers with same denominators: Insert four rational numbers between -4/7 and 2/7.

The 4 integers between -4 and 2 are -3, -2, -1, 0, 1.

The four rational numbers -3/7, -2/7, -1/7, 0/7 and 1/7 lies between -4/7 and 2/7.

ª Inserting rational numbers between rational numbers with different denominators:

Insert 6 Rational numbers between 3/8 & 3/5.

ª Inserting n rational numbers between two consecutive rational numbers : First multiply each rational number by (n+1).

L.C.M. of the denominators = 40.

So, 3/8 and 3/5 = 15/40 & 24/40.

The integers between 15 and 24 are 16, 17, 18, 19, 20, 21, 22, 23. So, any 6 rational numbers are:-

16/40, 17/40, 18/40, 19/40, 20/40, 21/40.

Find five rational numbers between 3/5 and 4/5.

Answer:- First method to find rational number between two numbers: Since we have

to find 5 rational numbers between 3/5 and 4/5, so we will multiply the numerator and denominator of the given numbers by 5+1, i.e. equal to 6.

We get,

3/5 = (3 × 6)/(5 × 6) = 18/30 and 4/5 = (4 × 6)/(5 × 6) = 24/30

Now, we have to find 5 rational numbers between 18/30 and 24/30, which will come as

(19/30), (20/30), (21/30), (22/30), (23/30)

= (19/30), (2×10/3×10), (7×3/10×3), (11×2/15×2), (23/30)

= (19/30), (2/3), (7/10), (11/15)

Thus, five rational numbers between 3/5 and 4/5 are = (19/30), (2/3), (7/10), (11/15) and (23/30)   (Ans.)