LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

INSERTING 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.**

**Answer:-**

**
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.**

**Then we follow usual procedure.**

**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.)**