Page 30 - Start Up Mathematics_8 (Non CCE)

1 110× 10 5 52 10×× 100
II. Now convert to equivalent fractions. = = = =
6 610× 60 3 32 10×× 60
III. Now we have a bigger range of numerators to choose for in-between rational numbers,

i.e., 11,12, 13, ..., 98, 99
IV. The in-between rational numbers with 60 as the common denominator are
11 12 13 98 99 10 11 12 13 98 99 100
, , , ..., , or < < < < ... < < <
60 60 60 60 60 60 60 60 60 60 60 60

To find more in-between rational numbers, you can choose the LCM as multiple of 100, 1000, ... and so on.

Simpler Method of Finding Rational Numbers Between Two Rational Numbers
p r p r p 1  p r  r
“If q and are any two rational numbers such that q < s , then q < 2   q + s  < . ”
s
s

Thus, to find rational numbers between any two given rational numbers, follow these steps:
Step 1: Add the rational numbers.
1 p r
Step 2: Multiply the result by to get one rational number between q and . This is the arithmetic mean
s
p r 2
of q and .
s
p
Step 3: Add q and the rational number obtained.
1
Step 4: Multiply the result by to get one more rational number.
2
Repeat this method to get as many rational numbers as required.

2 4
Example 46: Find three rational numbers between and .
3 5
2 4 52× + 34× 10 12+ 22
Solution: + = = =
3 5 15 15 15
22 1 22 1× 11 1× 11
× = = =
15 2 15 2× 15 1 15×
11 2 4 2 11 4
So, is the first rational number between and , i.e., < < .
15 3 5 3 15 5
2 11 52 111 10 11× + × + 21
+ = = =
3 15 15 15 15

21 1 21
× =
15 2 30
2 21 11 4
So, < < <
3 30 15 5
11 + 4 = 11 12+ = 23
15 5 15 15
23 1 23
× =
15 2 30

2 21 11 23 4
So, < < < <
3 30 15 30 5

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