Rational numbers between 3/5 &2/3
a=3/5 , b=2/3
(a+b)/2
(3/5+2/3)/2
((9+10)/15)/2
(19/15)/(2/1) (reciprocal)
(19/15)×(1/2)
=19/30 _____(1)
rational numbers between 3/5 & 19/30
a=3/5 b=19/30
(a+b)/2
(3/5+19/30)/2
((18+19)/30)/2
(37/30)/2/1 (reciprocal)
(37/30)×1/2
=37/60___(2)
rational numbers between 3/5 & 37/60
a=3/5 b=37/60
(a+b)/2
(3/5+37/60)/2
((36+37/60)/2
(73/60)/2/1
=73/120___(3)

so,we use (a+b/2) formula for finding the rational numbers

Introduction

In this question, we are asked to find rational numbers between 3/5 and 2/3.

Method

To find rational numbers between two given rational numbers, we can use the formula:

If a/b and c/d are two given rational numbers such that a/b < c/d,="" then="" the="" rational="" numbers="" between="" a/b="" and="" c/d="" are="" of="" the="" form="" (ad="" +="" bc)/bd,="" where="" d="" is="" the="" least="" common="" multiple="" of="" b="" and="" />

Explanation

Given rational numbers are 3/5 and 2/3.

Let us find the least common multiple of 5 and 3.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60

The least common multiple of 5 and 3 is 15.

Now, we can find the rational numbers between 3/5 and 2/3 using the formula:

(ad + bc)/bd, where d is the least common multiple of b and d.

Let's substitute the values in the formula.

(3 x 3 + 2 x 5)/(5 x 3) = 19/15

Therefore, one rational number between 3/5 and 2/3 is 19/15.

We can also find another rational number by changing the order of the given rational numbers.

(2 x 5 + 3 x 3)/(5 x 3) = 21/15

Therefore, another rational number between 3/5 and 2/3 is 21/15 or 7/5.

Conclusion

Using the formula, we have found two rational numbers between 3/5 and 2/3, which are 19/15 and 7/5.