Question

A rational number lying between 3and4 is

Answer 1

Answer:

3.5

Step-by-step explanation:

If a and b are any two real numbers with a < b, then a<a+b/2<b. Also, the sum of two rational numbers is a rational number, and the ratio of two rational numbers is also a rational number. Hence if a and b are rational, then a+b/2 is also rational.

Rational Numbers: Numbers which can be expressed in the form of p/q where “p” and “q” are integers and q is non-zero, are called rational numbers. Every integer n is a rational number since it can be expressed in the form n/1, where both n and 1 are integers, and 1 is non-zero. The sum of two rational numbers is always a rational number, and the ratio of two rational numbers is also a rational number where the denominator is non-zero.

We know that if a and b are any two real numbers with a < b, then a<a+b/2<b.

Hence if a = 3 and b = 4.

We have 3<3+4/2<4⇒3<7/2<4

Hence 3 < 3.5 < 4.

Hence a rational number between 3 and 4 is 3.5.

Note: We can insert n rational numbers between a and b by making use of properties of arithmetic mean.

If we insert n A.Ms between a and b, then a,A1,A2,...,An,b form an A.P.

Since a < b, we have d > 0. So a<A1<A2<⋯<An<b

Also, b=a+(n+1)d

Hence d=b−a/n+1

Hence the n rational numbers between two rational numbers a and b are

a+ b−a/n+1,a+ 2b−a/n+1,⋯,a+ nb−a/n+1

Consider the case of inserting one rational number between 3 and 4.

We have a = 3, b=4 and n =1.

Hene the number is 3+4−31+1=3+12=3.5

Hence a rational number between 3 and 4 is 3.5