Find a rational number between 1/2 and 3/4

All the numbers expressed in the form p/q where q ≠ 0, and p and q are integers are rational numbers. Rational numbers have two forms of representations, namely,

1. Terminating decimal form: These rational numbers when expressed in decimal form, terminate after a finite number of digits. Generally, here the denominator of the rational number in its simplest form must be in powers of 10 to ensure that the decimal representation terminates after a finite number of digits. For example, 13/100 = 0.13, a rational number that terminates after two digits.
2. Non-terminating but recurring decimal forms: These rational numbers when expressed in decimal form, go on indefinitely without terminating but repeating a certain pattern of digits infinitely. For example, 1/3 = 0.3333… here the digit 3 repeats infinitely and does not terminate.

Methods to Find a Rational Number Between Two Rational Numbers

There exist infinitely many rational numbers between the given pair of rational numbers. Below are the two methods to find a rational number between two given rational numbers,

Approach 1: Averaging Method

In this method, we calculate the average of the two given rational numbers to find a rational number between them. Let’s represent the first rational number by a/b and the second rational number by c/d then, a rational number between a/b and c/d by averaging method is given by, ((a/b) + (c/d)) / 2.

Approach 2: Median Method

In this method, we calculate the median of two given rational numbers and that median is a rational number between the given pair of rational numbers. Let’s represent the first rational number by a/b and the second rational number by c/d. Now, the mediant of a/b and c/d is a rational number given by, (a + c) / (b + d) which is larger than a/b and smaller than c/d (provided, a/b is less than c/d).

Approach 3: Using Decimal Expansion

In this approach, we convert the given rational numbers to decimal expansion and then write numbers between those decimal expansions.

Find a rational number between 1/2 and 3/4

Solution:

Approach 1: Averaging Method

Let the first rational number be, 1/2

Let the second rational number be, 3/4

By averaging method, the rational number between 1/2 and 3/4 is given by, ((1/2) + (3/4))/2 = (5/4)/2 = 5/8

Hence, the rational number between 1/2 and 3/4 according to averaging method is 5/8

Approach 2: Mediant Method

Let the first rational number be, 1/2

Let the second rational number be, 3/4

By mediant method, the rational number between 1/2 and 3/4 is given by, (1+3)/(2+4) = 4/6 = 2/3.

Hence, the rational number between 1/2 and 3/4 according to mediant method is 2/3

Approach 3: Using Decimal Expansion

Convert both fractions to decimals. Then, find a rational number between these decimal values. For instance, 1/2 =0.5 and 3/4 =0.75 A number between these two could be 0.6, which corresponds to 3/5.

Similar Questions

Problem 1: What is the rational number between 1/5 and 1/4?

Solution:

Here, 1/5 and 1/4 are the given two rational numbers

So, by averaging method,

((1 ⁄ 5) + (1 ⁄ 4)) / 2 = 9 / 40 is a rational number between 1/5 and 1/4

Problem 2: What is the rational number between 1/3 and 3/8?

Solution:

Here, 1/3 and 3/8 are the given two rational numbers

So, by mediant method,

((3+1) / (8+3)) = 4/11