Rational Numbers Between Two Unequal Rational Numbers

As we know that rational numbers are the numbers which are represented in the form of p/q where ‘p’ and ‘q’ are integers and ‘q’ is not equal to zero. So, we can call rational numbers as fractions too. So, in this topic we will get to know how to find rational numbers between two unequal rational numbers.

Let us suppose ‘x’ and ‘y’ to be two unequal rational numbers. Now, if we are told to find a rational number lying in the mid- way of ‘x’ and ’y’, we can easily find that rational number by using the below given formula:

\(\frac{1}{2}\)(x + y), where ‘x’ and ‘y’ are the two unequal rational numbers between which we need to find the rational number.

Rational numbers are ordered, i.e., given two rational numbers x, y either x > y, x < y or x = y.

Also, between two rational numbers there are infinite number of rational numbers.

Let x, y (x < y) be two rational numbers. Then

\(\frac{x + y}{2}\) - x = \(\frac{y - x}{2}\) > 0; Therefore, x < \(\frac{x + y}{2}\)

y - \(\frac{x + y}{2}\) = \(\frac{y - x}{2}\) = \(\frac{y - x}{2}\) > 0; Therefore, \(\frac{x + y}{2}\) < y.

Therefore, x < \(\frac{x + y}{2}\) < y.

Thus, \(\frac{x + y}{2}\) is a rational number between the rational numbers x and y.

For understanding it much better let us have a look at some of the below mentioned examples:

1. Find a rational number lying mid- way between \(\frac{-4}{3}\) and \(\frac{-10}{3}\).

Solution:

Let us assume x = \(\frac{-4}{3}\)

                                       y = \(\frac{-10}{3}\)

If we try to solve the problem using formula mentioned above in the text, then it can be solved as:

\(\frac{1}{2}\){( \(\frac{-4}{3}\))+ (\(\frac{-10}{3}\))}

⟹ \(\frac{1}{2}\){( \(\frac{-14}{3}\))}

⟹ \(\frac{-14}{6}\)

⟹ \(\frac{-7}{6}\)

Hence, (\(\frac{-7}{6}\)) or (\(\frac{-14}{3}\)) is the rational number lying mid- way between \(\frac{-4}{3}\)and \(\frac{-10}{3}\).

2. Find a rational number in the mid- way of \(\frac{7}{8}\) and \(\frac{-13}{8}\)

Solution: 

Let us assume the given rational fractions as:

x = \(\frac{7}{8}\), 

y = \(\frac{-13}{8}\)

Now we see that the two given rational fractions are unequal and we have to find a rational number in the mid- way of these unequal rational fractional. So, by using above mentioned formula in the text we can find the required number. Hence,

From the given formula:

\(\frac{1}{2}\)(x + y) is the required mid- way number.

So, \(\frac{1}{2}\){ \(\frac{7}{8}\)+ (\(\frac{-13}{8}\))}

⟹ \(\frac{1}{2}\)( \(\frac{-6}{8}\))

⟹ \(\frac{-6}{16}\)

⟹  (\(\frac{-3}{8}\))

Hence, (\(\frac{-3}{8}\)) or (\(\frac{-6}{16}\)) is the required number between the given unequal rational numbers.

In the above examples, we saw how to find the rational number lying mid- way between two unequal rational numbers. Now we would see how to find a given amount of unknown numbers between two unequal rational numbers.

The process can be better understood by having a look at following example:

1. Find 20 rational numbers in between (\(\frac{-2}{5}\)) and \(\frac{4}{5}\).

Solution:

To find 20 rational numbers in between (\(\frac{-2}{5}\)) and \(\frac{4}{5}\), following steps must be followed:

Step I: (\(\frac{-2}{5}\)) = \(\frac{(-2) × 5}{5 × 5}\) = \(\frac{-10}{25}\)

Step II: \(\frac{4 × 5}{5 × 5}\) = \(\frac{20}{25}\)

Step III: Since, -10 < -9 < -8 < -7 < -6 < -5 < -4 ...… < 16 < 17 < 18 < 19 < 20

Step IV: So, \(\frac{-10}{25}\) < \(\frac{-9}{25}\) < \(\frac{-8}{25}\) < …… <  \(\frac{16}{25}\) < \(\frac{17}{25}\) < \(\frac{18}{25}\) <  \(\frac{19}{25}\).

Step V: Hence, 20 rational numbers between \(\frac{-2}{5}\) and \(\frac{4}{5}\) are:

\(\frac{-9}{25}\), \(\frac{-8}{25}\), \(\frac{-7}{25}\), \(\frac{-6}{25}\), \(\frac{-5}{25}\), \(\frac{4}{25}\) ……., \(\frac{2}{25}\), \(\frac{3}{25}\), \(\frac{4}{25}\), \(\frac{5}{25}\), \(\frac{6}{25}\), \(\frac{7}{25}\), \(\frac{8}{25}\), \(\frac{9}{25}\), \(\frac{10}{25}\).

All the questions of this type can be solved using above steps.

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