Subscribe to our YouTube channel for the latest videos, updates, and tips.


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.

Rational Numbers

Rational Numbers

Decimal Representation of Rational Numbers

Rational Numbers in Terminating and Non-Terminating Decimals

Recurring Decimals as Rational Numbers

Laws of Algebra for Rational Numbers

Comparison between Two Rational Numbers

Rational Numbers Between Two Unequal Rational Numbers

Representation of Rational Numbers on Number Line

Problems on Rational numbers as Decimal Numbers

Problems Based On Recurring Decimals as Rational Numbers

Problems on Comparison Between Rational Numbers

Problems on Representation of Rational Numbers on Number Line

Worksheet on Comparison between Rational Numbers

Worksheet on Representation of Rational Numbers on the Number Line







9th Grade Math

From Rational Numbers Between Two Unequal Rational Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Terms Related to Simple Interest | Simple Interest Formula | Principal

    Jun 18, 25 02:57 PM

    In terms related to simple interest we will learn all the terms related to simple interest. The terms related to simple interest are Interest, Principal, Amount, Simple Interest, Time or period of tim…

    Read More

  2. Introduction to Simple Interest | Definition | Formula | Examples

    Jun 18, 25 01:50 AM

    Simple Interest
    In simple interest we will learn and identify about the terms like Principal, Time, Rate, Amount, etc. PRINCIPAL (P): The money you deposit or put in the bank is called the PRINCIPAL.

    Read More

  3. 5th Grade Profit and Loss Percentage Worksheet | Profit and Loss | Ans

    Jun 18, 25 01:33 AM

    5th Grade Profit and Loss Percentage Worksheet
    In 5th grade profit and loss percentage worksheet you will get different types of problems on finding the profit or loss percentage when cost price and selling price are given, finding the selling pri…

    Read More

  4. Worksheet on Profit and Loss | Word Problem on Profit and Loss | Math

    Jun 18, 25 01:29 AM

    Worksheet on Profit and Loss
    In worksheet on profit and loss, we can see below there are some different types of questions which we can practice in our homework.

    Read More

  5. Calculating Profit Percent and Loss Percent | Profit and Loss Formulas

    Jun 15, 25 04:06 PM

    In calculating profit percent and loss percent we will learn about the basic concepts of profit and loss. We will recall facts and formula while calculating profit percent and loss percent. Now we wil

    Read More