Rational numbers between two given rational numbers

A rational number is a number which can be written in the form of $\frac{p}{q}, q\neq 0$ . We can find infinitely many rational numbers between any two rational numbers. This property of rational numbers is known as the dense property. Methods to find rational numbers between any two rational numbers are given below..

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Method-1 (Average Technique)

Suppose we are required to find rational numbers between two rational numbers $a$

and $b$ such that $a< b$ . Since the average of two numbers always lying between the numbers, so $\frac{a+b}{2}$ is a rational number lying between $a$ and $b$ . We continously find out the averages of two numbers to find a number in between the first two numbers. We continue this method until we find out as many rational numbers as we need.

e.g. Find 4 rational numbers between 1 and 2 .

sol. Let $a=1$ and $b=2$ then $\frac{a+b}{2}=\frac{1+2}{2}=\frac{3}{2}$ is a number between 1 and 2.

Now a number between 1 and $\frac{3}{2}$ is $\frac{1+\frac{3}{2}}{2}=\frac{5}{4}$ . We can proceed in this manner to find two more numbers between 1 and 2. Thus four rational numbers between 1 and 2 is $\frac{3}{2}, \, \frac{5}{4},\, \frac{7}{4}$ and $\frac{11}{8}$ .

Method 2 (Gap Method)

Finding $n$ rational numbers between any two rational numbers $a$ and $b$ when $a< b$ . Use the following steps –

Step 1- Find the gap between the given rational numbers $a$ and $b$ ( $a< b$ ). Gap = $b-a$

Step 2- Divide the gap by $n+1$ .

Step 3- Multiply $\frac{b-a}{n+1}$ by $1,2,3,4.........n$ and add each product to $a$ .

Thus $n$ rational numbers between the given rational numbers $a$ and $b$ are

$a+\frac{b-a}{n+1}, \, a+2\frac{b-a}{n+1}\, a+3\frac{b-a}{n+1},\, ........., a+n\frac{b-a}{n+1}$ .

e.g. Find 6 rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ .

Sol.Let $a=\frac{3}{5}$ and $b=\frac{4}{5}$ , then $\frac{3}{5}< \frac{4}{5}$ . Gap between $\frac{3}{5}$ and $\frac{4}{5}$ = $\frac{4}{5}-\frac{3}{5}=\frac{1}{5}$ .

To find 6 rational numbers , divide by $6+1=7$ .

Dividing the gap by 7, we get $\frac{b-a}{n+1}=\frac{1}{35}$

Thus the 6 rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are

$\frac{3}{5}+1\cdot \frac{1}{35}$ , $\frac{3}{5}+2\cdot \frac{1}{35}$ , $\frac{3}{5}+3\cdot \frac{1}{35}$ , $\frac{3}{5}+4\cdot \frac{1}{35}$ , $\frac{3}{5}+5\cdot \frac{1}{35}$ , and $\frac{3}{5}+6\cdot \frac{1}{35}$ ,

i.e. $\frac{22}{35}, \frac{23}{35}, \frac{24}{35}, \frac{25}{35}, \frac{26}{35}\, \, and\, \, \, \frac{27}{35}$ .

Method 3 : To find Rational Numbers between Two Given Rational Numbers with the Same Denominator

(i) If the numerators differ by a large value then you can simply write the rational numbers with an increment of one while keeping the denominator part unchanged.

(ii) If the numerators differ by a smaller value than the number of rational numbers to be found simply multiply the numerators and denominators by multiples of 10.

e.g. Suppose we have to find rational numbers between $\frac{2}{8}$ and $\frac{7}{8}$ .

Obviously $\frac{3}{8},\frac{4}{8}, \frac{5}{8}, \frac{6}{8}$ are rational numbers between the given numbers. But we can write $\frac{2}{8}=\frac{20}{80}$ and $\frac{7}{8}=\frac{70}{80}$ . Now the numbers $\frac{21}{80}, \frac{22}{80}, \frac{23}{80}, .........., \frac{68}{80},\frac{69}{80}$ all are between $\frac{2}{8}$ and $\frac{7}{8}$ .

Also $\frac{2}{8}$ can be expressed as $\frac{200}{800}$ and $\frac{7}{8}$ as $\frac{700}{800}$ . Now we see that $\frac{201}{800},\frac{202}{800}, \frac{203}{800},.............., \frac{698}{800},\frac{699}{800}$ are between $\frac{2}{8}$ and $\frac{7}{8}$ . In this way, we can go on inserting more and more rational numbers between $\frac{2}{8}$ and $\frac{7}{8}$ .

So we can find countless rational numbers between any two given rational numbers.

Method 4: To find Rational Numbers between Two given Rational Numbers with the Different Denominators

To find Rational Numbers between Two Rational Numbers with the Different Denominators you need to equate the Denominators firstly.
You can Equate the Denominators by finding their LCM or by multiplying the denominators of one to another one’s numerator and denominator.

e.g. Find any 10 rational numbers between $-\frac{5}{6}$ and $\frac{5}{8}$ .

Sol. We first convert $-\frac{5}{6}$ and $\frac{5}{8}$ to rational numbers with the same denominator. $-\frac{5\times 4}{6\times 4}=-\frac{20}{24}$ and $\frac{5\times 3}{8\times 3}=\frac{15}{24}$ . Thus we have $-\frac{19}{24},-\frac{18}{24},-\frac{17}{24},.......,\frac{14}{24}$ as rational numbers between $-\frac{5}{6}$ and $\frac{5}{8}$ .

Some more examples:

e.g. How many rational numbers lie between -1/4 and 1/4?

Sol. We can write $\small -\frac{1}{4}=-\frac{10}{40}$ and $\small \frac{1}{4}=\frac{10}{40}$ . Now the numbers $\small -\frac{9}{40}, -\frac{8}{40},-\frac{7}{40},-\frac{6}{40},............., \frac{6}{40},\frac{7}{40},\frac{8}{40},\frac{9}{40}$ all are between $\small -\frac{1}{4}$ and $\small \frac{1}{4}$ .

Also $-\frac{1}{4}$ can be expressed as $\small -\frac{100}{400}$ and $\frac{1}{4}$ as $\frac{100}{400}$ . Now we see that $-\frac{99}{400},-\frac{98}{400},- \frac{97}{400},.............., \frac{98}{400},\frac{99}{400}$ are between $-\frac{1}{4}$ and $\frac{1}{4}$ . In this way, we can go on inserting more and more rational numbers between $-\frac{1}{4}$ and $\frac{1}{4}$ .

So we can find countless rational numbers between any two given rational numbers.

e.g.(ii) list three rational numbers between -3 and -2.

sol. Let $\small a=-3$ and $\small b=-2$ . Then $\small \frac{-3+(-2)}{2}=-\frac{5}{2}$ is a rational number between -3 and -2.

Now a number between -3 and $\small -\frac{5}{2}$ is $\small \frac{-3+(-\frac{5}{2})}{2}=-\frac{11}{4}$ . We can proceed in this manner to find one more numbers between -3 and -2. Thus three rational numbers between -3 and -2 are $\small -\frac{5}{2},-\frac{11}{4}$ and $\small -\frac{9}{4}$ .