Question

Find three rational numbers between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] .

Answer 1

Hint: In this question, we have to find three rational numbers between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] . We will first add the given two numbers and divide the result by \[2\] to get a rational number between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] . We will then again repeat the same process to find three rational numbers between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\].

Complete step-by-step answer:
This question is based on rational numbers. Rational Number is defined as any number of the form \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and denominators \[q \ne 0\] . For example, \[\dfrac{2}{3}\] is a rational number since \[2\] and \[3\] are integers and the denomination \[3\] is not equal to zero.
Consider two rational numbers \[a\] and \[b\] .
To find a rational number between \[a\] and \[b\]. We simply add the two numbers \[a\] and \[b\], and then divide the sum by \[2\] .
Then \[{a_1} = \dfrac{{a + b}}{2}\] Where \[{a_1}\] is a rational number between \[a\] and \[b\].
Now to find a more rational number we simply repeat the above process with \[{a_1}\] and \[a\] .
Consider the given question,
We have to find three rational numbers between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\].
First rational number between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] is \[\dfrac{{(\dfrac{2}{3} + \dfrac{3}{4})}}{2}\]
On solving , we get \[\dfrac{{(\dfrac{2}{3} + \dfrac{3}{4})}}{2} = \dfrac{{(\dfrac{{8 + 9}}{{12}})}}{2} = \dfrac{{(\dfrac{{17}}{{12}})}}{2} = \dfrac{{17}}{{24}}\]
Hence , we have \[\dfrac{2}{3} < \dfrac{{17}}{{24}} < \dfrac{3}{4}\]
Now we find a rational number between \[\dfrac{2}{3}\] and \[\dfrac{{17}}{{24}}\] .
A rational number between \[\dfrac{2}{3}\] and \[\dfrac{{17}}{{24}}\] is \[\dfrac{{(\dfrac{2}{3} + \dfrac{{17}}{{24}})}}{2}\].
On solving we get
\[
  \dfrac{{(\dfrac{2}{3} + \dfrac{{17}}{{24}})}}{2} = \dfrac{{(\dfrac{{16 + 17}}{{24}})}}{2} \\
   = \dfrac{{(\dfrac{{33}}{{24}})}}{2} = \dfrac{{33}}{{48}} \;
 \]
Hence, we have \[\dfrac{2}{3} < \dfrac{{33}}{{48}} < \dfrac{{17}}{{24}} < \dfrac{3}{4}\]
Similarly we find rational number between \[\dfrac{{17}}{{24}}\] and \[\dfrac{3}{4}\]
A rational number between \[\dfrac{{17}}{{24}}\] and \[\dfrac{3}{4}\] is \[\dfrac{{(\dfrac{3}{4} + \dfrac{{17}}{{24}})}}{2}\].
On solving we get,
\[
  \dfrac{{(\dfrac{3}{4} + \dfrac{{17}}{{24}})}}{2} = \dfrac{{(\dfrac{{18 + 17}}{{24}})}}{2} \\
   = \dfrac{{(\dfrac{{35}}{{24}})}}{2} = \dfrac{{35}}{{48}} \;
 \]
Therefore, \[\dfrac{2}{3} < \dfrac{{33}}{{48}} < \dfrac{{17}}{{24}} < \dfrac{{35}}{{48}} < \dfrac{3}{4}\] is three rational number between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] .
Hence three rational number between \[\dfrac{2}{3}\] and \[\dfrac{3}{4}\] are \[\dfrac{{33}}{{48}}\], \[\dfrac{{17}}{{24}}\] and \[\dfrac{{35}}{{48}}\]
So, the correct answer is “ \[\dfrac{{33}}{{48}}\], \[\dfrac{{17}}{{24}}\] and \[\dfrac{{35}}{{48}}\] ”.

Note: We can add , subtract, multiply and divide two rational numbers to get a rational number.
To add two numbers in fraction we take lcm of denominator and then make the denominator equal to LCM . For example, \[\dfrac{2}{3} + \dfrac{3}{4} = \dfrac{{2 \times 4 + 3 \times 3}}{{12}} = \dfrac{{17}}{{12}}\] here we take LCM of denominator, then divide the LCM by denominator and multiply the result by numerator.