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Q5) Find five rational numbers between.

(i)\frac{2}{3}\ \&\ \frac{4}{5}

(ii)\frac{-3}{2}\ \&\ \frac{5}{3}

(iii)\frac{1}{4}\ \&\ \frac{1}{2}

Answer:

Solution 5:

(i): Following numbers can be written as,

Equalizing the denomintor,

\frac{2\times15}{3\times15}\ and\ \frac{4\times9}{5\times9}=\ \frac{30}{45}\ and\ \frac{36}{45}

Therefore, \frac{31}{45},\ \frac{32}{45},\ \frac{33}{45},\ \frac{34}{45},\ \frac{35}{45} are the five numbers.

(ii): Following numbers can be written as,

Equalizing the denominator,

\frac{-3\ \times3}{2\times3}\ and\ \frac{5\times2}{3\times2}=\ \frac{-9}{6}\ and\ \frac{10}{6}

Therefore, \frac{-8}{6},\ \frac{-7}{6},\ -1,\ \frac{-5}{6}and\ \frac{1}{6} are the five numbers.

(iii): Following numbers can be written as,

equalizing denominator,

\frac{1\times5}{4\times5}\ and\ \frac{1\times10}{2\times10}\ =\ \frac{5}{20}\ and\ \frac{10}{20}

Therefore, \frac{6}{20},\ \frac{7}{20},\ \frac{8}{20},\ \frac{9}{20} are the following numbers.

Hi everyone welcomes to today's session in this session we will learn how to find rational numbers between any two given rational numbers so the questions for today find five rational numbers between one, two by three and four by five two minus three by two and five by three and three one by four and one bytes before I give you the solution I want you to try solving it on your own now I know it will take you a minute to do that so please pause the video at this point go solve it and come back to the video all right I hope you have tried solving it on your own now it's my turn to explain let's start with the first question which is 2 by 3 and 4 by 5. now to solve this we first need to convert these rational numbers to equivalent rational numbers with the same denominator and for that, we are going to take the lcm of the denominators 3 and 5 which is 15. Now we are going to find the equivalent fraction of 2 by 3 that has the denominator 15 to do that we are going to multiply both the numerator and denominator with yes you are right 5 so this will give us 10 by 50. Similarly, we are going to take 4 by 5 and convert it into an equivalent rational number with the denominator 10 sorry 15. to do that we are going to multiply both the numerator and denominator by the number 3 which will give us 12 by 50. Now, if we look at this, we know that the rational number that lies between 11 by 15 and 12 by 15. Sorry, the rational number that lies between 10 by 15 and 12 by 15 is 11 by 50. however; the question requires us to find five rational numbers between these two numbers how do we do that we take the equivalent rational numbers that we have and we multiply it with 10 on both numerator and denominator so that will give us 100 by 150 and 120 by 150 so now it's easy for us to find out the rational numbers that lie between 100 by 150 and 120 by 150 let's start with 101 by 150 102 by 150 103 by 150 104 by 150 and 105 by 150 now you can go on doing this until you reach 119 by 150. so any number between these two numbers is a correct answer to this question now let's solve the second one again we have two rational numbers minus 3 by 2 and 5 by 3 again we need to convert this into equivalent rational numbers of the same denominator so we are going to take the lcm of 2 and 3 which will give us 6. now let's take the first rational number.
minus 3 by 2 and convert it to equivalent rational number with the denominator six to do that we are going to multiply both numerator and denominator with the number three so that will give us minus nine by six let's go to the next rational number which is 5 by 3 now we need to convert this to an equivalent rational number whose denominator is 6. to do that we are going to multiply both the numerator and the denominator by number 2 so we will get plus 10 by 6.
now if you need to find out the rational numbers that lie between minus 9 by 6 and plus 10 by 6 it's very easy it will go on from minus 8 by 6 minus 7 by 6 there is 1 then there is 1 by 6. and 2 by 6 yes so any number is between minus 9 by 6
and 10 by 6 is the correct answer to this question with that we come to our last question which is to find the rational numbers between1 by 4 and 1 byte again we are going to take the lcm of 4 and 2 and we know that the lcm is 4
So we will take the first rational number we know that there is no need to convert it to an equivalent rational number because its denominator is already 4 so let's take the next one here we are going to multiply both the numerator and denominator with 2 so we will get 2 by 4. now here this becomes a little tricky because the first number is 1 by 4 and the second number is 2 by 4 and we can't find a number between 1 by 4 and 2 by 4. so now to make our lives a little easier, what we are going to do is we are going to multiply both the rational number numerator and denominator by 10. so that will give us the rational number 10 by 40. and 20 by 40. now it's easy for us to find rational numbers that lie between 10 by 40 and 20 by 40. so we are going to write 11 by 40 12 by 40 13 by 40 14 by 40. and 15 by 40. All right, so with that we come to the end of today's session, I hope you find this helpful and enjoyable I will see you in the next section before then goodbye Up next.

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