Question

Find three rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\] .

Answer 1

Hint: Here we are given two numbers and we need to find three rational numbers between these numbers. For that, first of all, we need to make their denominators the same so that we can compare them. Once the denominators are made equal, the number with a greater numerator will be greater than the other number. Now, we will see how many numbers are there between both the numerators. If the numbers are less than what we require i.e. three, we will then multiply and divide both the fractions by \[\dfrac{{10}}{{10}}\left( { = 1} \right)\] . Now, we will have two fractions with same denominators and numerators with more than ten numbers in between. We can choose any three numbers between these two fractions, which will be required numbers.

Complete step-by-step answer:
We are given two numbers i.e. \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\] .
Now, we need to make their denominators the same. We do this by taking LCM
 \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\] \[ \Rightarrow \dfrac{9}{{15}} and \dfrac{{10}}{{15}}\]
Hence, we got the two numbers to be \[\dfrac{9}{{15}}and\dfrac{{10}}{{15}}\] .
Now we don’t have any whole number between \[9\] and \[10\] .
We will now Multiply the two numbers by \[\dfrac{{10}}{{10}}\] .
So, the numbers become \[\dfrac{9}{{15}} \times \dfrac{{10}}{{10}}\] and \[\dfrac{{10}}{{15}} \times \dfrac{{10}}{{10}}\] \[ \Rightarrow \dfrac{{90}}{{150}}\] and \[\dfrac{{100}}{{150}}\]
Now, we have to find three rational numbers between \[\dfrac{{90}}{{150}}\] and \[\dfrac{{100}}{{150}}\] .
Denominators of both the numbers are same and \[9\] whole numbers between \[90\] and \[100\] .
We can choose any three whole numbers between \[90\] and \[100\] with denominator \[ = 150\] for each number.
Hence, the three required numbers are \[\dfrac{{92}}{{150}},\dfrac{{95}}{{150}},\dfrac{{97}}{{150}}\] .
So, \[\dfrac{{92}}{{150}},\dfrac{{95}}{{150}},\dfrac{{97}}{{150}}\] are three rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\] .
So, the correct answer is “Option B”.

Note: Basically, we need to make the denominator the same so that we can compare numbers. We can do it in any possible way. Here we made the denominator the same by taking LCM but we could not find any whole number between the numerators of both the numbers. Then, we multiplied it with \[\dfrac{{10}}{{10}}\] . Instead of 10, we can take any number greater than how many numbers we have to find.