Question

List four rational numbers between \[\dfrac{5}{7} \text{ and }\dfrac{7}{8}\].

Answer 1

Hint: We need to find four rational numbers between \[\dfrac{5}{7}and\dfrac{7}{8}\]. For that, first of all we will make their denominators the same so that we can compare. We will do this by taking LCM. Then, we will see the difference between the numerators of both the numbers. If we have sufficient whole numbers between the numerators of both the numbers we will write those numbers with same denominator and if the difference is less than the required number, we will then multiply the number by \[\dfrac{{10}}{{10}}( = 1)\]. After this, we will have the same denominators and at least ten whole numbers between their numerators and so we can choose any four numbers.

Complete step-by-step solution:
We need to find four rational numbers between \[\dfrac{5}{7} \text{ and }\dfrac{7}{8}\].
First of all, we will take LCM to make their denominator the same.
\[\dfrac{5}{7}\text{ and }\dfrac{7}{8} \Rightarrow \dfrac{{(5 \times 8)\text{ and }(7 \times 7)}}{{56}}\]
\[ \Rightarrow \dfrac{{40\text{ and }49}}{{56}}\]
Hence, the numbers become \[ \dfrac{{40}}{{56}}\text{ and }\dfrac{{49}}{{56}}\].
i.e. We, now, have to find four rational numbers between \[\dfrac{{40}}{{56}}\text{ and }\dfrac{{49}}{{56}}\].
For that, first we see the difference between the numerators of both the numbers.
Difference between the numerators\[ = 49 - 40 = 9\]
And we require only four numbers. So, we now choose any four whole numbers between \[49\text{ and } 40\] and the denominator remains the same i.e. \[56\].
Let us choose the numbers to be \[42,44,46,47\] as the numerators and denominator \[ = 56\].
So, the required four rational numbers between \[\dfrac{5}{7} \text{ and }\dfrac{7}{8}\] are \[\dfrac{{42}}{{56}},\dfrac{{44}}{{56}},\dfrac{{46}}{{56}},\dfrac{{47}}{{56}}\].

Note: We need to make sure that after making the denominators the same, we see how many whole numbers are there between their numerators. In this question there were already \[8\] numbers between their numerators, So we didn’t modify it. Otherwise, we have to multiply and divide both the numbers by such a number that there are at least as many numbers as we require. While doing calculations, we should be very attentive.