Question

Insert three rational numbers between $ \dfrac{1}{3} $ and $ \dfrac{4}{5} $ . And arrange in descending order.

Answer 1

Hint: Numbers which can be written as p/q form where p and q are integer and there is no common factor between p and q called rational number (q can’t be equal to 0). The numbers which are not rational are called irrational numbers ( $ \pi ,\sqrt{3},\sqrt{2} $ are irrational numbers). Here we need to insert three numbers between them so first we make both denominators equal so that we can compare quantities.

Complete step-by-step answer:
Take LCM of denominators. Now, Divide LCM by each numerator then multiply and divide by following quantity to fractions.
Multiply first quantity by $ \dfrac{5}{5} $ and second quantity by $ \dfrac{3}{3} $ in order to make both denominators equal.
 $ \dfrac{1}{3}=\dfrac{1}{3}\times \dfrac{5}{5}=\dfrac{5}{15} $
 $ \dfrac{4}{5}=\dfrac{4}{5}\times \dfrac{3}{3}=\dfrac{12}{15} $

Here we have two quantities between these numbers. If we take only numerators then there is a difference of 7 between them so we will add 1, 2 and 3 in the first quantity’s numerator consecutively. Then we will have three numbers which can be inserted between given quantities.
 $ \dfrac{5+1}{15}=\dfrac{6}{15},\dfrac{5+2}{15}=\dfrac{7}{15},\dfrac{5+3}{15}=\dfrac{8}{15} $

Now, we have three numbers which can be inserted between $ \dfrac{1}{3} $ and $ \dfrac{4}{5} $ .
 $ \dfrac{1}{3}=\dfrac{5}{15},\dfrac{6}{15},\dfrac{7}{15},\dfrac{8}{15},\dfrac{4}{5}=\dfrac{12}{15} $

Write it in descending order.
 $ \dfrac{4}{5},\dfrac{8}{15},\dfrac{7}{15},\dfrac{2}{5},\dfrac{1}{3} $

Note:If the denominators are equal then the value of fraction depends on the numerator. Numerator having greater value will be a big quantity. If we want to make a given dfraction’s denominator equal then take LCM of every denominator and divide LCM by each numerator and then multiply numerator and denominator by that quantity. Now we will have fractions which have equal denominators. If the difference between numerators is less than 1, then we will add 0.01 in order to insert rational numbers between the given quantities.