Question

List five rational numbers between:
(i) -1 and 0 (ii) -2 and -1 (iii) $-\dfrac{4}{5}$ and $-\dfrac{2}{3}$ (iv) $-\dfrac{1}{2}$ and $\dfrac{2}{3}$

Answer 1

Hint: We apply some basic rules to find the intermediate rational numbers between two given numbers. We convert them to fractions with the same denominator value. Then we multiply the fractions with the number which is 1 greater than the number of rational numbers we have been asked to find. Finally, we find middle values based on the numerator of those fractions.

Complete step by step answer:
To find the intermediate rational numbers between two given numbers we first need to match their denominator. If integers are given then we take their denominator as 1. To find the same denominator value we take the L.C.M of the actual denominators.
After matching the denominator, we multiply the fractions with the number which is 1 greater than the number of rational numbers we have been asked to find. This means if we have been asked to find r rational numbers then we multiply with $\left( r+1 \right)$.
At the end we find the middle values based on the numerator of those fractions.
For finding 5 rational numbers between -1 and 0, we convert them in $\dfrac{-1}{1}$ and $\dfrac{0}{1}$.
Now we multiply with 6. We get $\dfrac{\left( -1 \right)\times 6}{1\times 6}=\dfrac{-6}{6}$ and $\dfrac{0\times 6}{1\times 6}=\dfrac{0}{6}$.
The 5 rational numbers are $\dfrac{-5}{6},\dfrac{-4}{6},\dfrac{-3}{6},\dfrac{-2}{6},\dfrac{-1}{6}$.
For finding 5 rational numbers between -2 and -1, we convert them in $\dfrac{-2}{1}$ and $\dfrac{-1}{1}$.
Now we multiply with 6. We get $\dfrac{\left( -2 \right)\times 6}{1\times 6}=\dfrac{-12}{6}$ and $\dfrac{\left( -1 \right)\times 6}{1\times 6}=\dfrac{-6}{6}$.
The 5 rational numbers are $\dfrac{-11}{6},\dfrac{-10}{6},\dfrac{-9}{6},\dfrac{-8}{6},\dfrac{-7}{6}$.
For finding 5 rational numbers between $-\dfrac{4}{5}$ and $-\dfrac{2}{3}$, we find L.C.M of 5 and 3 which is $3\times 5=15$. We convert them in $\dfrac{\left( -4 \right)\times 3}{5\times 3}=\dfrac{-12}{15}$ and $\dfrac{\left( -2 \right)\times 5}{3\times 5}=\dfrac{-10}{15}$.
Now we multiply with 6. We get $\dfrac{\left( -12 \right)\times 6}{15\times 6}=\dfrac{-72}{90}$ and $\dfrac{\left( -10 \right)\times 6}{15\times 6}=\dfrac{-60}{90}$.
The 5 rational numbers are $\dfrac{-70}{90},\dfrac{-68}{90},\dfrac{-66}{90}\dfrac{-65}{90},\dfrac{-61}{90}$.
For finding 5 rational numbers between $-\dfrac{1}{2}$ and $\dfrac{2}{3}$, we find L.C.M of 2 and 3 which is $3\times 2=6$. We convert them in $\dfrac{\left( -1 \right)\times 3}{2\times 3}=\dfrac{-3}{6}$ and $\dfrac{2\times 2}{3\times 2}=\dfrac{4}{6}$.
Now we multiply with 6. We get $\dfrac{\left( -3 \right)\times 6}{6\times 6}=\dfrac{-18}{36}$ and $\dfrac{4\times 6}{6\times 6}=\dfrac{24}{36}$.
The 5 rational numbers are $\dfrac{-10}{36},\dfrac{-7}{36},0,\dfrac{9}{36},\dfrac{15}{36}$.

Note: There is no hard & fast rule to multiply with the number which is 1 greater than the number of rational numbers we have been asked to find. If we can find the required middle numbers only after making the denominators the same, we can skip the process of multiplying. That process is only to stretch the distance by expanding the numerator.