QUESTION

# Use the method of mean to find 3 rational numbers between -1 and -2.

Hint: First assume a mean between given two numbers and use the formula $Mean=\dfrac{Sum\text{ }of\text{ }the\text{ }terms}{number\text{ }of\text{ }terms}$ to get its value. Now you have three numbers therefore assume one mean in first and second number and another mean in second and third number and then use the same formula to get their values.

To solve the above question we will write the given data first,
We will assume,
a = -1, e = -2 …………………………………………………….. (1)
We will assume ‘c’ as the mean between ‘a’ and ‘b’ and to find the value of ‘c’ we should know the formula of mean given below,
Formula:
$Mean=\dfrac{Sum\text{ }of\text{ }the\text{ }terms}{number\text{ }of\text{ }terms}$ ……………………………………………….. (2)
By using the above formula we can write the formula for ‘c’ as,
$c=\dfrac{a+e}{2}$
If we put the values of equation (1) in the above equation we will get,
$c=\dfrac{-1-2}{2}$
By simplifying the above equation we will get,
$c=-\dfrac{3}{2}$ …………………………………………………………. (3)
As $-\dfrac{3}{2}$ is a rational number therefore we have inserted the first rational number using the method of mean.
Now we will assume ‘b’ as the mean between ‘a’ and ‘c’ and by referring to the formula given in equation (2) we will get,
$\therefore b=\dfrac{a+c}{2}$
If we put the values of equation (1) and equation (3) in the above equation we will get,
$b=\dfrac{-1-\dfrac{3}{2}}{2}$
By simplifying the above equation we will get,
$\therefore b=\dfrac{\dfrac{-2-3}{2}}{2}$
$\therefore b=\dfrac{\dfrac{-5}{2}}{2}$
Further simplification in the above equation will give,
$\therefore b=\dfrac{-5}{2\times 2}$
$\therefore b=-\dfrac{5}{4}$ …………………………………………………….. (4)
As $-\dfrac{5}{4}$ is a rational number therefore we have inserted the second rational number between given two numbers by using the method of mean.
We will again consider ‘d’ as the mean between ‘c’ and ‘e’ therefore by referring the formula given in equation (2) we will write the formula for ‘d’ as,
$d=\dfrac{c+e}{2}$
If we put the values of equation (1) and equation (3) in the above equation we will get,
$\therefore d=\dfrac{-\dfrac{3}{2}-2}{2}$
By simplifying the above equation we will get,
$\therefore d=\dfrac{\dfrac{-3-4}{2}}{2}$
$\therefore d=\dfrac{\dfrac{-7}{2}}{2}$
$\therefore d=\dfrac{-7}{2\times 2}$
$\therefore d=-\dfrac{7}{4}$ ………………………………………………………… (5)
As $-\dfrac{7}{4}$ is a rational number therefore we have inserted the third rational number between given two numbers by using the method of mean.
From equation (3), equation (4) and equation (5) we can say,
$-\dfrac{3}{2},-\dfrac{5}{4},-\dfrac{7}{4}$ are the three rational numbers inserted between -1 and -2.

Note: When you are assuming the means between two numbers then keep in mind that you are using the same numbers in the formula otherwise you will get wrong answers as in this case we have to find three means therefore we have to be careful while writing the formula.