Question

Out to three numbers, the first is twice the second and three times the third. If the average of the given three numbers is 88, then find the first number?

Answer 1

Hint: We start solving the problem by assigning the variables for the first, second and third numbers. We then use the conditions that the first is twice the second and three times the third to convert second and third numbers in terms of first number. We then use the condition that the average of the given three numbers is 88 and substitute the values of second and third numbers in terms of the first number that we just got. We make subsequent calculations to get the required result.

Complete step-by-step answer:
According to the problem, we have three numbers in which the first number is twice the second number and three times the third number. We need to find the first number if the average of the three numbers is given as 88.
Let us assume the first, second and third numbers be x, y and z.
According to the problem, we have given that the first number is twice the second number. So, we have $x=2y$.
$\Rightarrow y=\dfrac{x}{2}$ ---(1).
According to the problem, we have given that the first number is three times the third number. So, we have $x=3z$.
$\Rightarrow z=\dfrac{x}{3}$ ---(2).
According to the problem, we have given that the average of the three numbers is given as 88.
We know that the average of any three numbers ${{x}_{1}}$, ${{x}_{2}}$ and ${{x}_{3}}$ is $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3}$. So, we have $\dfrac{x+y+z}{3}=88$.
From equations (1) and (2) we get,
$\Rightarrow \dfrac{x+\dfrac{x}{2}+\dfrac{x}{3}}{3}=88$.
$\Rightarrow \dfrac{\dfrac{6x+3x+2x}{6}}{3}=88$.
$\Rightarrow \dfrac{11x}{3\times 6}=88$.
$\Rightarrow \dfrac{11x}{18}=88$.
$\Rightarrow x=88\times \dfrac{18}{11}$.
$\Rightarrow x=8\times 18$.
$\Rightarrow x=144$.
So, we have found the value of the first number as 144.
∴ The value of the first number is 144.

Note: Whenever we get this type of problems, it is better to start by assigning the variables for all the unknowns present. We should use all the conditions that were given in the problem perfectly in order to get the required results. We should not make mistakes while making addition, multiplication and division operations. Similarly, we can expect problems that may say that the second number is twice the first and third is thrice the first number.