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Of the three numbers, the first one is one third of the second and twice the third. The average of these numbers is 27. The largest of these three numbers is:
(a) 18
(b) 36
(c) 54
(d) 108

seo-qna
Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: Let us assume the first number as x then the second number is three times x and the third number is half of x. Now, we have given the average of three numbers. And we know the formula for average is $\dfrac{\text{Sum of all observations}}{\text{Total number of observations}}$. And then add all the three numbers and divide the addition of three numbers to 3. After that equate this average formula to 27 and solve the value of x. And then find the three numbers and hence will find the largest numbers.

Complete step-by-step solution:
We have given a relation between the first number with the second number and the third number. So, let us write the mathematical expression of these three relations.
First of all, we have given that the first number is one third of the second number. If the first number is one third of the second number, it means the second number is 3 times that of the first number so let us assume that first number is x so the second number is 3 times of x.
Second number $=3x$
After that, we are given that the first number is twice that of the third number. This means that the third number is one half of the first number and as we have assumed earlier that the first number is x so the third number is one half of x.
Third number $=\dfrac{x}{2}$
Now, we have given the average of the three numbers as 27. The formula for average is as follows:
$=\dfrac{\text{Sum of all observations}}{\text{Total number of observations}}$
Now, we are going to find the sum of the three numbers by adding the three numbers which we have written above:
$\begin{align}
  & \Rightarrow x+3x+\dfrac{x}{2} \\
 & =4x+\dfrac{x}{2} \\
\end{align}$
Taking x as common in the above expression we get,
$\Rightarrow x\left( 4+\dfrac{1}{2} \right)$
Taking 2 as L.C.M in the above expression written in the bracket we get,
\[\begin{align}
  & \Rightarrow x\left( \dfrac{4\left( 2 \right)+1}{2} \right) \\
 & =x\left( \dfrac{8+1}{2} \right) \\
 & =x\left( \dfrac{9}{2} \right) \\
\end{align}\]
From the above, we got the addition of three numbers as $\dfrac{9x}{2}$.
As the numbers given above are 3 so total the number of observations is 3. Now, substituting the sum of observations as $\dfrac{9x}{2}$ and total number of observations as 3 in the above average formula we get,
$\begin{align}
  & =\dfrac{\text{Sum of all observations}}{\text{Total number of observations}} \\
 & =\dfrac{\dfrac{9x}{2}}{3} \\
 & =\dfrac{9x}{2\left( 3 \right)} \\
\end{align}$
In the above expression, 3 will be cancelled out in the numerator by 3 times and denominator by 1 times we get,
$\Rightarrow \dfrac{3x}{2}$
Now, equating the above average to 27 we get,
$\Rightarrow \dfrac{3x}{2}=27$
Cross multiplying the above equation we get,
$\Rightarrow 3x=27\left( 2 \right)$
Dividing 3 on both the sides we get,
$\begin{align}
  & \Rightarrow \dfrac{3x}{3}=\dfrac{27\left( 2 \right)}{3} \\
 & \Rightarrow x=9\left( 2 \right) \\
 & \Rightarrow x=18 \\
\end{align}$
From the above we have got the value of x as 18 so the first number is 18.
And second number is three times 18 so multiplying 3 by 18 we get,
$\Rightarrow 3\left( 18 \right)=54$
And the third number is one half of x means multiplying 18 by one half we get,
$\begin{align}
  & \Rightarrow \dfrac{18}{2} \\
 & =9 \\
\end{align}$
From the above, the first number is 18, second is 54 and third is 9 so the largest number among these three numbers is 54. Hence, the correct option is (c).

Note: The possible mistake which could be possible in the above problem is that after getting the value of x. In the hurry of solving questions in examination you think that the value of x has arrived and it is given in the options also (which is 18) and you tick the option having the number 18 so make sure you won’t make this mistake and solve the question carefully.