Question

The sum of 3 numbers is 264. If the first number is twice the second and third number be one-third of the first, then the second number is:
(a) 48
(b) 54
(c) 72
(d) 84

Answer 1

Hint: Consider x , y and z be the 3 numbers then according to condition form an equation. Form equations based on the conditions given. This will give us 3 equations, one for each condition and then these three equations can be solved using algebraic methods to get the values of x, y and z.

Complete step-by-step answer:
This is a question of linear algebra and we proceed to solve this question by first assigning variable names to the numbers given in the question.
We let the first number be x, the second number be y and the third number be z. Now we’ll form equations sequentially.
The first sentence says:
$x+y+z=264....................(i)$
The second sentence says:
$x=2y.....................(ii)$
$z=\dfrac{1}{3}x....................(iii)$
Now we use the three equations we have formed to solve the given problem.
From equation (ii) by dividing 2 on both sides we can write:
$y=\dfrac{1}{2}x..................(iv)$
Now we will substitute the value of z and y in equation (i) from equation (iii) and (iv) respectively to get:
$x+\dfrac{1}{2}x+\dfrac{1}{3}x=264$
By taking x common from L.H.S we get,
$\Rightarrow x\left( 1+\dfrac{1}{2}+\dfrac{1}{3} \right)=264$
Taking L.C.M of the term in bracket we get,
$\Rightarrow x\left( \dfrac{6+3+2}{6} \right)=264$
Solving further we get,
$\Rightarrow x\times\dfrac{11}{6}=264$
$\Rightarrow x=\dfrac{6\times264}{11}$
$\Rightarrow x=144$
$x$ is our first number but the question has asked us for the second number so need to find out $y$
From equation (iv) we can find $y$.
\[y=\dfrac{1}{2}x\]
$\Rightarrow y=\dfrac{144}{2}$
$\Rightarrow y=72$
Hence, our answer is $72$.

Note: The chances of error highly increase when there are multiple variables involved, as in this case. We have three conditions that have been given in the form of first number, second number and third number. While forming the equations, it should be kept in mind which number was assumed to be the first number, second number and third number respectively and form equations accordingly. Incorrect framing of equations will lead to incorrect solutions.Similarly we can find third number by substituting value of $x$ in equation 3 we get $z=\dfrac{1}{3}\times x$ i.e $z=\dfrac{1}{3}\times144$=$48$.Therefore three numbers are 144, 72 and 48.