Question

Three numbers are in ratio $4:5:6$. If the sum of the largest and smallest number equal to the third number and 55. Find the numbers.

Answer 1

Hint: We first take a variable for the ratio constant. Using the constant, we get the numbers as $4x,5x,6x$. Using the given relation, we get the equation $4x+6x=5x+55$. We solve it to find the solution for $x$ and find the numbers.

Complete step-by-step solution:
It is given that three numbers are in ratio $4:5:6$. We can assume the numbers as $4x,5x,6x$. We are taking $x$ as the ratio constant.
Now it is given a sum of the largest and smallest number equal to the third number and 55.
The sum of largest and smallest number will be $4x+6x$ and the sum of remaining number and 55 is $5x+55$.
They are equal and so the mathematical expression is $4x+6x=5x+55$.
We now have to simplify them.
All the terms in the equation of $4x+6x=5x+55$ are either variable of $x$ or a constant. We first separate the variables and the constants.
We take the variables to get $4x+6x-5x=55$.
The binary operation of subtraction gives $4x+6x-5x=5x$.
Now we apply the binary operation of subtraction to get $5x=55$.
Now we divide both sides of the equation with 5 to get
\[\begin{align}
  & 5x=55 \\
 & \Rightarrow \dfrac{5x}{5}=\dfrac{55}{5} \\
 & \Rightarrow x=11 \\
\end{align}\]
Therefore, the ratio constant becomes \[x=11\].
The numbers are \[4x=44,5x=55,6x=66\].

Note: We can verify the result of the equation $4x+6x=5x+55$ by taking the value of as \[x=11\].
Therefore, the left-hand side of the equation becomes $4x+6x=4\times 11+6\times 11=110$.
The right-hand side of the equation becomes $5x+55=5\times 11+55=110$.
Thus, verified for the equation $4x+6x=5x+55$ the solution is \[x=11\].