Question

24 is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. Find each part.

Answer 1

Hint: We first assume the two parts of 24 as two variables whose sum would give 24. The second condition about those variables being 7 times the first part added to 5 times the second part makes 146. We get two linear equations of two variables. We solve them to find the solution of the problem.

Complete step by step answer:
24 is divided into two parts. Let’s assume the parts are a and b. The sum of a and b is 24 which means $a+b=24............(i)$
We try to form the equations based on the conditions of 7 times the first part added to 5 times the second part makes 146.
7 times a which is $7\times a=7a$ added to 5 times of b which is $5\times b=5b$ makes 146.
The equation becomes $7a+5b=146............(ii)$
We got two equations and we solve them by multiplying 5 with equation (i) and subtracting from equation (ii).
$5\left( a+b \right)=5\times 24\Rightarrow 5a+5b=120$.
We do the subtraction now and get
$\begin{align}
  & \left( 7a+5b \right)-\left( 5a+5b \right)=146-120 \\
 & \Rightarrow 2a=26 \\
 & \Rightarrow a=\dfrac{26}{2}=13 \\
\end{align}$
From the value of a we find the value of b.
$a+b=24\Rightarrow b=24-a=24-13=11$

Therefore, the parts are 13 and 11.

Note: We also could have solved it from just assuming only one variable. We would have taken as one part and other part being $24-a$, as the sum of those two parts is 24. The rest of the solution is similar by replacing $24-a$ in place of b. The difference is the equations would have been one variable equation.