Question

Divide \[75\] into two parts such that two times one part falls short of \[100\] by the same amount as \[200\] falls short of \[5\] times the other part.

Answer 1

Hint: We will consider one part as \[x\] and the other part as \[75 - x\]. Then since we are given some condition in the question, we will form equations based on the condition given on the existing two parts. Then after solving the question we will get one of the parts, then since the sum is \[75\], we can find the other part as well.

Complete step by step solution:
First, we are going to consider one of the parts as \[x\] and then the other part automatically will become \[75 - x\]. First let us form the LHS, then RHS and then equate them to find the parts. So, for the LHS, two times of the part which means \[2x\] and is short of \[100\], it means that we add a \[100\] to it, which forms the LHS. Which is
\[2x + 100\]
Now, coming to the RHS. It says the number \[200\] falls short of 5 times the other part.
The five times of other part is
\[5(75 - x)\]
It is given that the number \[200\] is short of five times the other part. We get
\[200 - 5(75 - x)\]
Now, we equate the RHS and LHS to find the parts
$2x + 100 = 200 - 5(75 - x) \\
\Rightarrow 2x - 100 = 200 - 375 + 5x \\
\Rightarrow 5x - 2x = - 100 + 175 \\
\Rightarrow 3x = 75 \\
\Rightarrow x = \dfrac{{75}}{3} \\
\therefore x = 25 $
So, other part is \[75 - x = 75 - 25 = 50\].

Hence, the two parts of \[75\] are 25 and 50.

Note: We have to be careful when constructing the equation, when LHS is short we are going to add and when RHS is short of any number then we are going to subtract it on the RHS. Hence while simplifying the equation, we have to be careful during multiplication and check the simplification for mistakes.