Question

One says, “Give me a hundred, friend! I shall then become twice as rich as you.” The other replies, “If you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their respective capital?

Answer 1

Hint- Here, we will proceed by assuming the capitals of two friends as two different variables i.e., x and y respectively. Then, we will form two linear equations in two variables and will solve them algebraically.

Complete Step-by-Step solution:
Let us suppose the capital of two friends be x and y respectively
Given, if the second friend gives a hundred out of his capital to the first one then, the first one becomes twice as rich as the second one
i.e., Capital of first person = x + 100
Capital of second person = y – 100
So, Capital of first person = 2(Capital of second person)
$
   \Rightarrow x + 100 = 2\left( {y - 100} \right) \\
   \Rightarrow x + 100 = 2y - 200 \\
   \Rightarrow x = 2y - 300{\text{ }} \to {\text{(1)}} \\
 $
When ten is given by the first person to the second person, we have
Capital of first person = x – 10
Capital of second person = y + 10
So, Capital of second person = 6(Capital of first person)
$
   \Rightarrow y + 10 = 6\left( {x - 10} \right) \\
   \Rightarrow y + 10 = 6x - 60 \\
   \Rightarrow y = 6x - 70 \\
 $
By substituting the value of x from equation (1) in the above equation, we get
$
   \Rightarrow y = 6\left( {2y - 300} \right) - 70 \\
   \Rightarrow y = 12y - 1800 - 70 \\
   \Rightarrow 12y - y = 1800 + 70 \\
   \Rightarrow 11y = 1870 \\
   \Rightarrow y = \dfrac{{1870}}{{11}} = 170 \\
 $
Put y = 170 in equation (1), we get
$ \Rightarrow x = 2\left( {170} \right) - 300 = 340 - 300 = 40$
Therefore, the amount of the capitals of two friends are 40 and 170 respectively.

Note- In this particular problem, in order to find the values of the two variables i.e., x and y we have used a substitution method. Instead of a substitution method, we can also solve with the help of elimination method in which we will make the coefficients of any one variable the same by multiplying the two equations with some number and then will subtract these two obtained equations.