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: First we understand what is in the question and try to make a mathematical expression.
Let us assume the two person are two variables like \[x\]and \[y\]
Then we formed an equation by using the data and we have to find one by one solution.
Finally, we will get the required solution.

Complete step-by-step answer:
It is given that the friend says, “Give me a hundred, friend! I shall then become twice as rich as you.”
Let the amount of the first person =$x$
When the first person takes $100$ from the second person
Money with first person =$x + 100$
Money with second person $ = y - 100$
Given that the first person takes $100$ from the second person, he becomes twice as rich as second person,
So we can write in the equation form $x + 100 = 2(y - 100)$
Let us we do the multiplication by open bracket
$x + 100 = 2y - 200$
Taking variables in the Left hand side,
$x - 2y = - 200 - 100$
Subtracting the integers
$x - 2y = - 300....\left( 1 \right)$
Also we make a second expression, from the given data that is;
When the second person takes $10$from the first person
Money with first person =$x - 10$
Money with the second person $ = y + 10$
According to the given data, When second person takes $10$from the first person, he becomes $6$times as rich as the first person.
So we can write it as the equation form,$y + 10 = 6(x - 10)$
We need to multiply the open bracket to solve equation
$y + 10 = 6x - 60$
We collect the variable terms in left hand side and integers on the right hand side.
$y - 6x = - 60 - 10$
On subtracting the integers we get,
$y - 6x = - 70....(2)$
Therefore, from the equation$(1)$, $x - 2y = - 300$
We can write it as, $x = - 300 + 2y........(3)$
Putting equation$(3)$ in the equation $(2)$we get,
$6x - y = 10$
Putting the value of $x$in it, we get
$6( - 300 + 2y) - y = 70$
Open the bracket by multiplying the terms
$ - 1800 + 12y - y = 70$
Subtracting the variable terms we get,
$11y = 70 + 1800$
By adding the right handed terms
$11y = 1870$
By divide,
$y = \dfrac{{1870}}{{11}}$
$y = 170$
Putting the value of $y$in $(1)$we get,
$x - (2 \times 170) = - 300$
By multiplying the bracket terms,
$x - 340 = - 300$
Separate variable term,
$x = - 300 + 340$
$x = 40$

Thus, Amount of money with first person $,x = 40$
Amount of money with second person $,y = 170$

Note: In these types of questions we have to use the concept of linear equations of two variables.
We can use the methods of cross-product method, substitution method, elimination method etc.
Here we use a substitution method, to check whether the obtained solution is correct or not, substitute the values of x and y in any equations.
Verification:
Take the equation\[\left( 2 \right)\],$ \Rightarrow y - 6x = - 70$
Substitute the value of \[x\]and \[y\]in\[\left( 2 \right)\], we get
\[170 - 6\left( {40} \right) = - 70\]
On multiplying the brackets terms, we get
\[170 - 240 = - 70\]
Let us subtract
\[ - 70 = - 70\]
Hence proved.