Question

A and B have some money with them . A said to B , if you give me Rs.100 , my money will become 75% of the money left with you. B said to A, ”Instead if you give me Rs.100, your money will become 40% of my money.How much money did A and B have originally?

Answer 1

Hint:
By the property that $a\% $ of a number b is $\dfrac{a}{{100}}\times b$ or $\dfrac{a}{{100}}$ times of b . we have the two given conditions to be when B gives 100 to A then the amount of A will be equal to 0.75 times of the money left with B and when A gives 100 to B then the amount of A will be equal to 0.4 times of the money with B. From this we get a pair of linear equations and solving which we get the values of x and y.

Complete step by step solution:
Let the amount of money which A have be x
And let the amount of money which B have be y
We know that $a\% $ of a number b is $\dfrac{a}{{100}}\times b$ or $\dfrac{a}{{100}}$ times of b .
We have two conditions here
A said to B , if you give me Rs.100 , my money will become 75% = 0.75 times of the money left with you
B said to A,”Instead if you give me Rs.100, your money will become 40% = 0.40 times of my money
By the first condition when B gives 100 to A then the amount of A will be equal to 0.75 times of the money left with B
So ,when A gets 100 from B the amount with A will be $x + 100$
And the amount with B will be $y - 100$
And we are given that the amount with A is 0.75 times of money with B
$
   \Rightarrow x + 100 = 0.75\left( {y - 100} \right) \\
   \Rightarrow x + 100 = 0.75y - 75 \\
   \Rightarrow x - 0.75y = - 75 - 100 \\
   \Rightarrow x - 0.75y = - 175 \\
 $
Let this equation be (1)
By the second condition when A gives 100 to B then the amount of A will be equal to 0.4 times of the money with B
So ,when B gets 100 from A the amount with B will be $y + 100$
And the amount with A will be $x - 100$
And we are given that the amount with A is 0.40 times of money with B
$
   \Rightarrow x - 100 = 0.4\left( {y + 100} \right) \\
   \Rightarrow x - 100 = 0.4y + 40 \\
   \Rightarrow x - 0.4y = 40 + 100 \\
   \Rightarrow x - 0.4y = 140 \\
 $
Let this be equation (2)
From equation (1) we get
$ \Rightarrow x = 0.75y - 175$
Substituting in equation (2) we get
$
   \Rightarrow 0.75y - 175 - 0.4y = 140 \\
   \Rightarrow 0.35y - 175 = 140 \\
   \Rightarrow 0.35y = 140 + 175 \\
   \Rightarrow 0.35y = 315 \\
   \Rightarrow y = \dfrac{{315}}{{0.35}} = 900 \\
 $
Using this in equation (1) we get
 $
   \Rightarrow x = 0.75\left( {900} \right) - 175 \\
   \Rightarrow x = 675 - 175 \\
   \Rightarrow x = 500 \\
 $

Hence the amount with A is Rs.500 and the amount with B is Rs.900

Note:
Here the pair of linear equations can be solved by using elimination method also
The equations are $x - 0.75y = - 175$ and $x - 0.4y = 140$
At first let's subtract (2) from (1)
 $
  {\text{ }}0 - 0.35y = - 315 \\
 $
From this
$
   \Rightarrow 0.35y = 315 \\
   \Rightarrow y = \dfrac{{315}}{{0.35}} = 900 \\
$
And once again substituting this in any one of the equations we can get the value of x.