Question

Madhav started playing a card game. In the first round, he doubled his amount and gave away p rupees to his friend. In the second round, he tripled the amount left with him after the first round and gave away 2p rupees to his friend. In the third round, he quadrupled the amount left with him after the second round and gave away p rupees to his friend and was finally left with no money. If he gave away a total of Rs.160 to his friend, then what was the amount that he started with?
A) Rs.21
B) Rs.40
C) Rs.24
D) Rs.35

Answer 1

Hint:
We can take the amount of money Madhav is having as x. Then we can calculate the money left with him after each round and after giving it to his friend. Then we can equate the sum of the money he gave to his friend to Rs.160 as given in the question and solve for p. Then we can equate the money he got after the ${3}^{\text{rd}}$ round to the money he gave to his friend after that round. Then using the value of p, we can solve for the value of x to get the required answer.

Complete step by step solution:
Let x be the amount of money with Madhav when he started playing.
It is given that he doubled his amount.
Then the money he got is given by, 2x.
From this he gave p rupees to his friend. Then the amount of money left with him is given by, $2x - p$
It is given that in the second round, he tripled the amount left with him after the first round. So, the money he has after ${2}^{\text{nd}}$ round is given by,
 $3\left( {2x - p} \right)$
It is given that from this he gave away an amount of 2p rupees to his friend.
So, the money left with him after the 2nd round is given by,
 $3\left( {2x - p} \right) - 2p$ .
On simplification we can write it as,
 $ \Rightarrow 3\left( {2x - p} \right) - 2p = 6x - 3p - 2p$
 $ \Rightarrow 3\left( {2x - p} \right) - 2p = 6x - 5p$
It is given that in the third round, he quadrupled the amount left with him after the second round. So, the money he has after 3rd round is given by,
 $4\left( {6x - 5p} \right)$
From this he gave an amount of p rupees to his friend. So, we can write the money left with him as,
 $4\left( {6x - 5p} \right) - p$
On simplification, we get,
 $ \Rightarrow 4\left( {6x - 5p} \right) - p = 24x - 20p - p$
 $ \Rightarrow 4\left( {6x - 5p} \right) - p = 24x - 21p$
It is given that no money is left with him after giving the money to his friend. So, we can equate this expression to zero.
 $ \Rightarrow 24x - 21p = 0$
On dividing throughout with 3, we get,
 $ \Rightarrow 8x - 7p = 0$
 $ \Rightarrow 8x = 7p$ … (1)
It is given that he gave away a total of Rs.160 to his friend. So, we can write it as an equation.
 $ \Rightarrow p + 2p + p = 160$
 $ \Rightarrow 4p = 160$
On dividing throughout with 4, we get,
 \[ \Rightarrow p = \dfrac{{160}}{4}\]
 \[ \Rightarrow p = 40\]
On substituting this in equation (1), we get,
 $ \Rightarrow 8x = 7 \times 40$
 $ \Rightarrow x = \dfrac{{7 \times 40}}{8}$
 $ \Rightarrow x = 7 \times 5$
 $ \Rightarrow x = 35$

Therefore, the amount of money that he started with is Rs.35.
So, the correct answer is option D.

Note:
The method of forming mathematical equations from a given statement is known as mathematical modelling. We must read the statement carefully before writing each expression. We must double or triple only the amount left with Madhav after giving it to his friend each time. We only need to take the expression of money left with him after the 3rd round and giving p rupees to his friend to equate it to zero. For doubling we need to multiply the amount with 2, for tripling we need to multiply the amount with 3 and for quadrupling, we need to multiply the amount with 4.