A bag contains some one-rupee coins and some fifty-paisa coins. The total amount is ₹ 140. If half of the one-rupee coins are replaced by fifty-paisa coins, then the amount becomes ₹ 115. The coins of each type in the bag initially, were :

1. one-rupee coins = 100 and fifty-paisa coins = 80

2. one-rupee coins = 80 and fifty-paisa coins = 100

3. one-rupee coins = 110 and fifty-paisa coins = 80

4. one-rupee coins = 70 and fifty-paisa coins = 90

Let x be the number of one rupee coins and y be the number of 50 paisa coins in the bag initially.

Given,

Initial total amount = ₹ 140.

⇒ x + 0.5y = 140

⇒ x = 140 - 0.5y     …….(1)

Given,

After replacing half of the 1-rupee coins with 50-paisa coins, the amount becomes ₹ 115.

⇒ $\dfrac{x}{2} \times 1$ + 0.5y + 0.5 $\Big(\dfrac{x}{2}\Big)$ = 115

⇒ 0.5x + 0.5y + 0.25x = 115

⇒ 0.75x + 0.5y = 115     ……..(2)

Substituting value of x from equation (1) in (2), we get :

⇒ 0.75(140 - 0.5y) + 0.5y = 115

⇒ 105 - 0.375y + 0.5y = 115

⇒ 0.125y = 115 - 105

⇒ 0.125y = 10

⇒ y = $\dfrac{10}{0.125}$ = 80.

Substituting value of y in equation (1), we get :

⇒ x = 140 - 0.5y

⇒ x = 140 - 0.5(80)

⇒ x = 140 - 40

⇒ x = 100.

∴ Number of one rupee coins = 100 and number of fifty paisa coins = 80.

Hence, option 1 is the correct option.