A vending machine contains ₹1, 50 paise, and 25 paise coins. In one tray, the number of coins of these denominations is in the ratio 2:3:4, and their total value is ₹180. In another tray, the values of ₹1, 50 paise, and 25 paise coins are in the ratio 13:11:7, and the total number of coins in this tray is 378. How many 50-paise coins are there in total across both trays?

The correct answer is Option 252.

To find the total number of 50-paise coins across both trays, we will calculate the number of 50-paise coins in each tray separately and then add them together.

Step 1: Analyze the First Tray

In the first tray, the ratio of the number of ₹1, 50 paise, and 25 paise coins is 2:3:4.
Let the number of coins of these denominations be 2x, 3x, and 4x, respectively.
We need to convert these denominations to a common unit, which is Rupees (₹):
- Value of a ₹1 coin = ₹1
- Value of a 50 paise coin = ₹0.50
- Value of a 25 paise coin = ₹0.25

The total value of the coins in the first tray is ₹180. We can write the equation for the total value as:

$$(2x\times 1)+(3x\times 0.50)+(4x\times 0.25)=180$$

Simplifying the equation:

$$2x+1.5x+1x=180$$

$$4.5x=180$$

$$x=\frac{180}{4.5}=40$$

Now, we can find the number of 50-paise coins in the first tray:
Number of 50-paise coins in the first tray = 3x = 3 × 40 = 120.

Step 2: Analyze the Second Tray

In the second tray, the ratio of the values of ₹1, 50 paise, and 25 paise coins is 13:11:7.
Let the values of these coins be 13y, 11y, and 7y, respectively.
To find the number of coins, we divide the value of each denomination by the value of a single coin of that denomination:
- Number of ₹1 coins = 13y / 1 = 13y
- Number of 50 paise coins = 11y / 0.50 = 22y
- Number of 25 paise coins = 7y / 0.25 = 28y

The total number of coins in the second tray is 378. We can write the equation for the total number of coins as:

$$13y+22y+28y=378$$

Simplifying the equation:

$$63y=378$$

$$y=\frac{378}{63}=6$$

Now, we can find the number of 50-paise coins in the second tray:
Number of 50-paise coins in the second tray = 22y = 22 × 6 = 132.

Step 3: Calculate the Total Number of 50-Paise Coins

Total number of 50-paise coins across both trays = Coins in Tray 1 + Coins in Tray 2
Total = 120 + 132 = 252.