Nine friends had a tea party. All boys took only coffee and all girls took only tea. The cost per cup of coffee in rupees is numerically 2 less than the number of girls and the cost per cup of tea in rupees is numerically 2 less than the number of boys. If the ratio of the total expenses of the boys and the girls is ` 5 : 6`, then what is the cost of each coffee? (In Rs.)

Options

A. 2
B. 5
C. 7
D. 3

Explanation

To solve the problem step by step, we will define the variables and use the information provided in the question.

### Step 1: Define Variables
Let:
- \( X \) = number of boys
- \( Y \) = number of girls

Since there are 9 friends in total, we can write our first equation as:
\[ X + Y = 9 \]

### Step 2: Express Costs
According to the problem:
- The cost per cup of coffee (for boys) is \( Y - 2 \) (2 less than the number of girls).
- The cost per cup of tea (for girls) is \( X - 2 \) (2 less than the number of boys).

### Step 3: Write Total Expenses
The total expenses for boys (who drink coffee) can be expressed as:
\[ \text{Total expenses of boys} = X \times (Y - 2) \]
The total expenses for girls (who drink tea) can be expressed as:
\[ \text{Total expenses of girls} = Y \times (X - 2) \]

### Step 4: Set Up the Ratio of Expenses
According to the problem, the ratio of the total expenses of boys to girls is \( 5:6 \). Therefore, we can write:
\[ \frac{X(Y - 2)}{Y(X - 2)} = \frac{5}{6} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ 6X(Y - 2) = 5Y(X - 2) \]

### Step 6: Expand Both Sides
Expanding both sides:
\[ 6XY - 12X = 5XY - 10Y \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 6XY - 5XY = 12X - 10Y \]
\[ XY = 12X - 10Y \]

### Step 8: Substitute \( Y \)
From the first equation \( X + Y = 9 \), we can express \( Y \) in terms of \( X \):
\[ Y = 9 - X \]
Substituting this into the equation \( XY = 12X - 10Y \):
\[ X(9 - X) = 12X - 10(9 - X) \]

### Step 9: Simplify
Expanding and simplifying:
\[ 9X - X^2 = 12X - 90 + 10X \]
\[ 9X - X^2 = 22X - 90 \]
Rearranging gives:
\[ X^2 - 13X + 90 = 0 \]

### Step 10: Factor the Quadratic Equation
Factoring the quadratic:
\[ (X - 9)(X - 10) = 0 \]
Thus, \( X = 9 \) or \( X = 10 \). Since \( X + Y = 9 \), \( X \) cannot be 10. Therefore, \( X = 9 \) and \( Y = 0 \) is not valid.

### Step 11: Solve for Valid Values
Since \( X \) must be less than 9, we can try \( X = 6 \) and \( Y = 3 \):
- If \( X = 6 \), then \( Y = 3 \).

### Step 12: Calculate Cost of Coffee
Now we can find the cost of coffee:
\[ \text{Cost of coffee} = Y - 2 = 3 - 2 = 1 \]

### Final Answer
The cost of each cup of coffee is **1 Rs**.