Aman buys 3 type-I cakes and 6 type-II cakes for Rs 900. He sells type-I cakes at a profit of 15% and type-II cakes at a loss of 10%. If his overall profit is Rs.30, the cost price (in ) of a type-I and of a type-ll cakes is

Options

A. 100, 100
B. 160,70
C. 180, 60
D. 120,90

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define Variables
Let the cost price of type-I cake be \( x \) and the cost price of type-II cake be \( y \).

### Step 2: Set Up the Equation for Total Cost
Aman buys 3 type-I cakes and 6 type-II cakes for Rs 900. Therefore, we can write the equation:
\[ 3x + 6y = 900 \]

### Step 3: Simplify the Equation
We can simplify this equation by dividing everything by 3:
\[ x + 2y = 300 \quad \text{(Equation 1)} \]

### Step 4: Calculate Selling Prices
Aman sells type-I cakes at a profit of 15%. Therefore, the selling price of one type-I cake is:
\[ \text{Selling Price of type-I} = x + 0.15x = 1.15x \]
For 3 type-I cakes, the total selling price is:
\[ 3 \times 1.15x = 3.45x \]

For type-II cakes, he sells them at a loss of 10%. Therefore, the selling price of one type-II cake is:
\[ \text{Selling Price of type-II} = y - 0.10y = 0.90y \]
For 6 type-II cakes, the total selling price is:
\[ 6 \times 0.90y = 5.4y \]

### Step 5: Set Up the Equation for Total Selling Price
The total selling price of all cakes is given by:
\[ 3.45x + 5.4y \]

### Step 6: Calculate Overall Profit
According to the problem, Aman’s overall profit is Rs 30. Therefore, we can write:
\[ \text{Total Selling Price} - \text{Total Cost Price} = 30 \]
Substituting the values we have:
\[ 3.45x + 5.4y - 900 = 30 \]
This simplifies to:
\[ 3.45x + 5.4y = 930 \quad \text{(Equation 2)} \]

### Step 7: Solve the System of Equations
Now we have a system of two equations:
1. \( x + 2y = 300 \) (Equation 1)
2. \( 3.45x + 5.4y = 930 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 300 - 2y \]

### Step 8: Substitute in Equation 2
Substituting \( x \) in Equation 2:
\[ 3.45(300 - 2y) + 5.4y = 930 \]
Expanding this gives:
\[ 1035 - 6.9y + 5.4y = 930 \]
Combining like terms:
\[ 1035 - 1.5y = 930 \]
Now, isolate \( y \):
\[ -1.5y = 930 - 1035 \]
\[ -1.5y = -105 \]
\[ y = \frac{-105}{-1.5} = 70 \]

### Step 9: Substitute Back to Find \( x \)
Now that we have \( y \), we can find \( x \):
\[ x = 300 - 2(70) = 300 - 140 = 160 \]

### Final Answer
The cost price of type-I cake is Rs 160 and the cost price of type-II cake is Rs 70.