Solving the Cake Cost Price Problem

This problem involves a common scenario in mathematics where we deal with costs, profits, and losses on items. We need to find the individual cost prices of two different types of cakes based on the total cost and the overall profit made from selling them at specific profit or loss percentages.

Understanding the Problem Setup

We are given information about two types of cakes, Type-I and Type-II. We know the total quantity of each type bought and the total amount spent. We also know the profit percentage for Type-I cakes and the loss percentage for Type-II cakes, and the final overall profit.

• Number of Type-I cakes bought: 3
• Number of Type-II cakes bought: 6
• Total cost price of all cakes: Rs. 900
• Profit percentage on Type-I cakes: 15%
• Loss percentage on Type-II cakes: 10%
• Overall profit on the entire transaction: Rs. 30

Our goal is to find the cost price of one Type-I cake and the cost price of one Type-II cake.

Setting Up Equations Based on Cost and Profit/Loss

Let's assign variables to the unknown cost prices:

• Let \(C_1\) be the cost price (in Rs.) of one Type-I cake.
• Let \(C_2\) be the cost price (in Rs.) of one Type-II cake.

Based on the total cost, we can write the first equation:

The total cost of 3 Type-I cakes is \(3 \times C_1 = 3C_1\).

The total cost of 6 Type-II cakes is \(6 \times C_2 = 6C_2\).

The sum of these costs is the total amount spent, which is Rs. 900.

Equation 1: \(3C_1 + 6C_2 = 900\)

This equation can be simplified by dividing by 3:

\(C_1 + 2C_2 = 300\)

Now, let's consider the profit and loss. The overall profit is the total profit from selling Type-I cakes minus the total loss from selling Type-II cakes (since loss reduces the overall gain).

• Profit per Type-I cake: 15% of \(C_1\) = \(0.15 C_1\).
• Total profit from 3 Type-I cakes: \(3 \times 0.15 C_1 = 0.45 C_1\).
• Loss per Type-II cake: 10% of \(C_2\) = \(0.10 C_2\).
• Total loss from 6 Type-II cakes: \(6 \times 0.10 C_2 = 0.60 C_2\).

The overall profit is Rs. 30.

Equation 2: (Total profit from Type-I) - (Total loss from Type-II) = Overall Profit

\(0.45 C_1 - 0.60 C_2 = 30\)

We now have a system of two linear equations with two variables:

1. \(C_1 + 2C_2 = 300\)
2. \(0.45 C_1 - 0.60 C_2 = 30\)

Solving the System of Linear Equations

We can solve this system using substitution or elimination. Let's use substitution.

From Equation 1, we can express \(C_1\) in terms of \(C_2\):

\(C_1 = 300 - 2C_2\)

Now, substitute this expression for \(C_1\) into Equation 2:

\(0.45 (300 - 2C_2) - 0.60 C_2 = 30\)

Distribute 0.45:

\(0.45 \times 300 - 0.45 \times 2C_2 - 0.60 C_2 = 30\)

\(135 - 0.90 C_2 - 0.60 C_2 = 30\)

Combine the terms with \(C_2\):

\(135 - (0.90 + 0.60) C_2 = 30\)

\(135 - 1.50 C_2 = 30\)

Now, isolate the term with \(C_2\):

\(135 - 30 = 1.50 C_2\)

\(105 = 1.50 C_2\)

Solve for \(C_2\):

\(C_2 = \frac{105}{1.50} = \frac{105}{\frac{150}{100}} = \frac{105 \times 100}{150} = \frac{10500}{150}\)

\(C_2 = \frac{1050}{15}\)

\(C_2 = 70\)

So, the cost price of a Type-II cake is Rs. 70.

Now, substitute the value of \(C_2\) back into the expression for \(C_1\):

\(C_1 = 300 - 2C_2\)

\(C_1 = 300 - 2(70)\)

\(C_1 = 300 - 140\)

\(C_1 = 160\)

Thus, the cost price of a Type-I cake is Rs. 160.

The cost prices of a Type-I and a Type-II cake are Rs. 160 and Rs. 70, respectively.

Verification

Let's check if these values satisfy the original conditions:

• Total cost: \(3 \times 160 + 6 \times 70 = 480 + 420 = 900\). This matches the given total cost.
• Profit from Type-I cakes: 15% of Rs. 160 = \(0.15 \times 160 = 24\). Total profit from 3 cakes = \(3 \times 24 = 72\).
• Loss from Type-II cakes: 10% of Rs. 70 = \(0.10 \times 70 = 7\). Total loss from 6 cakes = \(6 \times 7 = 42\).
• Overall Profit: Total Profit - Total Loss = \(72 - 42 = 30\). This matches the given overall profit.

The calculated cost prices satisfy all the conditions in the problem.

The cost price of a Type-I cake is Rs. 160, and the cost price of a Type-II cake is Rs. 70.

Revision Table: Key Concepts

Additional Information: Profit and Loss Calculations

Understanding how to calculate profit and loss percentages is crucial for solving such problems.

• Profit % = \(\left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\)
• Loss % = \(\left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100\)

Conversely, if you know the cost price and the percentage profit or loss, you can find the amount of profit or loss:

• Amount of Profit = Cost Price \(\times \left( \frac{\text{Profit %}}{100} \right)\)
• Amount of Loss = Cost Price \(\times \left( \frac{\text{Loss %}}{100} \right)\)

The selling price can be calculated as:

• Selling Price = Cost Price + Amount of Profit
• Selling Price = Cost Price - Amount of Loss

In our problem, the total overall profit is the sum of profits and losses from individual items. A profit contributes positively, while a loss contributes negatively to the overall result.

Overall Profit/Loss = Sum of Profits - Sum of Losses

Setting up equations like \(0.45 C_1 - 0.60 C_2 = 30\) directly calculates the net effect of the profit on Type-I cakes and the loss on Type-II cakes.