The cost price and the marked price of an article are in the ratio 4 : 5. The article was sold after a discount. The discount percentage offered and the profit/loss percentage made on the article are in the ratio 4 : 5. Which of the following statements is true?

1   The article was sold at a loss.

2  The discount percentage offered is 20%.

3  The profit / loss is 12%.

4  The article was sold at a profit.

Let's solve this step by step:

### Given Information:
- The cost price (CP) and the marked price (MP) of an article are in the ratio 4:5.
- The discount percentage offered and the profit/loss percentage made on the article are in the ratio 4:5.

### Step 1: Express CP and MP in terms of variables
Let the cost price (CP) be \( 4x \) and the marked price (MP) be \( 5x \), where \( x \) is some constant multiplier.

### Step 2: Assume the discount percentage and profit/loss percentage
Let the discount percentage be \( 4y \) and the profit/loss percentage be \( 5y \).

### Step 3: Calculate selling price using discount percentage
The discount is \( 4y\% \) of the marked price \( 5x \). Therefore, the selling price (SP) after the discount is:
\[
SP = MP - \text{Discount} = 5x - \frac{4y}{100} \times 5x = 5x\left(1 - \frac{4y}{100}\right)
\]
\[
SP = 5x\left(\frac{100 - 4y}{100}\right) = \frac{5x(100 - 4y)}{100}
\]

### Step 4: Use profit/loss percentage to relate SP and CP
The profit/loss percentage is \( 5y\% \) of the cost price. Therefore:
\[
SP = CP \left(1 + \frac{5y}{100}\right)
\]
\[
SP = 4x \left(1 + \frac{5y}{100}\right) = 4x \left(\frac{100 + 5y}{100}\right) = \frac{4x(100 + 5y)}{100}
\]

### Step 5: Equate both expressions for SP
We now have two expressions for SP. Equate them to solve for \( y \):
\[
\frac{5x(100 - 4y)}{100} = \frac{4x(100 + 5y)}{100}
\]
Cancel out the common terms and simplify:
\[
5(100 - 4y) = 4(100 + 5y)
\]
\[
500 - 20y = 400 + 20y
\]
\[
500 - 400 = 20y + 20y
\]
\[
100 = 40y
\]
\[
y = \frac{100}{40} = 2.5
\]

### Step 6: Interpret the result
- The discount percentage is \( 4y = 4 \times 2.5 = 10\% \).
- The profit percentage is \( 5y = 5 \times 2.5 = 12.5\% \).

### Step 7: Conclusion
From the above calculations, we see that:
- The article was sold at a **profit** (12.5% profit).
- The discount percentage is **10%**, not 20%.
- The profit percentage is **12.5%**, not exactly 12%.

Thus, the **correct statement** is:
**4. The article was sold at a profit.**

Would you like further details or explanations on any step?

### Here are 5 related questions to consider:
1. What happens to the profit percentage if the discount increases?
2. How do we calculate the cost price if only profit percentage and selling price are known?
3. What would the selling price be if there were no discount?
4. Can you have both a discount and still incur a loss? Under what conditions?
5. How do different ratios between CP and MP affect profit or loss?

### Tip:
Always ensure you understand the relationship between cost price, marked price, and selling price when dealing with profit and discount calculations.

The cost price and the marked price of an article are in the ratio 4:5. The article was sold after a discount. The discount percentage offered and the profit/loss percentage made on the article are in the ratio 4:5. Which of the following statements is true?

1. The article was sold at a loss.
2. The discount percentage offered is 20%.
3. The profit / loss is 12%.
4. The article was sold at a profit.

Solve a problem involving the cost price and marked price in a ratio of 4:5, with discount and profit percentages also in a ratio of 4:5. This step-by-step solution covers key mathematical concepts such as ratios, percentages, and profit calculations. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation