After giving two successive discounts, each of x%, on the marked price of an article, total discount is Rs. 259.20. If the marked price of the article is Rs. 720, then the value of x is :

Options

A. 18
B. 24
C. 20
D. 25

Answer

To solve the problem step by step, we need to find the value of \( x \) given the total discount after two successive discounts on the marked price of an article.

### Step 1: Understand the Problem
We know that the marked price (MP) of the article is Rs. 720, and the total discount after applying two successive discounts of \( x\% \) each is Rs. 259.20.

### Step 2: Calculate the Total Discount Formula
When two successive discounts of \( x\% \) are applied, the total discount can be calculated using the formula:
\[ \text{Total Discount} = MP \times \left( \frac{x}{100} + \frac{x}{100} - \frac{x^2}{10000} \right) \]
This simplifies to:
\[ \text{Total Discount} = MP \times \left( \frac{2x - \frac{x^2}{100}}{100} \right) \]

### Step 3: Substitute the Known Values
Substituting the known values into the formula:
\[ 259.20 = 720 \times \left( \frac{2x - \frac{x^2}{100}}{100} \right) \]

### Step 4: Simplify the Equation
Multiply both sides by 100 to eliminate the fraction:
\[ 25920 = 720 \times (2x - \frac{x^2}{100}) \]
Now, divide both sides by 720:
\[ \frac{25920}{720} = 2x - \frac{x^2}{100} \]
Calculating the left side:
\[ 36 = 2x - \frac{x^2}{100} \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ \frac{x^2}{100} - 2x + 36 = 0 \]
Multiplying through by 100 to eliminate the fraction:
\[ x^2 - 200x + 3600 = 0 \]

### Step 6: Solve the Quadratic Equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -200 \), and \( c = 3600 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-200)^2 - 4 \times 1 \times 3600 = 40000 - 14400 = 25600 \]
Now, calculate \( \sqrt{25600} = 160 \).

Now substituting back into the quadratic formula:
\[ x = \frac{200 \pm 160}{2} \]
Calculating the two possible values:
1. \( x = \frac{200 + 160}{2} = \frac{360}{2} = 180 \)
2. \( x = \frac{200 - 160}{2} = \frac{40}{2} = 20 \)

Since \( x \) must be a percentage, we take \( x = 20\% \).

### Conclusion
The value of \( x \) is \( 20\% \).

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