The ratio of cost price and the marked price of an article is 2 : 3 and ratio of percentage profit and percentage discount is 3 : 2 . What is the discount percentage?

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Ratios
We are given two ratios:
1. The ratio of Cost Price (CP) to Marked Price (MP) is 2:3.
2. The ratio of Percentage Profit (P) to Percentage Discount (D) is 3:2.

### Step 2: Express CP and MP
Let the Cost Price (CP) be represented as \(2x\) and the Marked Price (MP) as \(3x\) for some value \(x\).

### Step 3: Calculate the Selling Price (SP)
The Selling Price (SP) can be calculated using the relationship between CP, Profit, and SP. The profit can be expressed as:
\[ P = SP - CP \]
Thus,
\[ SP = CP + P \]

### Step 4: Express Profit and Discount
From the second ratio, we can express Profit (P) in terms of Discount (D):
\[ P = \frac{3}{2}D \]

### Step 5: Substitute Values
Now, substituting the values of CP and P into the SP equation:
\[ SP = CP + P \]
\[ SP = 2x + \frac{3}{2}D \]

### Step 6: Relate SP to MP
Since SP can also be expressed in terms of MP and Discount:
\[ SP = MP - D \]
Substituting the value of MP:
\[ SP = 3x - D \]

### Step 7: Set the Two SP Equations Equal
Now we have two expressions for SP:
1. \( SP = 2x + \frac{3}{2}D \)
2. \( SP = 3x - D \)

Setting them equal gives:
\[ 2x + \frac{3}{2}D = 3x - D \]

### Step 8: Solve for D
Rearranging the equation:
\[ 3x - 2x = \frac{3}{2}D + D \]
\[ x = \frac{3}{2}D + D \]
\[ x = \frac{3}{2}D + \frac{2}{2}D \]
\[ x = \frac{5}{2}D \]

### Step 9: Substitute Back to Find D
Now, substituting \(x\) back into the equation \(x = \frac{5}{2}D\):
\[ D = \frac{2}{5}x \]

### Step 10: Calculate the Discount Percentage
To find the discount percentage, we need to express D as a percentage of MP:
\[ D = MP - SP \]
Using \(MP = 3x\) and \(SP = 2x + \frac{3}{2}D\), we can find D in terms of MP.

From the earlier equation, we can see that:
\[ D = \frac{20}{100} \times MP \]
Thus, the discount percentage is:
\[ D = 20\% \]

### Final Answer
The discount percentage is **20%**.

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