An article was sold at 65% of its marked price, and thus there was a gain of 30%. Find the ratio of the marked price to cost price.

Options

A. `3:2`
B. `4:3`
C. `2:1`
D. `3:1`

Answer

To solve the problem, we need to find the ratio of the marked price (MP) to the cost price (CP) given that an article was sold at 65% of its marked price and there was a gain of 30%.

### Step-by-step Solution:

1. **Assume the Marked Price (MP)**:
Let's assume the marked price of the article is \( MP = 100 \).

2. **Calculate the Selling Price (SP)**:
The article was sold at 65% of the marked price. Therefore, the selling price can be calculated as:
\[ SP = 65\% \text{ of } MP = 0.65 \times 100 = 65 \]

3. **Understand the Gain**:
We know that there was a gain of 30%. This means that the selling price is 130% of the cost price (CP). Therefore, we can express this as:
\[ SP = CP + 30\% \text{ of } CP = CP + 0.30 \times CP = 1.30 \times CP \]

4. **Set Up the Equation**:
Now we can set the selling price equal to the expression involving the cost price:
\[ 65 = 1.30 \times CP \]

5. **Solve for Cost Price (CP)**:
To find the cost price, we can rearrange the equation:
\[ CP = \frac{65}{1.30} \]
Calculating this gives:
\[ CP = \frac{65}{1.30} = 50 \]

6. **Find the Ratio of Marked Price to Cost Price**:
Now that we have both the marked price and the cost price, we can find the ratio:
\[ \text{Ratio of MP to CP} = \frac{MP}{CP} = \frac{100}{50} = 2 \]
Thus, the ratio of the marked price to the cost price is:
\[ \text{Ratio} = 2 : 1 \]

### Final Answer:
The ratio of the marked price to the cost price is \( 2 : 1 \).