After offering discount of 23% a seller gains profit of 21%. Find ratio of C.P. to M.P.

Answer

To solve the problem, we need to find the ratio of the Cost Price (C.P.) to the Marked Price (M.P.) given that after offering a discount of 23%, the seller gains a profit of 21%.

### Step-by-Step Solution:

1. **Understand the Terms**:
- Let the Marked Price (M.P.) be represented as \( M \).
- The Selling Price (S.P.) after the discount is calculated as:
\[ S.P. = M.P. - \text{Discount} \]
- The discount given is 23% of the M.P.:
\[ \text{Discount} = 23\% \text{ of } M = \frac{23}{100} \times M \]
- Therefore, the Selling Price (S.P.) can be expressed as:
\[ S.P. = M - \frac{23}{100}M = \left(1 - \frac{23}{100}\right)M = \frac{77}{100}M \]

2. **Relate Selling Price to Cost Price**:
- We know that the seller gains a profit of 21%. This means that the Selling Price (S.P.) is 121% of the Cost Price (C.P.):
\[ S.P. = C.P. + \text{Profit} = C.P. + 21\% \text{ of } C.P. = 121\% \text{ of } C.P. \]
- This can be written as:
\[ S.P. = \frac{121}{100} \times C.P. \]

3. **Set the Two Expressions for Selling Price Equal**:
- From the two expressions for S.P., we have:
\[ \frac{77}{100}M = \frac{121}{100} \times C.P. \]

4. **Solve for the Ratio of C.P. to M.P.**:
- Rearranging the equation gives:
\[ C.P. = \frac{77}{121}M \]
- To find the ratio of C.P. to M.P., we write:
\[ \frac{C.P.}{M.P.} = \frac{C.P.}{M} = \frac{77}{121} \]

5. **Final Ratio**:
- Therefore, the ratio of C.P. to M.P. is:
\[ \frac{C.P.}{M.P.} = \frac{77}{121} \]

### Conclusion:
The ratio of Cost Price (C.P.) to Marked Price (M.P.) is \( \frac{77}{121} \).