By selling 35 eggs for Rs. 30, a man losses 25%. How many eggs should he sell for Rs.144 to gain 20%?

The man is selling 35 eggs for Rs. 30 and incurs a loss of 25%. We can use this information to calculate the cost price of the eggs and then determine how many eggs he should sell for Rs. 144 to gain 20%.

### Step 1: Calculate the Cost Price (C.P.) of 35 eggs
He incurs a loss of 25%, which means the selling price (S.P.) is 75% of the cost price.

Let the cost price of 35 eggs be \( C.P. \). 
We know that selling price (S.P.) is Rs. 30, and:
\[
S.P. = C.P. \times \left(1 - \frac{\text{Loss \%}}{100}\right) 
\]
\[
30 = C.P. \times \left(1 - \frac{25}{100}\right)
\]
\[
30 = C.P. \times 0.75
\]
\[
C.P. = \frac{30}{0.75} = 40
\]

So, the cost price of 35 eggs is Rs. 40.

### Step 2: Calculate the Cost Price per egg
The cost price of 1 egg will be:
\[
C.P. \text{ per egg} = \frac{40}{35} = \frac{8}{7} \text{ Rs.}
\]

### Step 3: Find the Selling Price for a 20% gain
To gain 20%, the selling price should be 120% of the cost price.

Let the number of eggs to be sold for Rs. 144 be \( x \).

The cost price of \( x \) eggs will be:
\[
C.P. \text{ of } x \text{ eggs} = x \times \frac{8}{7}
\]
For a 20% gain, the selling price should be:
\[
S.P. = C.P. \times \left(1 + \frac{\text{Gain \%}}{100}\right)
\]
\[
144 = x \times \frac{8}{7} \times \left(1 + \frac{20}{100}\right)
\]
\[
144 = x \times \frac{8}{7} \times 1.2
\]
\[
144 = x \times \frac{9.6}{7}
\]
\[
144 \times 7 = 9.6 \times x
\]
\[
1008 = 9.6x
\]
\[
x = \frac{1008}{9.6} = 105
\]

Thus, the man should sell **105 eggs** for Rs. 144 to gain 20%.

Would you like more details or explanations on any step? Here are 5 related questions to consider:

1. What would the selling price be to break even?
2. How would the problem change if the loss were 15% instead of 25%?
3. What would happen if the gain were 10% instead of 20%?
4. How can we express this problem in terms of a general formula?
5. How does profit or loss affect pricing strategies in other contexts?

**Tip**: When solving percentage-based problems, always relate the percentages to the cost price or original value to simplify calculations.

Learn how to determine the number of eggs to sell to gain 20% profit after incurring a 25% loss by selling 35 eggs for Rs. 30. This problem involves calculating cost price, selling price, and understanding percentage-based profit and loss. Suitable for students in Grades 6-8.

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