A man purchsed some eggs at 3 for Rs. 16 and sold them at 5 for Rs.36 . Thus, he gained Rs. 168 in all. How many eggs did he purchase ?

Answer

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

### Step 1: Determine the Cost Price (CP) of One Egg
The cost price of 3 eggs is Rs. 16. Therefore, the cost price of 1 egg can be calculated as follows:

\[ \text{Cost Price of 1 egg} = \frac{16}{3} \text{ Rs.} \]

### Step 2: Determine the Selling Price (SP) of One Egg
The selling price of 5 eggs is Rs. 36. Therefore, the selling price of 1 egg can be calculated as follows:

\[ \text{Selling Price of 1 egg} = \frac{36}{5} \text{ Rs.} \]

### Step 3: Calculate the Profit per Egg
Now, we can find the profit made on each egg by subtracting the cost price from the selling price:

\[ \text{Profit per egg} = \text{Selling Price} - \text{Cost Price} \]
\[ \text{Profit per egg} = \left(\frac{36}{5} - \frac{16}{3}\right) \]

To subtract these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15.

Converting both fractions:
\[ \frac{36}{5} = \frac{36 \times 3}{5 \times 3} = \frac{108}{15} \]
\[ \frac{16}{3} = \frac{16 \times 5}{3 \times 5} = \frac{80}{15} \]

Now, substituting back:
\[ \text{Profit per egg} = \frac{108}{15} - \frac{80}{15} = \frac{28}{15} \text{ Rs.} \]

### Step 4: Relate Total Profit to Number of Eggs
We know the total profit is Rs. 168. Let the number of eggs purchased be \( X \). Therefore, we can express the total profit as:

\[ \text{Total Profit} = \text{Profit per egg} \times \text{Number of eggs} \]
\[ 168 = \frac{28}{15} \times X \]

### Step 5: Solve for \( X \)
To find \( X \), we can rearrange the equation:

\[ X = 168 \times \frac{15}{28} \]

Calculating \( X \):
\[ X = 168 \div 28 \times 15 \]
\[ X = 6 \times 15 = 90 \]

### Conclusion
Thus, the man purchased **90 eggs**.

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