Mayank buys some candies for Rs $15$ a dozen and an equal number of different candies for Rs $12$ a dozen. He sells all for Rs $16.50$ a dozen and makes a profit of Rs $150$ . How many dozens of candies did he buy altogether?

$50$

$30$

$25$

$45$

✅ Correct Option: 1

Mayank buys two types of candies:

Type 1: Some dozens at Rs 15 per dozen

Type 2: Equal number of dozens at Rs 12 per dozen

Selling price: Rs 16.50 per dozen for all candies

Total profit: Rs 150

We need to find the total dozens of candies bought.

Let's say Mayank buys x dozens of each type of candy.

The problem states "equal number of different candies," so if he buys x dozens of the first type, he also buys x dozens of the second type.

Total candies bought = x + x = 2x dozens

Since Mayank buys equal quantities of both types, we can find the average cost:

Average cost per dozen = $\frac{\text{Cost of Type 1} + \text{Cost of Type 2}}{2}$

Average cost per dozen = $\frac{15 + 12}{2} = \frac{27}{2} = \text{Rs } 13.50$

When buying equal quantities of two items, the average cost is simply the arithmetic mean of their individual costs.

Profit per dozen = Selling price - Average cost price

Profit per dozen = 16.50 - 13.50 = Rs 3 per dozen

We know:

Profit per dozen = Rs 3

Total profit = Rs 150

Using the formula: Total profit = Profit per dozen × Number of dozens sold

$150 = 3 \times \text{Number of dozens sold}$

Number of dozens sold = $\frac{150}{3} = 50$ dozens

Mayank bought 50 dozens of candies altogether.

When buying equal quantities at different prices, we find the average cost, calculate profit per unit, then divide total profit by profit per unit to get the total quantity.

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Mayank buys some candies for Rs $15$ a dozen and an equal number of different candies for Rs $12$ a dozen. He sells all for Rs $16.50$ a dozen and makes a profit of Rs $150$ . How many dozens of candies did he buy altogether?