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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?

1. $50$
2. $30$
3. $25$
4. $45$

Let he buys $n$ dozen of candies at Rs.$15$ a dozen and $n$ more dozen at Rs.$12$ a dozen.

The average cost of candies per dozen $= \left(\frac{15+12}{2}\right) = \frac{27}{2} = \text{Rs.}\;13.50$ a dozen

By selling a dozen Rs.$16.50$, he will make a profit $= 16.50-13.50 = \text{Rs.}\;3$ per dozen.

• $1$ dozen $\longrightarrow \text{Rs.}\;3$
• $2n$ dozen $\longrightarrow \text{Rs.}\;150$

$\Rightarrow 2n = \frac{150}{3} = 50$ dozens.

$\therefore$ Fifty dozen of candies he will buy altogether.

$\textbf{Short Method:}$ Using the allegation method.

• $1$ dozen $\longrightarrow \text{Rs.}\;4.5 – \text{Rs.}\;1.5$
• $1$ dozen $\longrightarrow \text{Rs.}\;3$
• Rs. $1 \longrightarrow \frac{1}{3}$ dozen
• Rs. $150 \longrightarrow \frac{1}{3}\times150$ dozen
• Rs. $150 \longrightarrow 50$ dozen

Correct Answer $:\text{A}$

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