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I was preparing for a standardized test and came across the following problem:

During a closeout sale 30 suits are sold at \$60 each.The price is the reduced to \$50 each and 20 suits are sold. At what average price must the remaining 10 suits be sold in order to obtain an average of \$55 for the 60 suits?

The answer is supposed to be \$50. Could anyone tell me how they got that? A little working shown would be appreciated..

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Up to now 50 suits are sold for \$2800 alltogether. Now find $x$ with $(2800 + 10x)/60 = 55$. – martini May 22 '12 at 15:24
up vote 3 down vote accepted

The first and most important step is to introduce a variable:

$\ \ \ \ \ \ $Let $x$ be the common price of each of the remaining $10$ suits.

The second step is to find an equation in $x$. Of course, we may obtain one by knowing that the average sale price of the twenty suits is $55$ dollars. Once we have our equation, we'll solve it for $x$.

The average sale price of the suits is obtained by computing the total revenue from the $60$ suits and dividing by $60$.

Let's find the total revenue first. Thirty suits were sold at $60$ dollars and twenty suits were sold at $50$ dollars. The revenue from these is $30\cdot60+20\cdot 50$ dollars. The remaining ten suits are sold at $x$ dollars and the revenue from these is $x\cdot10$ dollars. So, adding these together, the total revenue from the $60$ suits is $30\cdot 60+ 20 \cdot 50 +x\cdot 10$ dollars.

Thus, the average sale price of the twenty suits is $$\tag{1}{\text{total revenue}\over 60}={30\cdot 60+ 20 \cdot 50 +x\cdot 10\over 60}.$$

Now we find our equation. You want $(1)$ to turn out to be $55$. So set $$\tag{2} {30\cdot 60+ 20 \cdot 50 +x\cdot 10\over 60}=55. $$

Now the third and final step: we solve equation $(2)$ for $x$. Towards that end, first multiply both sides of $(2)$ by $60$: $$ {30\cdot 60+ 20 \cdot 50 +x\cdot 10 }=55\cdot 60. $$ Simplify the arithmetic: $$\tag{3} 2800+10x=3300, $$ or, subtracting $2800$ from both sides of $(3)$, $$\tag{4} 10x=500. $$ Finally, divide both sides of $(4)$ by $10$ to obtain $x=50$ dollars.

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OMG - Why couldn't I see this ... Thank you. – Rajeshwar May 22 '12 at 15:40
@Rajeshwar You're welcome; glad to help. – David Mitra May 22 '12 at 15:43

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