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16 Sep 2014, 00:48
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If the price of a certain item increased by \(x\%\) from 2001 to 2002, and then increased by \(y\%\) from 2002 to 2003, where both \(x\) and \(y\) are positive numbers, what is the overall percentage increase in price from 2001 to 2003?
(1) \(xy = 30\)
(2) \(100x + 100y + xy = 1330\)
(1) \(xy = 30\)
(2) \(100x + 100y + xy = 1330\)
16 Sep 2014, 00:48
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Official Solution:
If the price of a certain item increased by \(x\%\) from 2001 to 2002, and then increased by \(y\%\) from 2002 to 2003, where both \(x\) and \(y\) are positive numbers, what is the overall percentage increase in price from 2001 to 2003?
The price in 2001: \(p\);
The price in 2002: \(p*(1+\frac{x}{100})\);
The price in 2003: \(p*(1+\frac{x}{100})(1+\frac{y}{100})=p*(1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})\);
The percentage increase from 2001 to 2003 is \(\frac{2003-2001}{2001}*100=\)
\(=\frac{p(1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})-p}{p}*100=\)
\(=((1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})-1)*100=\)
\(=x+y+ \frac{xy}{100}\).
(1) \(xy=30\). Not sufficient.
(2) \(100x+100y+xy=1330\).
Dividing both sides by 100, we get \(x+y+\frac{xy}{100}=13.3\), which directly gives us the percentage increase in price. Therefore, statement (2) is sufficient to answer the question.
Answer: B
If the price of a certain item increased by \(x\%\) from 2001 to 2002, and then increased by \(y\%\) from 2002 to 2003, where both \(x\) and \(y\) are positive numbers, what is the overall percentage increase in price from 2001 to 2003?
The price in 2001: \(p\);
The price in 2002: \(p*(1+\frac{x}{100})\);
The price in 2003: \(p*(1+\frac{x}{100})(1+\frac{y}{100})=p*(1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})\);
The percentage increase from 2001 to 2003 is \(\frac{2003-2001}{2001}*100=\)
\(=\frac{p(1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})-p}{p}*100=\)
\(=((1+\frac{y}{100}+\frac{x}{100}+\frac{xy}{10,000})-1)*100=\)
\(=x+y+ \frac{xy}{100}\).
(1) \(xy=30\). Not sufficient.
(2) \(100x+100y+xy=1330\).
Dividing both sides by 100, we get \(x+y+\frac{xy}{100}=13.3\), which directly gives us the percentage increase in price. Therefore, statement (2) is sufficient to answer the question.
Answer: B
General Discussion
22 Mar 2023, 04:21
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
