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20 May 2017, 19:19
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Last month, a florist gave 7 percent of her sales revenue to her employees in bonuses. This month, she gave them 5 percent of her sales revenue, and sales were up 20 percent from last month. By approximately what percentage did the bonuses change from last month to this month?
(A) –14.3%
(B) –2%
(C) 0%
(D) 2%
(E) 6%
(A) –14.3%
(B) –2%
(C) 0%
(D) 2%
(E) 6%
Let's begin by giving you everything you need to think about this problem correctly. This is a classic question that illustrates the difference between "percentage of" and "percentage change". In order to solve this question, you need to understand both of these ideas.
The leverage phrase "percentage of" requires the use of the formula \(Amount = (\frac{\%}{100})Total\)
The leverage phrase "percentage change" requires the use of the formula \(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
Let's take a look at a couple of ways to attack this problem:
METHOD 1: DOING THE MATH
The question asks for a percentage change, where the initial value (or \(V_{Initial}\)) is "7 percent of" the revenue, or \(\frac{7}{100}R\). (In this case, \(R\) represents the revenue of last month.)
The question then tells us that the final value (or \(V_{Final}\)) is "5 percent of" of the month's revenue, but the month's revenue increased by 20% (indicating a percentage change in the revenue from last month to this month.) Thus, \(R_{Final}=R(1+\frac{1}{5})\). This month's bonus would therefore be \(\frac{5}{100}*R(\frac{6}{5})=(\frac{6}{100})R\). (Notice how, by translating the decimals into fractions, the math simplifies very quickly. I like to tell my classes "Fractions Are Your Friends.")
Once we plug these values into our percentage change formula, the math turns out to be quite easy:
\(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
\(\frac{6}{100}*R=(\frac{7}{100})R(1+\frac{\%}{100})\)
\(\frac{6}{7}*R=R(1+\frac{\%}{100})\)
\(\frac{6}{7}=1+\frac{\%}{100}\)
\(-\frac{1}{7}=\frac{\%}{100}\)
Now, don't actually do the long division here. This would just be a time-killer. If you look down at the answer choices, it is very easy to see what correct answer is. \(-\frac{1}{7}\) is clearly negative, which eliminates answer choices C, D, and E. And \(-2\%\) (or \(-\frac{1}{50}\)) is clearly going to be too small. Answer choice B can be quickly eliminated. (Leveraging your answer choices against each other is often quicker than doing messy math!) The answer is A.
METHOD 2: EASY NUMBERS
Because this problem gives you percentages and percentage changes without ever defining revenue totals, you can invent your own value for the revenue amounts. This turns the abstract algebra into concrete numbers that lend themselves to rapid calculation.
Because we are dealing with percentages in this question, a natural possibility for our imaginary revenue is \(100\). If \(R=100\), then "7 percent of" the revenue would be \(7\).
The percentage increase from month to month is 20%. Thus, if last month's revenue was \(100\), then this month's would be \(100 (1+1/5) = 120\).
The revenue paid on this month is "5 percent of" the increased revenue (in this case, \(120\)). Since 5% is equal to \(\frac{1}{20}\), the new revenue would be: \((120)(\frac{1}{20}) = 6\).
Now we just plug these into the percentage change formula from above:
\(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
\(6=7(1+\frac{\%}{100})\)
\(\frac{6}{7}=1+\frac{\%}{100}\)
\(-\frac{1}{7}=\frac{\%}{100}\)
Once again, we can look down at the answer choices to avoid doing this long division. Eliminating answer choices that don't work is often a faster strategy than doing unnecessary math. Answers C, D, and E are positive, and answer choice B is too small. No matter how you solve this question, the answer is A.
The leverage phrase "percentage of" requires the use of the formula \(Amount = (\frac{\%}{100})Total\)
The leverage phrase "percentage change" requires the use of the formula \(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
Let's take a look at a couple of ways to attack this problem:
METHOD 1: DOING THE MATH
The question asks for a percentage change, where the initial value (or \(V_{Initial}\)) is "7 percent of" the revenue, or \(\frac{7}{100}R\). (In this case, \(R\) represents the revenue of last month.)
The question then tells us that the final value (or \(V_{Final}\)) is "5 percent of" of the month's revenue, but the month's revenue increased by 20% (indicating a percentage change in the revenue from last month to this month.) Thus, \(R_{Final}=R(1+\frac{1}{5})\). This month's bonus would therefore be \(\frac{5}{100}*R(\frac{6}{5})=(\frac{6}{100})R\). (Notice how, by translating the decimals into fractions, the math simplifies very quickly. I like to tell my classes "Fractions Are Your Friends.")
Once we plug these values into our percentage change formula, the math turns out to be quite easy:
\(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
\(\frac{6}{100}*R=(\frac{7}{100})R(1+\frac{\%}{100})\)
\(\frac{6}{7}*R=R(1+\frac{\%}{100})\)
\(\frac{6}{7}=1+\frac{\%}{100}\)
\(-\frac{1}{7}=\frac{\%}{100}\)
Now, don't actually do the long division here. This would just be a time-killer. If you look down at the answer choices, it is very easy to see what correct answer is. \(-\frac{1}{7}\) is clearly negative, which eliminates answer choices C, D, and E. And \(-2\%\) (or \(-\frac{1}{50}\)) is clearly going to be too small. Answer choice B can be quickly eliminated. (Leveraging your answer choices against each other is often quicker than doing messy math!) The answer is A.
METHOD 2: EASY NUMBERS
Because this problem gives you percentages and percentage changes without ever defining revenue totals, you can invent your own value for the revenue amounts. This turns the abstract algebra into concrete numbers that lend themselves to rapid calculation.
Because we are dealing with percentages in this question, a natural possibility for our imaginary revenue is \(100\). If \(R=100\), then "7 percent of" the revenue would be \(7\).
The percentage increase from month to month is 20%. Thus, if last month's revenue was \(100\), then this month's would be \(100 (1+1/5) = 120\).
The revenue paid on this month is "5 percent of" the increased revenue (in this case, \(120\)). Since 5% is equal to \(\frac{1}{20}\), the new revenue would be: \((120)(\frac{1}{20}) = 6\).
Now we just plug these into the percentage change formula from above:
\(V_{Final}=V_{Initial}(1+\frac{\%}{100})\)
\(6=7(1+\frac{\%}{100})\)
\(\frac{6}{7}=1+\frac{\%}{100}\)
\(-\frac{1}{7}=\frac{\%}{100}\)
Once again, we can look down at the answer choices to avoid doing this long division. Eliminating answer choices that don't work is often a faster strategy than doing unnecessary math. Answers C, D, and E are positive, and answer choice B is too small. No matter how you solve this question, the answer is A.
General Discussion
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20 May 2017, 19:39
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hazelnut
2 ways of doing this:
1. Use the values in the question directly to get the following equations:
Case 1: bonus = 7% * revenues = 7%*r
Case 2: bonus' = b' = 5% of the new revenues = 5%*r' = 5%*(1.2)*r = 6%*r
Thus, the bonus has clearly decreased, thus ruling out options C-E. % Decrease = (6%-7%)/7% = a value more than 10%. Thus A is the correct answer. Dont waste time on actual values.
2. Assign value to the revenue.
Case 1: b = 7* 100 = 7 $
Case 2: b' = 5%*1.2*100 = 6$
% decrease = (6-7)/7 = 1/7 > 10% decrease.
Thus, A is the correct answer.
