Comprehension
Annual Percent Change in Dollar Amount of Sales at Five Retail Stores from 2006 to 2008
Question: 1
If the dollar amount of sales at Store P was $800,000 for 2006, what was the dollar amount of sales at that store for 2008?
If the dollar amount of sales at Store P was $800,000 for 2006, what was the dollar amount of sales at that store for 2008?
Show Hint
A common mistake is to simply add the percentages (+10% and -10% = 0% change). Successive percentage changes are multiplicative, not additive. The base changes after the first percent change, leading to a different final result. A 10% increase followed by a 10% decrease will always result in a net decrease from the original amount.
Updated On: Oct 6, 2025
- $727,200
- $792,000
- $800,000
- $880,000
- $968,000
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
This problem requires calculating the result of two consecutive percentage changes. The base for the second percentage change is the result after the first change, not the original amount.
Step 2: Key Approach:
1. Calculate the sales amount for 2007 based on the 2006 amount and the first percentage change.
2. Calculate the sales amount for 2008 based on the 2007 amount and the second percentage change.
Step 3: Detailed Explanation:
Calculate 2007 Sales:
The sales in 2006 for Store P were $800,000.
From the table, the percent change from 2006 to 2007 was +10%.
An increase of 10% is equivalent to multiplying by (1 + 0.10) = 1.10.
\[ \text{Sales in 2007} = $800,000 \times 1.10 = $880,000 \] Calculate 2008 Sales:
The base for the next calculation is the 2007 sales amount, which is $880,000.
From the table, the percent change from 2007 to 2008 was -10%.
A decrease of 10% is equivalent to multiplying by (1 - 0.10) = 0.90.
\[ \text{Sales in 2008} = $880,000 \times 0.90 = $792,000 \] Step 4: Final Answer:
The dollar amount of sales at Store P for 2008 was $792,000. Therefore, the correct answer is (B).
This problem requires calculating the result of two consecutive percentage changes. The base for the second percentage change is the result after the first change, not the original amount.
Step 2: Key Approach:
1. Calculate the sales amount for 2007 based on the 2006 amount and the first percentage change.
2. Calculate the sales amount for 2008 based on the 2007 amount and the second percentage change.
Step 3: Detailed Explanation:
Calculate 2007 Sales:
The sales in 2006 for Store P were $800,000.
From the table, the percent change from 2006 to 2007 was +10%.
An increase of 10% is equivalent to multiplying by (1 + 0.10) = 1.10.
\[ \text{Sales in 2007} = $800,000 \times 1.10 = $880,000 \] Calculate 2008 Sales:
The base for the next calculation is the 2007 sales amount, which is $880,000.
From the table, the percent change from 2007 to 2008 was -10%.
A decrease of 10% is equivalent to multiplying by (1 - 0.10) = 0.90.
\[ \text{Sales in 2008} = $880,000 \times 0.90 = $792,000 \] Step 4: Final Answer:
The dollar amount of sales at Store P for 2008 was $792,000. Therefore, the correct answer is (B).
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Question: 2
At Store T, the dollar amount of sales for 2007 was what percent of the dollar amount of sales for 2008?
Give your answer to the nearest 0.1%.
Enter your answer in the box: [ \ \ \ ] %
At Store T, the dollar amount of sales for 2007 was what percent of the dollar amount of sales for 2008?
Give your answer to the nearest 0.1%.
Enter your answer in the box: [ \ \ \ ] %
Give your answer to the nearest 0.1%.
Enter your answer in the box: [ \ \ \ ] %
Show Hint
When asked for "A is what percent of B?", the formula is always \((A/B) \times 100\). In this problem, A is the 2007 sales and B is the 2008 sales. Set up the relationship between B and A first (\(S_{2008} = 0.92 \times S_{2007}\)), then solve for the required ratio.
Updated On: Oct 6, 2025
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Solution and Explanation
Step 1: Understanding the Concept:
The question asks us to express the 2007 sales as a percentage of the 2008 sales for Store T. This is a "reverse percentage" problem.
Step 2: Key Formula or Approach:
Let \(S_{2007}\) be the sales in 2007 and \(S_{2008}\) be the sales in 2008.
The table gives the relationship: \(S_{2008}\) is a certain percentage change from \(S_{2007}\).
We need to calculate the ratio \(\left(\frac{S_{2007}}{S_{2008}}\right) \times 100%\).
Step 3: Detailed Explanation:
From the table for Store T, the percent change from 2007 to 2008 was -8%.
This means the sales in 2008 were 8% less than the sales in 2007.
We can write this relationship as an equation:
\[ S_{2008} = S_{2007} \times (1 - 0.08) \] \[ S_{2008} = 0.92 \times S_{2007} \] The question asks for \(S_{2007}\) as a percent of \(S_{2008}\). We need to find the value of \(\frac{S_{2007}}{S_{2008}}\). We can rearrange the equation above:
\[ \frac{S_{2007}}{S_{2008}} = \frac{1}{0.92} \] Now, we calculate the value of this fraction and convert it to a percentage:
\[ \frac{1}{0.92} \approx 1.086956... \] To express this as a percentage, we multiply by 100:
\[ 1.086956... \times 100% \approx 108.6956...% \] The question asks to round to the nearest 0.1%.
\[ 108.6956...% \approx 108.7% \] Step 4: Final Answer:
The dollar amount of sales for 2007 was 108.7% of the dollar amount of sales for 2008.
The question asks us to express the 2007 sales as a percentage of the 2008 sales for Store T. This is a "reverse percentage" problem.
Step 2: Key Formula or Approach:
Let \(S_{2007}\) be the sales in 2007 and \(S_{2008}\) be the sales in 2008.
The table gives the relationship: \(S_{2008}\) is a certain percentage change from \(S_{2007}\).
We need to calculate the ratio \(\left(\frac{S_{2007}}{S_{2008}}\right) \times 100%\).
Step 3: Detailed Explanation:
From the table for Store T, the percent change from 2007 to 2008 was -8%.
This means the sales in 2008 were 8% less than the sales in 2007.
We can write this relationship as an equation:
\[ S_{2008} = S_{2007} \times (1 - 0.08) \] \[ S_{2008} = 0.92 \times S_{2007} \] The question asks for \(S_{2007}\) as a percent of \(S_{2008}\). We need to find the value of \(\frac{S_{2007}}{S_{2008}}\). We can rearrange the equation above:
\[ \frac{S_{2007}}{S_{2008}} = \frac{1}{0.92} \] Now, we calculate the value of this fraction and convert it to a percentage:
\[ \frac{1}{0.92} \approx 1.086956... \] To express this as a percentage, we multiply by 100:
\[ 1.086956... \times 100% \approx 108.6956...% \] The question asks to round to the nearest 0.1%.
\[ 108.6956...% \approx 108.7% \] Step 4: Final Answer:
The dollar amount of sales for 2007 was 108.7% of the dollar amount of sales for 2008.
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Question: 3
Based on the information given, which of the following statements must be true?
Indicate {all such statements.}
Based on the information given, which of the following statements must be true?
Indicate {all such statements.}
Indicate {all such statements.}
Show Hint
To calculate the total percent change from two successive changes, \(p_1\) and \(p_2\), do not add them. Instead, use the formula: Total Change = \((1 + p_1/100)(1 + p_2/100) - 1\), and then multiply by 100. Remember to use negative values for percentage decreases.
Updated On: Oct 6, 2025
- For 2008 the dollar amount of sales at Store R was greater than that at each of the other four stores.
- The dollar amount of sales at Store S for 2008 was 22% less than that for 2006.
- The dollar amount of sales at Store R for 2008 was more than 17% greater than that for 2006.
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
This question asks us to evaluate three statements based only on the percent change data. Since we are not given any initial sales figures, we can only make conclusions about the relative percentage changes, not about the absolute dollar amounts.
Step 2: Key Approach:
We will analyze each statement individually to determine if it must be true. For statements involving a combined percent change from 2006 to 2008, we will calculate the effective multiplier.
Multiplier = \((1 + \frac{%\text{change}_1}{100}) \times (1 + \frac{%\text{change}_2}{100})\)
Step 3: Detailed Explanation:
Analysis of Statement A:
"For 2008 the dollar amount of sales at Store R was greater than that at each of the other four stores."
This statement compares the absolute dollar amounts of sales. The table only gives us percentage changes. A store could have a high percentage growth but start from a very low initial sales value, resulting in a lower final sales amount than a store with low growth but a high initial value. Since we don't know the initial sales for 2006 for any store, we cannot compare the final dollar amounts. Therefore, this statement does not have to be true.
Analysis of Statement B:
"The dollar amount of sales at Store S for 2008 was 22% less than that for 2006."
Let \(S_{2006}\) be the sales in 2006.
- Change from 2006 to 2007 is -7%. Multiplier = \(1 - 0.07 = 0.93\).
- Change from 2007 to 2008 is -15%. Multiplier = \(1 - 0.15 = 0.85\).
The combined multiplier from 2006 to 2008 is:
\[ \text{Multiplier} = 0.93 \times 0.85 = 0.7905 \] This means \(S_{2008} = 0.7905 \times S_{2006}\). The sales are 79.05% of the 2006 level.
The percent decrease is \(1 - 0.7905 = 0.2095\), which is a 20.95% decrease.
A 22% decrease is not the same as a 20.95% decrease. Therefore, this statement is false.
Analysis of Statement C:
"The dollar amount of sales at Store R for 2008 was more than 17% greater than that for 2006."
Let \(R_{2006}\) be the sales in 2006.
- Change from 2006 to 2007 is +5%. Multiplier = \(1 + 0.05 = 1.05\).
- Change from 2007 to 2008 is +12%. Multiplier = \(1 + 0.12 = 1.12\).
The combined multiplier from 2006 to 2008 is:
\[ \text{Multiplier} = 1.05 \times 1.12 = 1.176 \] This means \(R_{2008} = 1.176 \times R_{2006}\). This represents a 17.6% increase over the 2006 sales.
The statement says the increase was "more than 17%". Since 17.6% is indeed greater than 17%, this statement is true.
Step 4: Final Answer:
Only statement (C) must be true based on the given information.
This question asks us to evaluate three statements based only on the percent change data. Since we are not given any initial sales figures, we can only make conclusions about the relative percentage changes, not about the absolute dollar amounts.
Step 2: Key Approach:
We will analyze each statement individually to determine if it must be true. For statements involving a combined percent change from 2006 to 2008, we will calculate the effective multiplier.
Multiplier = \((1 + \frac{%\text{change}_1}{100}) \times (1 + \frac{%\text{change}_2}{100})\)
Step 3: Detailed Explanation:
Analysis of Statement A:
"For 2008 the dollar amount of sales at Store R was greater than that at each of the other four stores."
This statement compares the absolute dollar amounts of sales. The table only gives us percentage changes. A store could have a high percentage growth but start from a very low initial sales value, resulting in a lower final sales amount than a store with low growth but a high initial value. Since we don't know the initial sales for 2006 for any store, we cannot compare the final dollar amounts. Therefore, this statement does not have to be true.
Analysis of Statement B:
"The dollar amount of sales at Store S for 2008 was 22% less than that for 2006."
Let \(S_{2006}\) be the sales in 2006.
- Change from 2006 to 2007 is -7%. Multiplier = \(1 - 0.07 = 0.93\).
- Change from 2007 to 2008 is -15%. Multiplier = \(1 - 0.15 = 0.85\).
The combined multiplier from 2006 to 2008 is:
\[ \text{Multiplier} = 0.93 \times 0.85 = 0.7905 \] This means \(S_{2008} = 0.7905 \times S_{2006}\). The sales are 79.05% of the 2006 level.
The percent decrease is \(1 - 0.7905 = 0.2095\), which is a 20.95% decrease.
A 22% decrease is not the same as a 20.95% decrease. Therefore, this statement is false.
Analysis of Statement C:
"The dollar amount of sales at Store R for 2008 was more than 17% greater than that for 2006."
Let \(R_{2006}\) be the sales in 2006.
- Change from 2006 to 2007 is +5%. Multiplier = \(1 + 0.05 = 1.05\).
- Change from 2007 to 2008 is +12%. Multiplier = \(1 + 0.12 = 1.12\).
The combined multiplier from 2006 to 2008 is:
\[ \text{Multiplier} = 1.05 \times 1.12 = 1.176 \] This means \(R_{2008} = 1.176 \times R_{2006}\). This represents a 17.6% increase over the 2006 sales.
The statement says the increase was "more than 17%". Since 17.6% is indeed greater than 17%, this statement is true.
Step 4: Final Answer:
Only statement (C) must be true based on the given information.
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