Question:
In 1998, the list price of a home was \(\frac{1}{3}\) greater than the original price. In 2008, the list price of the home was \(\frac{1}{2}\) greater than the original price. By what percent did the list price of the home increase from 1998 to 2008? (Note: The fractions were missing in the text and have been inferred).
In 1998, the list price of a home was \(\frac{1}{3}\) greater than the original price. In 2008, the list price of the home was \(\frac{1}{2}\) greater than the original price. By what percent did the list price of the home increase from 1998 to 2008? (Note: The fractions were missing in the text and have been inferred).
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When dealing with percent changes of values that are themselves based on an initial value, you can often assume the initial value is a convenient number, like 100, to simplify calculations. For instance, if original price = $120 (a multiple of 3 and 2), then Price 1998 = $120 + (1/3)*120 = $160. Price 2008 = $120 + (1/2)*120 = $180. Percent increase = `(180-160)/160 * 100 = 20/160 * 100 = 1/8 * 100 = 12.5%`.
Updated On: Oct 3, 2025
- 10%
- 12.5%
- \(16\frac{2}{3}\)%
- \(33\frac{1}{3}\)%
- 50%
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
This problem requires calculating the percent increase between two values (the price in 2008 and the price in 1998), where both values are defined relative to a common "original price".
Step 2: Key Formula or Approach:
Let \(P_O\) be the original price of the home. The price in 1998 (\(P_{1998}\)) is \(P_O \times (1 + \frac{1}{3})\). The price in 2008 (\(P_{2008}\)) is \(P_O \times (1 + \frac{1}{2})\). The formula for percent increase from 1998 to 2008 is: \[ \text{Percent Increase} = \frac{P_{2008} - P_{1998}}{P_{1998}} \times 100% \] Step 3: Detailed Explanation:
Let's express the prices in 1998 and 2008 in terms of the original price, \(P_O\). \[ P_{1998} = P_O \left(1 + \frac{1}{3}\right) = \frac{4}{3} P_O \] \[ P_{2008} = P_O \left(1 + \frac{1}{2}\right) = \frac{3}{2} P_O \] Now, we can substitute these expressions into the percent increase formula. The \(P_O\) term will cancel out. \[ \text{Percent Increase} = \frac{\frac{3}{2} P_O - \frac{4}{3} P_O}{\frac{4}{3} P_O} \times 100% \] \[ = \frac{(\frac{3}{2} - \frac{4}{3}) P_O}{\frac{4}{3} P_O} \times 100% \] Find a common denominator for the fractions in the numerator: \[ = \frac{\frac{9}{6} - \frac{8}{6}}{\frac{4}{3}} \times 100% \] \[ = \frac{\frac{1}{6}}{\frac{4}{3}} \times 100% \] To divide by a fraction, we multiply by its reciprocal: \[ = \frac{1}{6} \times \frac{3}{4} \times 100% \] \[ = \frac{3}{24} \times 100% = \frac{1}{8} \times 100% \] \[ = 12.5% \] Step 4: Final Answer:
The list price of the home increased by 12.5% from 1998 to 2008.
This problem requires calculating the percent increase between two values (the price in 2008 and the price in 1998), where both values are defined relative to a common "original price".
Step 2: Key Formula or Approach:
Let \(P_O\) be the original price of the home. The price in 1998 (\(P_{1998}\)) is \(P_O \times (1 + \frac{1}{3})\). The price in 2008 (\(P_{2008}\)) is \(P_O \times (1 + \frac{1}{2})\). The formula for percent increase from 1998 to 2008 is: \[ \text{Percent Increase} = \frac{P_{2008} - P_{1998}}{P_{1998}} \times 100% \] Step 3: Detailed Explanation:
Let's express the prices in 1998 and 2008 in terms of the original price, \(P_O\). \[ P_{1998} = P_O \left(1 + \frac{1}{3}\right) = \frac{4}{3} P_O \] \[ P_{2008} = P_O \left(1 + \frac{1}{2}\right) = \frac{3}{2} P_O \] Now, we can substitute these expressions into the percent increase formula. The \(P_O\) term will cancel out. \[ \text{Percent Increase} = \frac{\frac{3}{2} P_O - \frac{4}{3} P_O}{\frac{4}{3} P_O} \times 100% \] \[ = \frac{(\frac{3}{2} - \frac{4}{3}) P_O}{\frac{4}{3} P_O} \times 100% \] Find a common denominator for the fractions in the numerator: \[ = \frac{\frac{9}{6} - \frac{8}{6}}{\frac{4}{3}} \times 100% \] \[ = \frac{\frac{1}{6}}{\frac{4}{3}} \times 100% \] To divide by a fraction, we multiply by its reciprocal: \[ = \frac{1}{6} \times \frac{3}{4} \times 100% \] \[ = \frac{3}{24} \times 100% = \frac{1}{8} \times 100% \] \[ = 12.5% \] Step 4: Final Answer:
The list price of the home increased by 12.5% from 1998 to 2008.
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