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In 1980, when the consumer price index (CPI) was 82.4, Ivan purchased a house for [tex]$71,900[/tex]. Assuming that the price of houses increased at the same rate as the CPI from 1970 to 1980, approximately how much would the house have cost in 1970, when the CPI was 38.8?

A. [tex]$38,800[/tex]
B. [tex]$82,400[/tex]
C. [tex]$38,000[/tex]
D. [tex]$33,900[/tex]

Answer :

To solve this problem, we need to determine the cost of the house in 1970 using the change in the Consumer Price Index (CPI) from 1970 to 1980. The CPI is a measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food, and medical care. Here's how you can approach the problem step-by-step:

  1. Identify the CPI for the two years:

    • In 1970, the CPI was 38.8.
    • In 1980, the CPI was 82.4.

  2. Determine the percentage increase in the CPI from 1970 to 1980:

    The percentage increase can be calculated using the formula:

    [tex]\text{Percentage Increase} = \left( \frac{\text{CPI in 1980} - \text{CPI in 1970}}{\text{CPI in 1970}} \right) \times 100[/tex]

    Plug in the values:

    [tex]\text{Percentage Increase} = \left( \frac{82.4 - 38.8}{38.8} \right) \times 100 \approx 112.37\%[/tex]

  3. Determine the original price of the house in 1970:

    Since the price of the house increased at the same rate as the CPI, we can use the percentage increase to find the price of the house in 1970. The formula for the original price is:

    [tex]\text{Original Price} = \frac{\text{Price in 1980}}{1 + \text{Percentage Increase (as a decimal)}}[/tex]

    [tex]\text{Original Price} = \frac{71,900}{1 + 1.1237} \approx \$33,900[/tex]

  4. Choose the closest option from the given choices:

    Looking at the options provided, the closest answer is:

    D. $33,900

Thus, the house would have cost approximately $33,900 in 1970.

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