Question

question 34 of 46
from 1980 to 1990, the consumer price index (cpi) increased from 82.4 to 130.7. if a bottle of dish soap cost $0.74 in 1980 and the price of dish soap increased at the same rate as the cpi from 1980 to 1990, by approximately how much did the price of a bottle of dish soap increase from 1980 to 1990?
a. $1.31
b. $1.17
c. $0.82
d. $0.43

Explanation:

Step 1: Calculate the CPI multiplier

The CPI increased from 82.4 (1980) to 130.7 (1990). The multiplier for the price (since the price of dish soap increases at the same rate as CPI) is calculated as the ratio of the 1990 CPI to the 1980 CPI. So, $r=\frac{130.7}{82.4}\approx1.586$.

Step 2: Find the 1990 price of dish soap

The price of dish soap in 1980 is $\$0.74$. To find the price in 1990, we multiply the 1980 price by the multiplier $r$. So, $P_{1990}=0.74\times r = 0.74\times\frac{130.7}{82.4}\approx0.74\times1.586\approx1.173$.

Step 3: Calculate the price increase

The increase in price is the 1990 price minus the 1980 price. So, $\Delta P=P_{1990}-0.74 = 1.173 - 0.74\approx0.43$.

Answer:

D. $0.43$ (Wait, correction: Let's re - calculate. First, find the rate of increase of CPI. The CPI increased from 82.4 to 130.7. The multiplier is $\frac{130.7}{82.4}$. Then, the 1990 price of dish soap is $0.74\times\frac{130.7}{82.4}$. Let's compute that: $0.74\times\frac{130.7}{82.4}\approx0.74\times1.586\approx1.1736$. The increase is $1.1736 - 0.74\approx0.4336$, which is approximately $0.43$? Wait, no, the options: A is 1.31, B is 1.17, C is 0.82, D is 0.43. Wait, maybe I misread the CPI numbers. Let's check again. If CPI went from 82.4 to 130.7. The factor is $130.7\div82.4\approx1.586$. Then 0.741.586≈1.17. So the increase is 1.17 - 0.74 = 0.43? Wait, no, 1.17 - 0.74 is 0.43? Wait, 0.741.586 is approximately 1.17, so the increase is 1.17 - 0.74 = 0.43? Wait, but option B is 1.17 (the new price) and D is 0.43 (the increase). The question is "by approximately how much did the price...increase", so we need to find the increase. So first, calculate the new price: $0.74\times\frac{130.7}{82.4}\approx0.74\times1.586\approx1.17$. Then the increase is $1.17 - 0.74 = 0.43$. So the answer is D. $0.43$? Wait, but let's do the calculation more accurately. $\frac{130.7}{82.4}=130.7\div82.4\approx1.586165$. Then $0.74\times1.586165 = 0.74\times1.586165$. Let's compute 0.71.586165 = 1.1103155, 0.041.586165 = 0.0634466, total is 1.1103155+0.0634466 = 1.1737621. Then 1.1737621 - 0.74 = 0.4337621, which is approximately $0.43$. So the answer is D. $0.43$? Wait, but the options: D is $0.43$. So the correct answer is D. $0.43$.

Wait, maybe I made a mistake earlier. Let's recast:

Step 1: Find the ratio of CPI in 1990 to CPI in 1980. That is $r=\frac{130.7}{82.4}$.

Step 2: The price of dish soap in 1990 is $P_{1990}=0.74\times r$.

Step 3: The increase in price $\Delta P = P_{1990}-0.74$.

Calculating $r=\frac{130.7}{82.4}\approx1.586$.

$P_{1990}=0.74\times1.586\approx1.173$.

$\Delta P = 1.173 - 0.74 = 0.433\approx0.43$.

So the answer is D. $\$0.43$.