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I am working through a math book and the question states:

The cost of living in a city increased by 20% in the 10 years from 1980 to 1990 and by 50% from 1980 to 2000. What was the percent increase in cost of living from 1990 to 2000?

The book shows the work involved to reach the answer, but does not explain how the problem is setup. Why is 120 placed in the denominator?

$$\frac{150 - 120}{120} * 100 = \frac{30}{120} * 100 = 25$$

Thanks for your help, sorry if this is too basic for this website.

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  • $\begingroup$ If for example it cost $\$100$ a week to live in the city in 1980, how much did it cost in 1990 and in 2000? If you knew that, then how would you work out the percentage change from 1990 to 2000? $\endgroup$ – Henry Feb 27 '16 at 1:17
  • $\begingroup$ It goes in the bottom because its the cost in 1990 and that's what we want the base to be. The percent is $\frac{C_{2000} - C_{1990}}{C_1990}*100$, where $C_x$ is cost of living in year x. We don't actually have any actual numbers but we know $C_{2000} = 1.5 C_{1980}$ and $C_{1990} = 1.2 C_{1980}$. But the book thinks it's easier to assume the basic $C_{1980} = 100$ and so $C_{1990} = 120$ and $C_2000} = 150$. We could use any amount. 50 dollars in 1980 is 60 dollars in 1990 and 75 dollars in 2000. 100 dollars is a convenient basis for percents. $\endgroup$ – fleablood Feb 27 '16 at 1:43
  • $\begingroup$ Thanks for your help everyone - I see why I was confused now. I am supposed to be finding the increase from 100% of the cost of living in 1990, $\frac{120}{120}$, to $\frac{150}{120}$ in 2000. $\endgroup$ – tylerkmw Feb 27 '16 at 1:46
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It should be seen as percentages. Compared to 1980 cost in 1990 is 120%. Compared to 1980 cost in 2000 is 150%.

So what percent increase is 150 to 120 percent? $\frac{\text{cost of living in 2000} - \text{cost of living in 1990}}{\text{cost of living in 1990}}*100 = \text{percentage increase from 1990 to 2000}$.

Plugging in 120% and 150%

$\frac{\text{150%} - \text{120%}}{\text{120%}}*100 = \frac {30}{120}*100= \text{percentage increase from 1990 to 2000}$.

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$P(2000)/P(1980)=150$%$=1.50.$....$P(1990)/P(1980)=120$%$=1.20.$... $P(2000)/P(1990)=[ P(2000)/P(1980) ]/ [P(1990)/P(1980)]= 1.50/1.20=1.25=125$%. The proportionate increase from 1990 to 2000 is the ratio $P(2000)/P(1990),$ not $[P(2000)-P(1990)]/P(1980).$ The amount of increase from 1990 to 2000 was 30% of the 1980 value of P but only 25% of the 1990 value of P.

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