SOLUTION: A population of a certain town increased by 50% every 50 years. If the population in 1950 was 810 in what year was the population 160?

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: A population of a certain town increased by 50% every 50 years. If the population in 1950 was 810 in what year was the population 160?      Log On


   

Question 1103313: A population of a certain town increased by 50% every 50 years. If the population in 1950 was 810 in what year was the population 160?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
1.5p=p%2A%28b%29%5E50
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b%5E50=1.5
50%2Alog%28%28b%29%29=log%28%281.5%29%29
log%28%28b%29%29=log%28%281.5%29%29%2F50

basetencommonlogs
log%2810%2Cb%29=0.00352

10%5E0.00352=b
highlight_green%28b=1.00814%29
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MODEL: highlight%28y=p%2A1.00814%5Ex%29

For the question, start as p=160 and find how many years x until population reached y=810.
160%2A1.00814%5Ex=810
1.00814%5Ex=81%2F16
log%28%281.00814%5Ex%29%29=log%28%2881%2F16%29%29
x=log%28%2881%2F16%29%29%2Flog%28%281.00814%29%29
0.704365%2F0.0035208
x=200------------years BEFORE 1950 year.

The year was 1750 when population was 160.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solution from the other tutor is fine; but it goes into much more detail than is necessary to solve this particular problem. It would have been the perfect solution if the numbers in the problem had been "ugly".

But with the numbers 810 and 160, it turns out a little bit of arithmetic leads to an answer with little effort.

We can work backwards if we want, starting with the 1950 population of 810.

An increase of 50% every 50 years means the population gets multiplied by 1.50, or 3/2. Working backwards in time, the population decreases by a factor of 2/3 every 50 years. So
1950: 810
1900: 2/3 of 810 = 540
1850: 2/3 of 540 = 360
1800: 2/3 of 360 = 240
1750: 2/3 of 240 = 160

The population of the town was 160 in 1750.