# Percentage Problem: Understanding the Solution

• Jan 20th 2011, 06:44 AM
dumluck
Percentage Problem: Understanding the Solution
Hi all,
I was wondering if you could help me with the following...

Q. The population of a certain town increases by 50 percent every 50 years. If the population in 1950 was 810, in what year was the population 160?

Ans: 1650 , 1700, 1750, 1800, 1850.

How I would have tackled this was to decrease 810 and the resulting amount by 50% and 50 years until I hit 160. But that's wrong. Here is the solution..

Sol

If the population increases by 50% every 50 years, the population in 1950 was 150% or 3/2 of the 1900 population. How; is it not just a 50% on 1900 so decreasing the value of 1950 by 50% should result in the 1900 figure?
This mean the 1900 population was 2/3 that of the 1950.... and so on. How is that? 2/3 would be 66% would it not?

A layman explanation so I can understand the principle here would be very much appreciated.

Paul
• Jan 20th 2011, 06:56 AM
Unknown008
Ok, I'm not sure I fully understand the solution you provided, but this is how I think about it.

Say in 1900, there was a certain population. That amount is 100%, you agree with me? Then, to get the population in 1950, you add 50% f that amount, or, you multiply by 150% (50% added to 100%).

The reverse, is to divide by 150%, right?

Now, each year, you will have to divide by 150%. This is a geometric progression and you can use the general formula:

$T_n = ar^{n-1}$

and convert it to:

$P_n = pr^{n-1}$

Where P_n is the population size after going back n times (that is, n = 50 years),
p the initial population size, in this case 810.
r is the common ratio, here it's 1/150%, or 1/(150/100) = 100/150 = 2/3\

Here is where the 2/3 comes from.

Then, at n times, the population became 160:

$160 = 810\left(\dfrac23\right)^{n-1}$

Solve for n.

Then, multiply this by 50 years, subtract this from the year 1950.
• Jan 20th 2011, 07:04 AM
dumluck
Thanks; so if the common ratio is 2/3, why is the first calculation performed using 3/2 do you know?

Could I come to the same conculsion by reducing the amounts by 150% at a time?
• Jan 20th 2011, 07:07 AM
Unknown008
Sure, it should bring you to the same result, because 2/3 is just a simplification of dividing by 150% (Smile)
• Jan 20th 2011, 07:09 AM
dumluck
Ok but is 150 not 3/2?
• Jan 20th 2011, 07:12 AM
Unknown008
Yes, 150% = 150/100 = 3/2

(Happy)
• Jan 20th 2011, 07:14 AM
Wilmer
Quote:

Originally Posted by dumluck
Q. The population of a certain town increases by 50 percent every 50 years. If the population in 1950 was 810, in what year was the population 160?
Ans: 1650 , 1700, 1750, 1800, 1850.

Or look at it this way:
160(1 + percentage_increase)^("n"umber of 50_year_periods) = 810
160(1 + .5)^n = 810
1.5^n = 810/160
n = log(810/160) / log(1.5)
n = 4

SO: 1950 - 4*50 = 1950 - 200 = 1750
• Jan 20th 2011, 07:14 AM
dumluck
Thanks. bare with me here. I'm sure we are on our way to a Eureka moment, so why are we multiplying each value by the reciprical? i.e. 2/3 if 150% = 3/2.
• Jan 20th 2011, 07:16 AM
Unknown008
If we are going 'backwards', you do the reverse, and the reverse of multiplying by 150% is dividing by 150%.

The fact that you have to divide several times by 150% might be tiresome, and what if you have to do that operation some dozen times! This is why you use the power.
• Jan 20th 2011, 07:20 AM
dumluck
ahhh ok. So if the question asked for an increase in value we would be multiplying by 3/2?

Good point. Thanks.
• Jan 20th 2011, 07:23 AM
Wilmer
Quote:

Originally Posted by dumluck
ahhh ok. So if the question asked for an increase in value we would be multiplying by 3/2?

RIGHT! Same as 1.5 which I show in my example-solution ... see it?
• Jan 20th 2011, 07:23 AM
Unknown008
Exactly right! (Nod)
• Jan 20th 2011, 07:40 AM
Wilmer
Going back to the solution I showed you, you can "re-assure" yourself that 4 periods of 50 years is correct:

160 * (1.5)^4 = 160 * 5.0625 = 810

You can look at this as a financial transaction:
if \$160 is deposited and earns interest of 50% annually (yikes!), what will be the value in 4 years?
F = A(1 + i)^n
F = 160(1 + .50)^4
F = 810 ... if no income tax!!
• Jan 20th 2011, 07:43 AM
Unknown008
If interest was that much, banks would be ruined in no time while people would get rich extremely quickly (Wink) (Rofl)