M01-09

Official Solution:

The population of Linterhast was 3,600 people in 1990 and 4,800 people in 1993. If the population growth rate per thousand is constant, then what will be the population in 1996?

A. 6,000
B. 6,400
C. 7,200
D. 8,000
E. 9,600


First of all, the population growth rate per thousand is constant means a constant percentage growth rate.

In 3 years, from 1990 to 1993, the population grew \(\frac{4,800}{3,600} = \frac{4}{3}\) times, and since the the rate of growth is constant, then in the 3 years, from 1993 to 1996, the population will grow by the same rate. Hence, in 1996, the population will be \(4,800*\frac{4}{3}=6,400\)


Answer: B
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Pop in 1990 = 3600, Pop in 1993 = 4800 Time = 3 yrs. Let rate of growth be r% per year

This implies \(4800 = 3600(1+\frac{r}{100})^3\)

This gives \((1+\frac{r}{100})^3 = \frac{4800}{3600}= \frac{4}{3}\)

Now Pop in 1996 i.e. 3 years after 1993 will be \(= 4800(1+\frac{r}{100})^3\)

\(=4800*\frac{4}{3} = 6400\)

It is mentioned in the question that
Population growth rate per thousand is constant
but @Buneul you took % of increase in population as constant.
Is this correct?
I think u need to correct the question.

The rate of increase in multiplying by some constant the same as the percentage increase.
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I solved this question using the concept of linear growth and got the answer as 6000 as follows:
y=mx+c where y is the final value,x is time,m is growth and c is the constant.
at x=0,y=3600 so c=3600
Now when y=4800,x=3 so 4800=m*3+3600 which implies m=400
Thus at x=6,i.e in 1996, y=400*6+3600=6000.

Can't we use linear growth for this question?Where am I going wrong exactly?Please help

It's exponential growth. So, it should be 4,800 = 3,600*n^3, which gives n^3 =4/3 --> 4,800*4/3 = 6,400.
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I think this is a high-quality question. my issue with the explanation is that i agree to the way it is explained but i m not fully convinced that this clears my doubt about why this question is solved in this particular manner..
i solved like this and want to know why my approach is wrong
4800-3600=1200/3=400
1990=3600
1991=3600+400=4000
1992=4000+400=4400
1993=4400+400=4800
(now when the rate is same for each year so i did this)
1994=4800+400=5200
1995=5200+400=5600
1996=5600+400=6000
according to this method i got 6000 as an answer please help me understand why i m wring what i m basically missing out...please explain

We are told that the population growth rate per thousand is constant, not that the population growth number is constant per year.
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Hi Bunuel,

Is there any difference between "growth rate is constant" and "growth rate per 1000 is constant" ? If yes, can you please provide any simple example on how it's works out differently?
I guess they both mean the same only but I'm not completely sure if that's true or if I've made any mistake below.

growth rate is (4800-3600 )/ 3600 X 100 = 33.33%
growth rate per year = 33.33 % / 3 = 11.11 %
AND
growth rate per thousand per year is ((4.8 - 3.6) / 3.6 X 100 ) /3 = 33.33% / 3 = 11.11%

Also, here the time period before and after is 3. Let's say if for the same data if we'd to compute the population in 1997, how would you solve it?
I tried and I get 3600 ( 10/9)^7 which is clearly wrong because 3600( 10/9)^3 is not equal to 4800. What's the mistake here?
Seeing the explanation of the unitary method should help understand the original question and it's solution in a much more better way.

This is good question and particularly confusing if someone is reading it for the first time.

Thanks.

1. If the growth rate is constant, then it's constant for any period of time.

2. The growth rate per 3 years is 1 + 1/3 = 4/3, so per year it's x^3 = 4/3 --> \(x = \sqrt[3]{\frac{4}{3}}\)

3. In 1993, the population would be \(3,600*(\sqrt[3]{\frac{4}{3}})^3 = 4,800\)

4. In 1997, the population would be \(3,600*(\sqrt[3]{\frac{4}{3}})^7 \approx 7,044\)

Hope it helps.
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Bunuel can you please elaborate a little more on your reply below? Its still not clear to me.
What does population growth per 1000 mean? and How to identify this problem as exponential growth instead of linear growth??

Take an example: say we are told that a population is growing 100 people per 1000 per year (10%). This means that every 1000 people "produce" 100 more. For example, if the population is 3,600 people in 1990, then in 1991, the growth will be 3*100 + 600/1000*100 = 360, which is 10% of 3,600.

Another case would be if we are told that a population is growing 100 people per year. So, if the population is 3,600 people in 1990, then in 1991 it will be 3,600 + 100, in 1992 it will be 3,600 + 100 + 100, so basically each year adds another 100 people.

Hope it's clear.
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I think this is a high-quality question and I agree with explanation. Hi Bunel,
Why is it mentioned that the population growth rater per thousand is constant?

Hello, Vihar123. I am not Bunuel, but your question deals with a point that I bring up all the time in my tutoring, which is to make sure you understand the question being asked. Without per thousand in the question stem, the question then becomes

If the population growth rate is constant...

and a linear growth rate could be assumed. Here, if the population grew by 1,200 in 3 years, then that would represent a linear growth rate of 1,200/3, or 400 people per year. I suspect that a fair number of test-takers would make this assumption and stumble into choice (A), which would complete a similar linear growth of 1,200 people in 3 years' time. But of course, such a solution ignores some of the keywords of the actual question, and considering a similar proportion from one year to another leads to the correct solution instead, per the official solution.

Good luck with your studies.

- Andrew
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for me it is much easier to use exponential growth formula:
F = p x k^n, where F - future growth, P - initial growth, k - growth factor and n - number of growth period.

1993: F = 3,600 x k^3 = 4,800. From this we can find k^3 = 4/3
1996: F = 3,600 x k^6 = 3,600 x k^3 x k^3 = 3,600 x 4/3 x 4/3 = 6,400

Seems that a lot of people had trouble understanding why the problem required us to use exponential growth instead of linear growth. The key here is terminology. According to Investopedia, growth rate is defined as a percentage change of a specific variable within a specific time period. So if we're told that the growth rate is constant then the percentage change between each period is the same i.e., the result of each next period is found by multiplying the value from the previous period by the same number. With linear growth, the difference between each period is the same BUT the percentage change between each period is NOT the same.

The takeaway:
(1) If you're told that the growth number or the difference between each period is the same, then apply linear growth. In Linear Growth, we add the same number each time.
(2) If you're told that the growth rate or the percentage change is the same, then apply exponential growth. In Exponential Growth, we multiply by the same number each time.

See screenshot below - notice that for linear growth the difference between each period is the same but the percentage change actually decreases (this makes sense because as we get larger values but add the same number each time, that same number will become a smaller proportion of the larger value).

This question can be solved pretty quickly -

You need to calculate the growth factor from 1990 to 1993 = (4800/3600) = (48/36) = (4/3)

Now this growth factor is constant .... so Population in 1996 will be (4/3) * 4800 = 6,400

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.
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