Annual growth rate %

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hopeless_mathematician

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Hello everyone, I am confused by this problem. Using a %change formula I got 6320% but the correct answer is 81%. Does anyone understand why and could you explain? Thank you!
Screenshot 2022-09-18 at 14.21.52.png
 
topsquark said:
How did you get 6320%? For all we know, this might even be correct!

Hello, thank you for the reply! I used the formula [(new value-old value)/old] * 100 -- [(3.21-0.05_/0.05)*100] = 6320% -- the answer posted on the website says that I could do the 7th root of (3.21/0.05)-1 but I don't understand how this works and even when I do it I do not get 81%
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hopeless_mathematician said:
Thank you for the reply. Yes, I was thinking the same thing, how would you go about solving this?
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I would guess that they want the average annual percentage growth rate. What have you learned about that? In effect, this would be asking, what constant annual percentage growth would go from 0.05 to 3.21 in 16 years?
 
Dr.Peterson said:
I would guess that they want the average annual percentage growth rate. What have you learned about that? In effect, this would be asking, what constant annual percentage growth would go from 0.05 to 3.21 in 16 years?
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How did I make that typo? I meant in 7 years!

I must have read 2010 as 2001.
 
hopeless_mathematician said:
Hello everyone, I am confused by this problem. Using a %change formula I got 6320% but the correct answer is 81%. Does anyone understand why and could you explain? Thank you!
View attachment 34075
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81% per annum is the correct answer (rounded to a whole number).
Here is a similar example to illustrate:-

The price of a loaf in 2016 was 16p
In 2020 it had increased to 81p
81÷16=5.0625
Taking (81-16) ÷16 = 4.0625 and multiplying by 100 (like you did) will give you the total percentage increase over the four years (406.25%), ie: the price has become more than five times bigger than it was four years previously (16×5.0625). ......................................edited

But you are not asked for that. You are asked to calculate what the (average?) percentage increase has been in each year. [NB: you can only calculate the average percentage increase because it might have gone up differing amounts each year with the same end result but given only the start & end points you can only calculate a single (ie: the same) annual percentage increase for each year.]

How?
Well you take the nth root of the overall multiplier (where n is the number of years).
To get an nth root there should be a \(\displaystyle \sqrt[x]{y} \text{or} \sqrt[y]{x}\) button on your calculator (if it's a half-decent one! ?)
Thus the Annual Percentage Increase = \(\displaystyle \sqrt[4]{5.0625}=1.5\)

So the (average) annual increase has been 50%

Verifying:
2016 – 16p (Starting price.)
2017 – 16p + 8p (50%) = 24p
2018 – 24p + 12p = 36p
2019 – 36p + 18p = 54p
2020 – 54p + 27p = 81p (End price.)

This is equivalent to multiplying by 1.5 each year (ie: four times)

And 16×1.5×1.5×1.5×1.5 ≡ 16×1.5⁴
("≡" is the Mathematical symbol for "Equivalent to" or "Exactly the same as")

Now 1.5⁴ = 5.0625 which is why \(\displaystyle \sqrt[4]{5.0625}=1.5\)

Do you now see how you might go about solving the problem (as it has been set) to arrive at the answer provided (regardless of how “flawed” it is)? ?

Please now show your attempt(s) to arrive at an answer and further guidance will be offered if necessary. ?
 
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hopeless_mathematician said:
Hello, thank you for the reply! I used the formula [(new value-old value)/old] * 100 -- [(3.21-0.05_/0.05)*100] = 6320% -- the answer posted on the website says that I could do the 7th root of (3.21/0.05)-1 but I don't understand how this works and even when I do it I do not get 81%
Click to expand...
That's because the website is crap at Maths! In addition to the flaws pointed out by both @Subhotosh Khan & myself (the APR is not constant over the years in question) they have also given you the wrong formula to use to get their 'correct' answer! There should not be a "-1" in that formula!
I expect the answer you got was: 80.8%? (though that would round to 81% ?)
 
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