Growing Everyday
Observation of a subject shows that it grows at a constant rate everyday. If it grows by 69% in 22 days, what percentage has it grown by in 11 days?
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7 solutions
I took the base as 100. I realized that in 22 days, it would grow to 169. As it is a constant rate, it grows by 69/22 everyday. In 11 days, it grows by (69/22)*11 = 34.5. The percentage increase is hence 34.5%, so I chose 35%.
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i guess there is something wrong with your method, if u divide 69 by 22, the "constant rate" u found is rather constant range of growth, which means everyday it grows 1/22 of 69. The value it grows everyday is actually different but not constant.
i also do the same ...
I do the same, why it is wrong?
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It is not a constant rate. Constant rate is like simple interest. The problem is like compound interest.
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@Niranjan Khanderia – So where is this formula come from "1.69 = ( 1 + r )^22, ? = ( 1 + r )^11 "? and those number too.
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@Hafizh Ahsan Permana – It is the formula for compound interest, 0r constant growth rate as explained by Calvin Lin, the staff, see below.
See my reply to Rahul Goel.
thats what i chose too...
It was COMPOUND INTEREST
That's not 'constant rate', that's 'constant amount'. If it's constant rate then the percentage of growth remains the same, say 5%. If it's constant amount, it'll grow by the same amount, let's take it to be 5. (per day, in both cases)
So at constant rate, growth from 100 After 1 day = 105 (= 100 + 5% of 100) After 2 days = 110.25 (= 105 + 5% of 105)
At constant amount After 1 day = 105 (= 100 + 5) After 2 days = 110 (= 105 + 5)
That's the difference.
Same method ;)
nice question
Constant growth rate could imply either a percentage increase or a numeric increase. Since it can easily be interpreted as an arithmetic progression, the question should really be reworded.
I am solving it by following the arithmetic progression.
Since it is constant growth, denoting a to be the initial number and r to be the rate of increase,
1st day = a
2nd day = a + r
3rd day = a + 2r
....
22nd day = a + 21r
So, a + 21r = (169/100) * a
Solving above equation will give you r = (3/100) * a
So, in 11th day, --> a + 10r --> a + (30/100) * a
As you can see, a has increased by 30% in 11 days.
It is constant growth rate, and not constant growth. As such, we have a geometric series instead of an arithmetic series.
Note that in your equation a + 2 1 r = ( 1 6 9 / 1 0 0 ) ∗ a , this does not give us r = ( 3 / 1 0 0 ) ∗ a as claimed. Instead, it gives us r = 2 1 × 1 0 0 6 9 ∗ a .
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ohh yah, u are right there. Must not have been thinking properly. Thanks for pointing this out!
As the growth rate is constant why cant we take dg/dt=K and integrate?
(1+r)^{22}=1.69\tag1
(1+r)^{11}=y\tag2
Dividing (1) by (2) we get
( 1 + r ) 1 1 = y 1 . 6 9 = y
⇒ y 2 = 1 . 6 9
⇒ y = + 1 . 6 9 = 1 . 3
Percentage Growth hence is ( 1 . 3 − 1 ) × 1 0 0 = 3 0
This is the best solution to this problem. Well, can you tell me other equations to solve this problem?
Really good than other but i don't understand this method cause i found 34.5 like Rahul Goel in upward.
the easiest sum.... If we count 69%=100+69 that means (100+rate)^22=1.69 for counting(100+rate)^11 we must get the sqrt of 1.69 thus we will get 1.3 and 1.3 = 130/100 and this shows increase of 30%
it's like formula of compound interest p*(1+r/100)^n =Amount by putting on this formula you'll grt answer 30
Um... you can do the 22nd root of 1.69, and then put it to the power of 11. That's 1.3. So 30%.
Using the formula for nth term of a geometric series;
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ar^{n-1}=1.69a (where r is the common ratio, a is the first term and n is the nth term, taken as 23 here because technically this is after the 22nd day)
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r^{22}=1.69 (dividing both sides by a)
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r^{11} = 1.3 (taking the square root of 1.69)
Thus at the 11th day the population will be 1.3 times the original, 30% more than at the beginning.
square root it and multiply by 10 and sbstract 100.........
Increased by 69% means..... 1.69 = ( 1 + r )^22, ? = ( 1 + r )^11
Taking square root of both sides, ? = square root (1.69 ) = 1.30 means 30%
A = P ( 1 + r )^t the formula for compound interest is used.