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Each year for 4 years, a farmer increased the number of trees in a

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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 04 Nov 2018, 03:28
yashna36 wrote:
Hi,

I tried using geometric progression formula for this ->
x is the no of trees at the end of 1st year, end of 2nd year no of trees is 5x/4. Therefore, common ratio (r) is 5/4

6250=x(r^n-1)
6250= x(5/4)^4-1
6250=x(5/4)^3
x=(6250*64)/125
x=3200

can you please explain to me where has my reasoning gone wrong?


Hey yashna36,
There is nothing wrong in your reasoning - you need to do one more step to get the final answer.

You have assumed x as the number of trees at the end of 1st year, whereas the question is asking us to find the number of trees at the beginning of the 1st year.

Therefore, you need to divide 3200 by 5/4 to get the final answer.

Hope this helps. :)
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 04 Nov 2018, 12:09
It's a compound interest with rate = 25% , time = 4 year.

6250 = x (1+ 25/100)^4

x = 2560
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 28 Jan 2019, 13:30
imhimanshu wrote:
Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard of the preceding year. If all of the trees thrived and there were 6250 trees in the orchard at the end of 4 year period, how many trees were in the orchard at the beginning of the 4 year period.

A. 1250
B. 1563
C. 2250
D. 2560
E. 2752

[/spoiler]


Useful to know your Powers of 4 \(5^4 = 625\) and \(4^4 = 256\)

"Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard of the preceding year"
This translate into:

\(Year 1 = X *\) \(\frac{5}{4}\)

\(Year 2 = X *\) \(\frac{5}{4}\) * \(\frac{5}{4}\)

\(Year 3 = X *\) \(\frac{5}{4}\) * \(\frac{5}{4}\) * \(\frac{5}{4}\)

\(Year 4\) or 6250 trees \(= X *\) \(\frac{5}{4}\) * \(\frac{5}{4}\) * \(\frac{5}{4}\) * \(\frac{5}{4}\)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------

\(6250 = X *\) \(\frac{5^4}{4^4}\)

\(625 * 10 = X *\) \(\frac{5^4}{4^4}\)

\(25^2 * 10 = X *\) \(\frac{5^4}{4^4}\)

\(5^4 * 10 = X *\) \(\frac{5^4}{4^4}\) (The \(5^4\) cancel out, leaving \(4^4\))

\(4^4 * 10 = X\)

\(256 * 10 = X\)

\(2560 = X\)

Answer is D
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 30 Jan 2019, 14:36
Hi,

I may not be 100% sure, but tell me if you agree: I've solved the problem using the compound rate formula for exercises about loans.

Final value = Initial value * (1+ i)^n, where "i" is the interest rate or growth in this case, and "n" number of years. Therefore:

6250= x*(1+1/4)^4 = x*(5/4)^4 --> x= 5^4*10/(5^4/4^4) =(5^4*10*4^4)/5^4=10*4^4=10*64*4=640*4=2560 --> Answer D
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 02 Jul 2019, 04:20
Bunuel wrote:
imhimanshu wrote:
Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard of the preceding year. If all of the trees thrived and there were 6250 trees in the orchard at the end of 4 year period, how many trees were in the orchard at the beginning of the 4 year period.

A. 1250
B. 1563
C. 2250
D. 2560
E. 2752

Can someone walk me through the logic behind this question. I am able to solve this by using options as well as by assuming the number of trees = x. However, had the question been, "If all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period". then it would have been difficult to solve.

Thanks


Say the number of trees at the beginning of the 4 year period was x, then:
At the end of the 1st year the number of trees would be \(x+\frac{1}{4}x=\frac{5}{4}*x\);
At the end of the 2nd year the number of trees would be \((\frac{5}{4})^2*x\);
At the end of the 3rd year the number of trees would be \((\frac{5}{4})^3*x\);
At the end of the 4th year the number of trees would be \((\frac{5}{4})^4*x\);
At the end of the \(n_{th}\) year the number of trees would be \((\frac{5}{4})^n*x\);

So, we have that \((\frac{5}{4})^4*x=6,250\) --> \(\frac{5^4}{4^4}*x=5^4*10\) --> \(x=4^4*10=2,560\).

Answer: D.

If the question were "if all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period", then we would have that: \((\frac{5}{4})^{15}*x=6,250\) --> \(x\neq{integer}\), so it would be a flawed question.

Hope it's clear.

Bunuel
In that time, the maker of the question will replace the figure 6250 with another one so that x has to be integer, right? :)
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 02 Jul 2019, 04:24
Asad wrote:
Bunuel wrote:
imhimanshu wrote:
Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard of the preceding year. If all of the trees thrived and there were 6250 trees in the orchard at the end of 4 year period, how many trees were in the orchard at the beginning of the 4 year period.

A. 1250
B. 1563
C. 2250
D. 2560
E. 2752

Can someone walk me through the logic behind this question. I am able to solve this by using options as well as by assuming the number of trees = x. However, had the question been, "If all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period". then it would have been difficult to solve.

Thanks


Say the number of trees at the beginning of the 4 year period was x, then:
At the end of the 1st year the number of trees would be \(x+\frac{1}{4}x=\frac{5}{4}*x\);
At the end of the 2nd year the number of trees would be \((\frac{5}{4})^2*x\);
At the end of the 3rd year the number of trees would be \((\frac{5}{4})^3*x\);
At the end of the 4th year the number of trees would be \((\frac{5}{4})^4*x\);
At the end of the \(n_{th}\) year the number of trees would be \((\frac{5}{4})^n*x\);

So, we have that \((\frac{5}{4})^4*x=6,250\) --> \(\frac{5^4}{4^4}*x=5^4*10\) --> \(x=4^4*10=2,560\).

Answer: D.

If the question were "if all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period", then we would have that: \((\frac{5}{4})^{15}*x=6,250\) --> \(x\neq{integer}\), so it would be a flawed question.

Hope it's clear.

Bunuel
In that time, the maker of the question will replace the figure 6250 with another one so that x has to be integer, right? :)
Thanks__


Since x represent the number of trees, then it must be an integers, so if you were to replace 6,250 with some other number, you should replace it so that x at the end turns out to be an integer.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 28 Jul 2019, 15:21
Bunuel wrote:
imhimanshu wrote:
Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard of the preceding year. If all of the trees thrived and there were 6250 trees in the orchard at the end of 4 year period, how many trees were in the orchard at the beginning of the 4 year period.

A. 1250
B. 1563
C. 2250
D. 2560
E. 2752

Can someone walk me through the logic behind this question. I am able to solve this by using options as well as by assuming the number of trees = x. However, had the question been, "If all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period". then it would have been difficult to solve.

Thanks


Say the number of trees at the beginning of the 4 year period was x, then:
At the end of the 1st year the number of trees would be \(x+\frac{1}{4}x=\frac{5}{4}*x\);
At the end of the 2nd year the number of trees would be \((\frac{5}{4})^2*x\);
At the end of the 3rd year the number of trees would be \((\frac{5}{4})^3*x\);
At the end of the 4th year the number of trees would be \((\frac{5}{4})^4*x\);
At the end of the \(n_{th}\) year the number of trees would be \((\frac{5}{4})^n*x\);

So, we have that \((\frac{5}{4})^4*x=6,250\) --> \(\frac{5^4}{4^4}*x=5^4*10\) --> \(x=4^4*10=2,560\).

Answer: D.

If the question were "if all of the trees thrived and there were 6250 trees in the orchard at the end of 15 year period, how many trees were in the orchard at the beginning of the 4 year period", then we would have that: \((\frac{5}{4})^{15}*x=6,250\) --> \(x\neq{integer}\), so it would be a flawed question.

Hope it's clear.


Hi.

I made the eqn of x+0.4 x+1.6x+6.4x=6250 but it did not work .Do you mind clarifying where the mistake is ? I am not also very clear on where we derived the (5/4)^n formula.

Thanks a lot in advance !!
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 23 Aug 2019, 08:30
Hi,
If we manually try to solve the question, it comes at the end 125x/64. This is equal to the given value i.e. 6250.
Steps:
1st year> x trees
2nd year> x+(1/4)x = (5/4)x
3rd year> (5/4)x + 1/4{(5/4)}x = (25/16)x
4th year> (25/16)x + 1/4{(25/16)}x = (125/64)x

Solving this, we get x/64 = 50. Thus, x = 3200. But the correct answer is 2560. Please let me know what went wrong.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 01 Sep 2019, 22:46
I solved it in less 45 seconds. The question says the growth happened by a factor of 1/4. The correct answer should be divisible by 4, otherwise the number of trees for the next year would not be an integer. Out of all the options only D is divisible by 4.
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New post 25 Nov 2019, 22:41
Hi, I am a still confused as to why we should consider the end of the first year. The question says that the number of trees is increasing every year. So if I take the first year say as one tree the second year will be 5/4. Can someone please explain why the first year itself is being considered as 5/4. What if the trees were planted at the year-end then they can be 5/4 in the end itself.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 25 Nov 2019, 22:51
Netijyata wrote:
Hi, I am a still confused as to why we should consider the end of the first year. The question says that the number of trees is increasing every year. So if I take the first year say as one tree the second year will be 5/4. Can someone please explain why the first year itself is being considered as 5/4. What if the trees were planted at the year-end then they can be 5/4 in the end itself.


Dear Netijyata ,
In Question itself, it has asked for number of trees at starting.
& trees increased by 1/4.
So if it is 1 in start, it will be 5/4 at the end of 1st year.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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New post 26 Nov 2019, 22:40
Netijyata wrote:
Hi, I am a still confused as to why we should consider the end of the first year. The question says that the number of trees is increasing every year. So if I take the first year say as one tree the second year will be 5/4. Can someone please explain why the first year itself is being considered as 5/4. What if the trees were planted at the year-end then they can be 5/4 in the end itself.


This is what the question says: "Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 ..."

The farmer increase the number of trees by 1/4 each year for 4 years. So he increases them by 1/4 in the first year, increases them by 1/4 in the second year, by 1/4 in the third year and by 1/4 in the fourth year. So say he increases them by 1/4 in 2016, 2017, 2018 and 2019.
To increase the number of trees by 1/4 in 2016, if there are 4 trees by the end of 2015, there will 5 trees by the end of 2016 because he increased them by 1/4 in 2016.
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Re: Each year for 4 years, a farmer increased the number of trees in a   [#permalink] 26 Nov 2019, 22:40

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