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Manager  Joined: 07 Sep 2010
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Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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Question Stats: 72% (02:37) correct 28% (03:05) wrong based on 2723 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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37
40
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 $$(\frac{5}{4})^4*x=6,250$$ --> $$\frac{5^4}{4^4}*x=5^4*10$$ --> $$x=4^4*10=2,560$$.

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

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20
11
it can easily solved by compound interest formula
R = 25%
P = original number of trees
N = no of times of compounding within a year = 1
T = total number of years = 4

P+I at the end of compounding years = 6250
hence,

6250 = (1 + 0.25/1)^(4*1) * P
5^4*10 = 1.25^4 * P
(5/1/25)^4 *10 = P
4^4 * 10 = P
2560 = P

Another way to think is to pick ans choice which is exactly divisible by 4. A, B and C are not. Only D and E are.
Pick D and increase it by 25% for 4 times, you should get 6250.
##### General Discussion
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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13
9
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

The number of trees increases by 1/4 i.e. 25% every year. It is just a matter of thinking in terms of successive percentage changes e.g. population increase. Here, we are talking about the increase of tree population.

If x increases by 25%, how we denote it? (5/4)*x
If next year, this new number increases by 25% again, how do we denote it? (5/4)*(5/4)*x
and so on...

For more on this, check: http://www.veritasprep.com/blog/2011/02 ... e-changes/

So if we are taking into account 4 years, we simply get (5/4)^4 * x = 6250

As for your next question, the numbers given would be such that the calculation will not be tough.

Say, you have 8 years and 100% increase every year (population doubles every year). The final population will be divisible by 2^8 i.e. 256.
Something like 2^8 * x = 2560

and if you meant what you wrote (though I considered that the 4 was a typo because of the language of the question) "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", note that you still need to work with

(5/4)^4 * x = 6250
since you need the number of trees 4 yrs back only. The only thing is that the answer (2560) needs to be divisible by $$5^{11}$$ which it isn't so there is a problem in this question. If it were an actual question, the answer would be divisible by $$5^{11}$$ but you wouldn't really need to bother about it.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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5
2
Trees increase 1/4th every year, which means 100 trees become 125 after 1 yr & so on
So, 125/100 = 5/4 is the resultant (after adding 1/4 as the growth) for 1 year
So, for 4 yrs is 5^4/(4^4)
From the condition given, inital trees were = 6250 x 4^4 / 5^4 = 2560
Manager  Joined: 12 Jan 2013
Posts: 142
Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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

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.

Isn't the question quite ambiguous, though? I mean the first scentence could be interpreted as "for the first year we have (4/4)x and for the second year (5/4)x and for the third..." etc.. With that reasoning one would have (5/4)^3 * x + x and then your approach doesnt work.

Obviously, I understand that this was a flaw in my reasoning but I cannot understand how they - with that wording - will assume that we totally understand that at the end of year one he has (5/4)x..

Is there a straightforward "word translation" way in knowing how to interpret wordings like this?
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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aeglorre 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 $$(\frac{5}{4})^4*x=6,250$$ --> $$\frac{5^4}{4^4}*x=5^4*10$$ --> $$x=4^4*10=2,560$$.

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.

Isn't the question quite ambiguous, though? I mean the first scentence could be interpreted as "for the first year we have (4/4)x and for the second year (5/4)x and for the third..." etc.. With that reasoning one would have (5/4)^3 * x + x and then your approach doesnt work.

Obviously, I understand that this was a flaw in my reasoning but I cannot understand how they - with that wording - will assume that we totally understand that at the end of year one he has (5/4)x..

Is there a straightforward "word translation" way in knowing how to interpret wordings like this?

Actually, it is not ambiguous. Read the statement:
Each year a farmer increased the number of trees by 1/4. He did this for 4 years. (In GMAT Verbal and Quant are integrated. You need Verbal skills (slash and burn) in Quant and Quant skills (Data Interpretation) in Verbal.

So in the first year, he increased it by 1/4
The next year, he again increased it by 1/4 (of preceding year)
Next year, again the same.
Next year, again the same.
So he did it for a total of 4 years.

So if initially the number of trees was x, in the first year he made them (5/4)x
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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Hi Karishma,

I am not able to solve it by below method, where am I wrong:

6250*(3/4)*(3/4)*(3/4)

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

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email2vm wrote:
Hi Karishma,

I am not able to solve it by below method, where am I wrong:

6250*(3/4)*(3/4)*(3/4)

Regards,
Ravi

If I increase A by 25% and get B, I will not get A if I reduce B by 25%. The bases are different in the two cases.

e.g. A = 80
25% of A is 20 so I increase A by 25% to get B = 100

Now if I decrease B by 25%, I will decrease B by 25 (25% of 100). This will give me 75 which is not the same as A (which is 80).

In the first step, I found 25% of 80 and added that. In the second step, I found 25% of 100 and subtracted that. These two numbers are different.

You will get the correct answer if you do 6250*(4/5)*(4/5)*(4/5)*(4/5)
4/5 is the inverse of 5/4 (which is 25% increase in x).
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Karishma
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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2
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You can use the formula A=p(1 + r/100)^n with r = 25%

You will get P as 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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Is it true that the number of trees in the beginning has to be a multiple of 4 (as additions are always 1/4 th of previous and will not be non-integers)? In that case I'm left with D or E.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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deeuk wrote:
Is it true that the number of trees in the beginning has to be a multiple of 4 (as additions are always 1/4 th of previous and will not be non-integers)? In that case I'm left with D or E.

Yes, that's correct. But you might still need to work with two options. On the other hand, working with 6250 will definitely yield the answer.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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2
1
Trees increase by 1/4 the number of trees in preceding year. Hence, correct answer must be divisible by 4. Based on divisibility rules, if last 2 digits are divisible by 4 then the number is divisible by 4. Thus, we can eliminate A, B, C. The answer has to be D or E.

Again, trees increase by 1/4 the number of trees in preceding year. Hence, the number of trees increase by 5/4 times the number of trees the preceding year.

If x = initial number of trees = Answer D or E = 2560 or 2752
Year 1 = 5/4x
Year 2 = (5/4)(5/4)x
Year 3 = (5/4)(5/4)(5/4)x
Year 4 = (5/4)(5/4)(5/4)(5/4)x

Only for Answer D: (5/4)(5/4)(5/4)(5/4) 2560 = 6250

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

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@VeritasPrepKarishma
VeritasPrepKarishma 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

The number of trees increases by 1/4 i.e. 25% every year. It is just a matter of thinking in terms of successive percentage changes e.g. population increase. Here, we are talking about the increase of tree population.

If x increases by 25%, how we denote it? (5/4)*x
If next year, this new number increases by 25% again, how do we denote it? (5/4)*(5/4)*x
and so on...

For more on this, check: http://www.veritasprep.com/blog/2011/02 ... e-changes/

So if we are taking into account 4 years, we simply get (5/4)^4 * x = 6250

As for your next question, the numbers given would be such that the calculation will not be tough.

Say, you have 8 years and 100% increase every year (population doubles every year). The final population will be divisible by 2^8 i.e. 256.
Something like 2^8 * x = 2560

and if you meant what you wrote (though I considered that the 4 was a typo because of the language of the question) "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", note that you still need to work with

(5/4)^4 * x = 6250
since you need the number of trees 4 yrs back only. The only thing is that the answer (2560) needs to be divisible by $$5^{11}$$ which it isn't so there is a problem in this question. If it were an actual question, the answer would be divisible by $$5^{11}$$ but you wouldn't really need to bother about it.

Karisma,

However, it still isn't clear to me why we are keeping the same base for the increase during the 4-year period time. To me, this seems to be an example of successive % in which the base is shifting every year. Indeed every year we have an increase of 25% more on the increase of the previous year.

Why is it incorrect to compute the increase of the successive year on the increased base of the previous year?

Thank you a lot for your help!

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

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elisabettaportioli wrote:
However, it still isn't clear to me why we are keeping the same base for the increase during the 4-year period time. To me, this seems to be an example of successive % in which the base is shifting every year. Indeed every year we have an increase of 25% more on the increase of the previous year.

Why is it incorrect to compute the increase of the successive year on the increased base of the previous year?

Thank you a lot for your help!

Regards,
Eli.

The base for each year changes.

Say, you have 100 trees and you increase them by 20% each year.

At the end of the first year, you will have 100 + (20/100)*100 = 100*(1 + 20/100) = 100*120/100 = 100*(6/5) = 120 trees.

So if you have to increase a number by 20%, you just need to multiply it by 6/5 every time no matter what the number is.

By the same logic, next year, you will have 120 * (6/5) = 144 trees when you increase them by 20%.

You can write 120 as 100 * (6/5) and then multiply it by another 6/5 to increase 120 by 20%.

This is the same as 100 * (6/5) * (6/5) = 144 trees.

So every time you multiply it by 6/5, you increase the base. The next time you multiply it by 6/5, you are increasing the new base by 20%. This is the concept of successive percentage changes and I have discussed it in the link I mentioned in my post above.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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elisabettaportioli wrote:

Why is it incorrect to compute the increase of the successive year on the increased base of the previous year?

Thank you a lot for your help!

Regards,
Eli.

hi,
you are correct that the increase is on the preceeding year and that is exactly what is happening when you are multiplying x by 5/4..
first year say it was x.. 2nd year it becomes 5/4*x.. now for next year when we write the value as 5/4*5/4*x, the increase is on 5/4*x and not x..
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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1
1/4 = 0.25 = 25%
using the formula of compound interest calculation
p(1+r)^n= A

p = principal amount (to be identified), r = rate (0.25), n = numbers years (4), A= amount at the end of term(6250 given)

p(1+0.25)^4=6250
p(1.25)^4= 25*25*10 = 5^2*5^2*10=5^4*10
p(125/100)^4 = 5^4 x 10
p(5/4)^4 = 5^4 x 10
p(5^4)/4^4 = 5^4 x 10
p= (5^4 x 10 x 4^4)/5^4
p= 10 x 4^4
p= 2560
Ans 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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we have here 4 (4 years) successive increases of 25% or *1,25 --> 1,25^4 * X = 6250, X = 2560

See MGMAT (Percents) for detailed explanation of such question types.....
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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BrainLab wrote:
we have here 4 (4 years) successive increases of 25% or *1,25 --> 1,25^4 * X = 6250, X = 2560

See MGMAT (Percents) for detailed explanation of such question types.....

Dear BrainLab

Perfect logic but for easier calculation, you may want to work with ratio here (1/4 increase per annum) instead of percentages (25% increase per annum). Both convey the same thing but the equation

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

will take lesser time to solve (especially if you know that $$5^4 = 625$$) than $$(1.25)^4*X = 6250$$

Hope this was useful!

Japinder
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