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Posts: 7971
Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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Sunil01 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 the preceding year. If all of the
trees thrived and there were 6,250 trees in the
orchard at the end of the 4-year period, how many
trees were in the orchard at the beginning of the
4-year period?
(A) 1,250
(B) 1,563
(C) 2,250
(D) 2,560
(E) 2,752

I am going wrong somewhere.
Could someone help me, is something wrong with my method to solve this question.
Not getting the correct answer.

My Approach:
Let number of trees at the beginning of the year be x.
2nd year no. of trees = x+x/4
3rd year no. of trees = (x+x/4) +1/4(x+x/4) 1.e. x + x/4 + x/4 + x/16 = x +x/2 + x/16
4th year no of trees = x + x/2 + x/16 + 1/4(x + x/2 + x/16)
6250 = x + x/2 + x/16 + x/4 + x/8 + x/64
6250 = (64x + 32x + 4x + 16x + 8x + x)/64
6250 = 125x/64
x = 3200
...

Hi,
you have gone wrong in that you have calculated only for three years..
each year it increases by 1/4, so calculate for the end of the year,...
in this case the first year end would be x+x/4, which you have shown for second year..
By your approach you should check for % year begining to get the answer..

the answer you have got is at teh end of one year..
if it was y in begining, it will become5y/4..
so here 5y/4=3200...
or y=3200*4/5=4*640=2560..

Hope it helps you..
you have missed out on one 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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1
let t=# of original trees
(t)(5/4)^4=6250
t=6250/(5/4)^4
t=2560 trees
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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isn't this a GP and the 6250 is sum of the GP?

Can't we use S=$$\frac{a(1-r^n)}{1-r}$$

where r=1/4
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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
bimalr9 wrote:
isn't this a GP and the 6250 is sum of the GP?

Can't we use S=$$\frac{a(1-r^n)}{1-r}$$

where r=1/4

Yes, there is a GP but r = 5/4. Also, 6250 is the last term, not sum of the GP.

a*(5/4), a*(5/4)*(5/4), ...

Every previous term is multiplied by 5/4 to get the next term. The 4th term is 6250.

$$6250 = a * (5/4)^4$$

$$a = 10 * 4^4$$

$$a = 2560$$

a ( r^n - 1)/(r - 1) = a ((5/4)^4 - 1)/(5/4 - 1)

6250 = a * (5^4 - 4^4)/4^3

2 * 5^4 * 4^3 = a * 369
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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converting 6250 into prime factorization form (625*10 = 5*5*5*5*5*2) can save some time in solving this question, as many 5's will cancel out nicely
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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What's wrong if we add up 1/4 for each year. Please help me understand this in detail.

First year begin: x------------- ending 5x/4
2nd year begin: 5x/4----------ending 3x/2(5x/4+x/4)
3year begin: 3x/2--------------ending 7x/4
4 year begin : 7x/4-----------ending 2x
I dont get the answer and iam not understanding what is wrong with this approch ... please help
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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
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

This problem is testing us on exponential growth. We are given that the number of trees increased by ¼ each year. To determine the number of trees in a certain year, we multiply the number of trees from the previous year by 1.25, or 5/4. Let’s let x equal the number of trees at the beginning (of the first year) of the 4-year period.

Start of year 1 = x

End of year 1 = x(5/4)

End of year 2 = x(5/4)(5/4)

End of year 3 = x(5/4)(5/4)(5/4)

End of year 4 = x(5/4)(5/4)(5/4)(5/4) = (625/256)x

We are given that there were 6,250 trees at the end of year 4, so we can set up the following equation:

(625/256)x = 6,250

625x = 6,250(256)

x = 10(256) = 2,560

Thus, there were 2,560 trees at the beginning of the 4-year period.

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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
muthappashivani wrote:
What's wrong if we add up 1/4 for each year. Please help me understand this in detail.

First year begin: x------------- ending 5x/4
2nd year begin: 5x/4----------ending 3x/2(5x/4+x/4)
3year begin: 3x/2--------------ending 7x/4
4 year begin : 7x/4-----------ending 2x
I dont get the answer and iam not understanding what is wrong with this approch ... please help

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 of the number of trees in the orchard of the preceding year"

He increases the trees by 1/4 of the number in the preceding year. So in 3rd year, he increases the trees by a fourth of the number of trees in the 2nd year, not one fourth of those in the 1st year.
Hence, 2nd year end/3rd year beg, the number of trees will be 5x/4 + (1/4)*(5x/4)
and so on...
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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
All the above responses have been around trying to solve this algebraically. But we can do plug and play with the answer choices. We know what the end result trees are and we know what the annual increase. Given that the annual increase is 1/4 --> we know that the answer choices must be a factor of 4. So (B) is eliminated automatically since its an odd number.

The rest of the answers choices with the exception of D are not factors of 4. so we can eliminate that way.

Now if we had two answers that were divisible by 4, then we can easily do another step and recalculate.

Personally I find this much easier then having to the algebra formula but both can be done under 2 min easily.
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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
Another approach would be test the answers:
Normally, starting with B or D, I have chosen D as it is a cleaner number.
Assuming the farmer had at the beginning 2,560 trees he will have one year later (2,560 *1,25) or calculate 1/4 of it and add it.
--> After 1 year: 3200 trees
--> 2. Year: 3200*1,25= 4000 trees
--> 3. Year: 4000*1,25= 5000 trees
--> 4. Year: 5000*1,25= 6250 trees - Bingo!
As we arrived to the desired value Answer D is correct! The hint was that after the first year the numbers cleaned up very nicely and the calculations involved less work.
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IMHO I think we can skip the calculation part in this question.

my reasoning behind solving this question was : since the number of trees are increasing in a fashion of 1/4 .
so by end of 4th year we will have a number which is of form X/64....( we are adding 1/4 three times in the initial number) Total number of trees in the orchard after 4 years = 6250, factor of which does not have 64 : so we must have a initial number which will have a factor of 64 or we can say the number must be divisible by 64.

Let's scan the numbers : let's start finding the number which is divisible by 8

A. 1250
B. 1563
C. 2250
D. 2560 --> divisible by 8
E. 2752 --> divisible by 8

so we have only two options to check for further divisibility: as we further divide the number by another 8 we will see only D fits our requirement
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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I arrived to the answer through an easier way (at least in my opinion), but one thing almost got me troubled in the end.

1. I started considering 6250 it was the number of trees in 3rd year + a fifth of this number.
2. So, I divided 6250 by 5 and multiplied the result by 4, to get the number of trees in the 3rd year = 5000.
3. Then, I did the same process with 5000 and arrived to the number of trees on 2nd year = 4000.
4. Did it once again and arrived to the number of trees in the 1st year, 3200.

I stopped here and fortunatelly there was no choice with that number, otherwise I would have gone for it. The tricky thing is that the author doesn't ask you the number of trees in the first year, but how many trees there were prior to that.

So, I picked the 3200 and did it all once again: (3200/5)*4 and finally got the letter (D) 2560. I think this approach is more straight to the point and doesn't require almost any algebra.
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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My solution was quick and simple. Each of the answer choices would need to be multiplied by 125% the first year and than that answer multiplied by 125% for the 2nd year, and so on until the 4th year equaled 6250.

I looked at each of the answer choices, and given that I wanted to find what 25% of each answer choice was, and then I would add that to one of the answer choices (which would give me 25% +100% = 125%). However, since 25% is 1/4 of 100% I looked at which answer choice could be divisible by 4. The only one was answer D.

Choice A:
100% of 1250 = 1250
25% of 1250 = 312.50 (Farmer isn't going to plant half a tree in the 2nd year so this answer is wrong. Thus all decimal answers are wrong.)

Choice B:
25% of 1563 = 390.75 (Farmer isn't going to plant 3/4ths of a tree in 2nd year so this is wrong.)

Choice C:
etc. etc...

To save time, I then divided all answer choices by 4. (All you need to do is look at the last 2 digits of each big number, if the last two digits are divisible by 4, then the overall big number is divisible by 4.) The only answer choice that was divisible by 4 was D. (The last two digits of answer choice D. are 60, which is divisible by 4).
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Re: Each year for 4 years, a farmer increased the number of trees in a  [#permalink]

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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 ratio of FV and PV is 5/4. In 4 years, the ratio will be (5/4)^4=625/256. The FV is given as 6250; the PV will be 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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VeritasPrepKarishma wrote:
aeglorre 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

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

Correct me if I am wrong, but it seems to me that there is a flaw in this GMAT problem. If you read the statement:
"by 1/4 of the number of trees in the orchard of the preceding year"
So you cannot increase the number of trees the first year as you do not know the number of tree of the year preceding that first year.
And the only way to increase the number of trees of the preceding year is to be in the second year, and then you know what is the number of trees of the preceding year.

So either you are assuming that the number of trees before that first year is = n, but this is not provided by the statement, or you are taking into account the fourth increase that happened actually at the beginning of the fifth year, and this is wrong as the number 6250 is the number of trees at the end of the 4-year period.

(And you cannot be between the first and the second year, either you are in the first year or in the second year)
Attachments GMAT flaw.png [ 125.63 KiB | Viewed 6021 times ]

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

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Let there be 256x(purposefully chosen, 256 as it as $$4^4$$) tree at the beginning of 4 years.

First year: (5/4) * 256x = 320x
second year: (5/4)* 320x = 400x
thrid year: (5/4)*400x = 500x
fourth year: (5/4) * 500x = 625x = 6250 => x = 10
So first year, number of trees 256x = 2560
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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?

I agree with this interpretation of the question, mostly because this is how I read it as well.
So, when I read : "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" I assume Year 1 starts with x trees. At the start of year 2 the farmer plants 1/4 x more than year 1, so the trees at the start of the 2nd year is 5x/4.
Then,
Year 0 (before the question): (4/5)x
Year 1: x
Year 2: (5/4)x
Year 3: (5/4)^2x
Year 4: (5/4)^3x
From the question I'm not sure if the increase happens during the year or at the start of it...
Can someone help me understand why this interpretation is entirely incorrect?

Edit: Never mind, I got it...Took me a while :/
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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

Important: if the number of trees increases by 1/4, then the new number is 5/4 times the original number.

Let x = the # of trees in the orchard at the beginning of the 4 year period.
(5/4)x = # of trees after 1 year
(5/4)(5/4)x = # of trees after 2 years
(5/4)(5/4)(5/4)x = # of trees after 3 years
(5/4)(5/4)(5/4)(5/4)x = # of trees after 4 years

We're told that, after 4 years, there are 6250 trees, so we now know that:
(5/4)(5/4)(5/4)(5/4)x = 6250
Simplify: (625/256)x = 6250
Multiply both sides by 256/625 to get: x = 6250(256/625)
Evaluate: x = 2560

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

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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.

Bunuel - No where it is mentioned that tree are increased towards end of the year. And questions says tree increased then it must be towards start. I have correct answer in hand but not happy with wording of question , I know it is OG to blame. Ideally tree count is x, 5x/4, 25x/16 and in 4th year 125x/64. By end of year 4 it should be 125x/64. If we enter into 5th year then we can consider 625x/64*4.
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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,

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? Re: Each year for 4 years, a farmer increased the number of trees in a   [#permalink] 04 Nov 2018, 02:07

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