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

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KarishmaB

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15 Sep 2023, 00:35

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Engineer1

Bunuel

imhimanshu

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.

Should not this be : (5/4)^3 * x trees at the beginning of year 4? x is the number of trees at the beginning of year 1. I am confused how the correct answer is value of x.

Consider 1st Jan 2019 as the beginning of the 4 year period. Say there were x trees.
In 2019, the farmer planted more trees such that by the end of 2019, there were 5x/4 trees (1/4 extra).

He starts 2020 with 5x/4 trees.
In 2020, he plants more trees such that at the end of 2020, he has (5/4)*(5x/4) trees.

HE does the same in 2021 and 2022 too.
By the end end of four years i.e. 31st dec 2022, he has (5/4)^4 * x trees. This is equal to 6250.
x is the number of trees at the beginning of the 4 year period.

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Here is an another way to solve this ques.
as we know that increase of 25% and decrease of 20% is equals to 0% of change,
so simply decrease
6250 by 20% four times ,
6250 - 20% = 5000
5000 - 20% = 4000
4000 - 20% = 3200
3200 - 20% = 2560
2560 is answer !
thats the answer D .

hope this helps

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pv=fv/(1+i)^n = 6250/(1+.25)^4= 2560

imhimanshu

Each year for 4 years, a farmer increased the number of trees in a certain orchard by one fourth of the number of trees in the orchard the preceding year. If all 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

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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

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thorinoakenshield

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

Hence, correct answer = D

Great approximation, but how did you assume to be divisible by 4 as start number can be an odd number

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Many ways to solve.

I personally like the idea of increase/decrease ratios.

5/4 times something is equal to 4/5 times the final value.

Hence 6250*(4/5)^3 = 2560.

Answer: Option D

imhimanshu

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