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

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.

STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can test the answer choices. In fact, when I scan the answer choices, I see that 3 of them are automatically disqualified.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we can also use algebra to solve the question.
Since I already know that I need to test just one answer choice, I'll go that route.

Let's first examine how I know that 3 answer choices are disqualified:
We’re told the number of trees increases by ¼ each year. Since answer choices A, B, and C aren’t divisible by 4, we can eliminate them immediately.
For example, check out what happens when we test choice C: 2250
If there were 2250 trees at the beginning, then the number of trees after 1 year = 2250 + (1/4 of 2250) = 2812.5, which makes no sense, since we can’t have half a tree.
Useful property: An integer is divisible by 4 if and only if the number created by its last two digits is divisible by 4.
Since 50 and 63 (the two-digit numbers created by the last two digits of A, B and C) aren’t divisible by 4, we can eliminate A, B and C.

This leaves us with D and E. From here, we’ll just test one option. If it works, we’re done. If it doesn’t work, the other option must be correct.
I’ll test choice D (2560), since calculating 1/4 of 2560 looks easier than calculating 1/4 of 2752:
- Number of trees at beginning = 2560
- Number of trees after 1 year = 2560 + (1/4 of 2560) = 2560 + 640 = 3200
Tip: We can calculate ¼ of k by dividing k by 2 twice
- Number of trees after 2 years = 3200 + (1/4 of 3200) = 3200 + 800 = 4000
- Number of trees after 3 years = 4000 + (1/4 of 4000) = 4000 + 1000 = 5000
- Number of trees after 4 years = 5000 + (1/4 of 5000) = 5000 + 1250 = 6250 Perfect!
Answer: E


Alternate approach
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

Answer: D

Cheers,
Brent

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.

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.

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.

Thanks! But how do you know when to solve difficult fractions, rather than to use a shortcut? I guessed on the question because I figured I had missed a strategy to avoid solving (1.25)^4 or (5/4)^4

adkor95

You don't have to solve (5/4)^4 here.

Note that you have
(5/4)^4 * x = 6250

sNow recognise that 5^4 = 625 (you should know the first few exponents)

x/4^4 = 10

x = 4^4 * 10 = 2^8 * 10 = 256 * 10

Again note that 2^8 = 256 is something you should know.

You should know powers till 2^10, 3^6, 5^4, squares all numbers till 20 and cubes of all numbers till 10.

Can someone please explain to me why this answer is wrong:

1st year = x
2nd year = 5x/4
3rd year = 25x/16
4th year = 125x/64

125x/64 = 6250

Why am I not starting and finishing year 1 with X and then beginning the increase at year 2? What language am I missing that is making me think the process to solve is the one immediately above?

VeritasKarishma

"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."

For 4 years, every year the farmer increased number of trees by 1/4th.
So in year 1, he increased the number of trees by 1/4th. In year 2, he did the same again and so on in years 3 and 4 too.
You have missed the increase in year 1. Before year 1 if number of trees was x, in year 1, it increased by 1/4 to become 5x/4.

Video solution from Quant Reasoning:
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Answer: Option D

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KarishmaB
I have two questions:
-When you say "So if initially the number of trees was x, in the first year he made them (5/4)x" to clarify... at time zero the number of trees was x, correct?
-At first, I was solving for the number of trees for year 1. I see how it can be easy to mistake year 1 for time zero when the question asks for "at the beginning of the 4-year period"... are there other Official Guide problems that you recommend looking at in this nature for more practice?

Thank you!

Yes, you start with number x. Think of it as the principal you have in compound interest problems.
We begin with a certain value and then apply successive percentage changes to it over time periods.

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.