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## Farmers Orchard Problem

tagged by: Brent@GMATPrepNow

This topic has 4 expert replies and 2 member replies
NeilWatson Senior | Next Rank: 100 Posts
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#### Farmers Orchard Problem

Sat Mar 01, 2014 7:38 pm
Elapsed Time: 00:00
• Lap #[LAPCOUNT] ([LAPTIME])
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 previous year. If all of the trees thrived and there were 6,250 trees in the end of the 4 year period, how many trees did he start with in the first year?

While I understand the logic of this problem (each year the amount increased by 1.25) I am confused by the explanation. I broke the problem down as follows:
1st year - x
2nd year - 1.25x
3rd year - 1.25^2 x
4th year - 1.25^3 x

According to the textbook, my reasoning is wrong. Can someone explain?

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GMATGuruNY GMAT Instructor
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Sat Mar 01, 2014 7:53 pm
Quote:
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
We can PLUG IN THE ANSWERS, which represent the original number of trees.
Since the number of trees increases by 1/4 each year, the correct answer must be a multiple of 4.
The last 2 digits of a multiple of 4 must themselves form a multiple of 4.
Eliminate A (1250) and C (2250), since 50 is not a multiple of 4.
Eliminate B (1563), since 63 is not a multiple of 4.

After the 1st year, the number of trees = 2560 + (1/4)2560 = 3200.
After the 2nd year, the number of trees = 3200 + (1/4)3200 = 4000.
After the 3rd year, the number of trees = 4000 + (1/4)4000 = 5000.
After the 4th year, the number of trees = 5000 + (1/4)5000 = 6250.
Success!

Note that we had to try only ONE answer choice -- a very efficient way to solve the problem.

Quote:
While I understand the logic of this problem (each year the amount increased by 1.25) I am confused by the explanation. I broke the problem down as follows:
1st year - x
2nd year - 1.25x
3rd year - 1.25^2 x
4th year - 1.25^3 x
The number of trees increases by 1/4 EACH YEAR FOR 4 YEARS.
Thus, it increases by 1/4 a total of 4 times.
To increase a value by 1/4 is the same as multiplying the value by 5/4.
Thus, the original number of trees -- x -- is multiplied by 5/4 four times, yielding a final total of 6250 trees:
(5/4)(5/4)(5/4)(5/4)x = 6250
x = (6250*4*4*4*4)/(5*5*5*5) = 10*256 = 2560.

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Rich.C@EMPOWERgmat.com Elite Legendary Member
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Sat Mar 01, 2014 8:05 pm
Hi NeilWatson,

Mitch has properly explained a couple of different approaches to solving this problem (including TESTING the answers - which is a tactic that I'm a big fan of).

Your approach would have been correct if you had completed the task. The number of trees increases for 4 YEARS. This would translate into:

Start = X
1st yr = 1.25(X)
2nd yr = 1.25^2(X)
3rd yr = 1.25^3(X)
4th yr = 1.25^4(X)

The drawback to using decimals is that you'd have to do LOTS of work to calculate 1.25^4, whereas using fractions (or the answers) would have probably been faster.

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Abhishek009 Master | Next Rank: 500 Posts
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Mon Mar 03, 2014 9:14 am
NeilWatson 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 previous year. If all of the trees thrived and there were 6,250 trees in the end of the 4 year period, how many trees did he start with in the first year?
A = P ( 1 + r /100)^n

6250 = P ( 1 + 1/4)^4

6250 = P ( 5/4)^4

P = 6250 * (4/5)* (4/5)* (4/5)* (4/5)

P = 6250/625 * 256

P = 2560

So the Farmer started with 2560 trees answer is (c)

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Scott@TargetTestPrep GMAT Instructor
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Fri Jun 26, 2015 8:51 am
NeilWatson 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 previous year. If all of the trees thrived and there were 6,250 trees in the end of the 4 year period, how many trees did he start with in the first year?

While I understand the logic of this problem (each year the amount increased by 1.25) I am confused by the explanation. I broke the problem down as follows:
1st year - x
2nd year - 1.25x
3rd year - 1.25^2 x
4th year - 1.25^3 x

According to the textbook, my reasoning is wrong. Can someone explain?
Solution:

This problem is testing us on exponential growth. A good way to navigate this problem is to set up a growth chart. We are given that the number of trees increased by ¼, or 25 percent, each year. To obtain the number of trees this year, we multiply last year's number of trees by 125%, or 1.25, which can be fractionally expressed as 5/4.

Let’s let x be the number of trees at the beginning (of the first year) of the 4-year period.

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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Brent@GMATPrepNow GMAT Instructor
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Fri Jun 26, 2015 9:29 am
Quote:
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
Here's another approach:

First notice that, 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
(625/256)x = 6250
x = 6250(256/625)
x = 2560 = D

Cheers,
Brent

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Mon Jun 29, 2015 3:38 pm
Neil, you've got the right idea: the hiccup is that your numbers are the numbers for the BEGINNING of each year. Since we have the number AFTER four years, you would need to add a fifth year, since the number at the beginning of the fifth year = the number after four years.

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