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?
Farmers Orchard Problem
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- NeilWatson
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We can PLUG IN THE ANSWERS, which represent the original number of trees.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
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.
Answer choice D: 2560
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!
The correct answer is D.
Note that we had to try only ONE answer choice -- a very efficient way to solve the problem.
The number of trees increases by 1/4 EACH YEAR FOR 4 YEARS.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
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.
The correct answer is D.
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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.
GMAT assassins aren't born, they're made,
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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.
GMAT assassins aren't born, they're made,
Rich
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A = P ( 1 + r /100)^nNeilWatson 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?
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)
Abhishek
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Solution: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?
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.
Answer:D
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Here's another approach: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
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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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.