Official Guide
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
OA D.
Each year for 4 years, a farmer increased the number of
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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
Answer: D
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Brent
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We can PLUG IN THE ANSWERS, which represent the original number of trees.AAPL wrote:Official Guide
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 test only ONE answer choice -- a very efficient way to solve the problem.
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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.AAPL wrote:Official Guide
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
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.
Answer: D
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