imhimanshu 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 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
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 approachImportant: 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 =
6250Simplify: (625/256)x = 6250
Multiply both sides by 256/625 to get: x = 6250(256/625)
Evaluate: x = 2560
Answer: DCheers,
Brent
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Brent Hanneson – Creator of gmatprepnow.com
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