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by confuse mind » Sat Jul 21, 2012 7:22 pm
27. A lady grows cabbages in her garden that is in the shape of a square. Each cabbage takes 1 square feet of area in her garden. This year, she has increased her output by 211 cabbages as compared to last year. The shape of the area used for growing the cabbages has remained a square in both these years. How many cabbages did she produce this year?
A. 11236
B. 11025
C. 14400
D. 12696
E. Cannot be determined


IMO - E
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by theCEO » Sat Jul 21, 2012 7:48 pm
ans = a
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by Anurag@Gurome » Sat Jul 21, 2012 8:54 pm
confuse mind wrote:27. A lady grows cabbages in her garden that is in the shape of a square. Each cabbage takes 1 square feet of area in her garden. This year, she has increased her output by 211 cabbages as compared to last year. The shape of the area used for growing the cabbages has remained a square in both these years. How many cabbages did she produce this year?
Say, the area of the place used for growing cabbage was S² square feet last year.
Hence, last year she has grown S²/1 = S² cabbages

Hence, this year she has grown (S² + 211) cabbages
According to the question, (S² + 211) must be a square.

Let us assume (S² + 211) = T²
Hence, (T² - S²) = (T - S)(T + S) = 211
As 211 is a prime number, only way it can expressed as a product of two positive integers is 1*211
Hence, (T - S) = 1 and (T + S) = 211
Hence, T = (211 + 1)/2 = 106
Hence, T² = 106² = An integer that ends with 36 --> Option A

The correct answer is A.
Last edited by Anurag@Gurome on Sat Jul 21, 2012 9:17 pm, edited 1 time in total.
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by theCEO » Sat Jul 21, 2012 9:09 pm
confuse mind wrote:27. A lady grows cabbages in her garden that is in the shape of a square. Each cabbage takes 1 square feet of area in her garden. This year, she has increased her output by 211 cabbages as compared to last year. The shape of the area used for growing the cabbages has remained a square in both these years. How many cabbages did she produce this year?
A. 11236
B. 11025
C. 14400
D. 12696
E. Cannot be determined


IMO - E
Whats wrong with this approach?

number of this year's cabbage = y
number of last year's cabbage = x

y^2 - x^2 = 211
(y-x)(y+x) = 211 --- factors of 211 are 1 and 211
y-x = 1 and y+x = 211
y = 106 and x = 105

Amount she produced this year = y^2 = 106^2 = 11236 = - a
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