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Mensuration
December 11, 2003
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?
| (1) | 11236 |
| (2) | 11025 |
| (3) | 14400 |
| (4) | 12696 |
| (5) | Cannot be determined |
Correct Answer Choice (1)
Solution:
The shape of the area used for growing cabbages has remained a square in both the years.
Let the side of the square area used for growing cabbages this year be X ft.
Therefore, the area of the ground this year = X2 sq.ft.
And let the side of the square area used for growing cabbages last year be Y ft.
Therefore, the area of the ground used last year = Y2 sq.ft.
As the number of cabbages grown has increased by 211, the area would have increased by 211 sq ft as each cabbage takes 1 sq ft space.
Hence X2 - Y2 = 211 => (X + Y)(X – Y) = 211.
211 is a prime number and hence it will have only two factors. i.e., 211 = 211*1.
This can be represented as (106 + 105)*(106-105).
(X + Y)(X – Y) = (106 + 105)(106 – 105).
From this we can deduce that X = 106 and Y = 105.
Therefore, number of cabbages produced this year = X2 = 1062 = 11236.
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