Problem Solving

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

q,Q are outputs for diff. years q=r^2 and Q=R^2
r,R are diff. sides for two areas inside the bigger square (garden)
Q-q=211, find Q?

(R^2-r^2)=211

It's clear that the difference between two squares may mean that R and r may be assigned any values. The output Q will depend on the value of R, hence not clear.

e

bharti.2010 wrote: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

Imagine 2 concentric squares,

Given Area(bigger) - Area(smaller) = 211

We are asked Area(bigger) = ?

Looking at the answer choices, we know side(bigger) > 100

A is 106^2

now since units digit is 1, Area(smaller) should end in 5, consider 105

106^2 - 105^2 = 211 - Success

A IMO

bharti.2010 wrote: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

Let T² = the area of the garden this year.
Let L² = the area of the garden last year.

Since each cabbage takes up 1 square foot, the area of the garden = the number of cabbages produced.
Since the number of cabbages produced increases by 211:
T² - L² = 211.
(T+L)(T-L) = 211.

211 is a prime number. It's only factors are 211 and 1.
Since the area = the number of cabbages, T and L -- the dimensions of the garden this year and last year -- are integers.
Thus, T+L = 211 and T-L = 1.

Adding the two equations:
(T+L) + (T-L) = 211+1.
2T = 212.
T = 106.

Thus, the area of the garden this year = 106² = 11236.

The correct answer is A.

Forgive me if this turns out to be a silly question.

The question says the lady grows cabbages in a SQUARE garden and each cabbage takes 1 SQ FEET of area in the garden. The first question that came to my mind after reading this is, "Is the garden completely packed with end-to-end with cabbages?". If the answer is yes, then the number of cabbages = X*X (X=length of square). If the answer is no, for example if only 1/4th of the garden was filled with cabbages, we'd end up with choice E.

Is it ok to assume the garden is packed?

bharti.2010 wrote:The shape of the area used for growing the cabbages has remained a square in both these years.

we are having squares for both years and not other shape quadrilaterals to answer your question about packing

pemdas wrote:

bharti.2010 wrote:The shape of the area used for growing the cabbages has remained a square in both these years.

we are having squares for both years and not other shape quadrilaterals to answer your question about packing

I think, I didn't get my question through clearly last time. Let me try to do it with more clarity.

The question says the shape remained a square.

Case 1: Not fully packed.
=========================
Let us assume she grew 10 cabbages last year. This year the number of cabbages is 10 + 211 = 221. If the garden (SQUARE) originally had a side of say 100feet, its area would be have been 10,000 sqft. Enough to grow 10 cabbages last year AND enough to grow 221 cabbages this year. So in this case the garden was/is a square in both years, but it is the same square