# GMAT Quant Questions | Geometry #13

#### Area of a square | Number properties - difference between square of numbers

This GMAT practice question is a quant problem solving question from Solid Geometry (Mensuration). Concept: computing circumference between area of two squares. Includes elementary number properties concept relating to difference between square of two numbers. A GMAT 600 to 650 level, medium difficulty quantitative reasoning question.

Question 13: 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

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### Explanatory Answer | GMAT Geometry

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 used for cultivation this year = X2 sq.ft.

Let the side of the square area used for growing cabbages last year be Y ft.
Therefore, the area of the ground used for cultivation last year = Y2 sq.ft.

As the number of cabbages grown has increased by 211, the area would have increased by 211 sq ft because 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 and 1.
Therefore, 211 can be expressed as product of 2 numbers in only way = 211 × 1
i.e., (X + Y)(X - Y) = 211 × 1

So, (X + Y) should be 211 and (X - Y) should be 1.
Solving the two equations we get X = 106 and Y = 105.

Therefore, number of cabbages produced this year = X2 = 1062 = 11236.

#### Alternative Approach: Use answer choices

The area in both the years are squares of two numbers.
That rules out choice D. 12696 is not the square of any number.

>

Check Choice A: If this year's produce is 11236, last year's produce would have been 11236 - 211 = 11025
11025 is the square of 105.

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