Area & Perimeter Combination Problem

Math Puzzle: Area of a soccer field is 12 sq.m. The perimeter of the soccer field is 14 m. What are the length & width of the soccer field if they are all whole numbers?

Solving:

Let us arrange the clues in a logical order:

• Area = 12 sq.m
• Perimeter = 14 m
• Length & width are Whole Numbers (like 1,2,3,4 etc & not fractions like 1.5, 2.26 etc)

Let us assume that the length of the soccer field is L meters & width of the soccer field is B meters.

So, 

1. Area of the soccer field = L*B sq.m = 12 sq.m
2. Perimeter = L+B+L+B = 2*(L+B) m = 14 m, so L+B = 14/2 Or, L+B = 7

Because Area is the product of the Length & Width, L & B must be one combination of those factors. So, let us find the factors of the Area.

Factors of 12 are: 2,2,3

Let us create a table with all possible ways in which these factors can make up 12. Unique numbers are underlined:

3. So all possible factors, all unique numbers, are 2,3,4,6 & 9.

Hence any one of these could be the Length or the Width. Let us say that one of these numbers is the Length (you can also assume that it is the Width)

Now, to find the the other side, we have to choose another number from this table (3), so that their Sum = 7 (from point 2 above), let us create a table & tally the sum.

So from this table, Combination 2 & Combination 2 are adding up to 7 (from point (2)). We can ignore Combo 5 since the Width cannot be a fraction (from the 3rd clue).

Hence, the Length & Width must be 4 & 3.

Now let us go back to the clues & see if the values of length & width of 4 & 3 satisfy the clues:

1. Area of the soccer field with Length 4 & Width 3 = 4*3 = 12 sq.m
2. Perimeter of the soccer field with Length 4 & Width 3 = 2*(L+B) = 2*(4+3) = 2*7 = 14 m

So our answer is: Length = 4 meters, Width = 3 meters