Math Puzzle – Area & Perimeter of Rectangles with Charts

Math Puzzle: The perimeter of a rectangle is 14 m & the area of the rectangle is 12 sq. m. What is the length of the longest side of the rectangle if both the length & the width are integers.

Solving: Let us consider the Length of the rectangle = L meters & Width = W meters. We will solve for L & W below. Since Length is greater than Width, our answer would be the Length. So we have to solve for L.

Arranging the clues in a logical order:

1. Perimeter = 2*Length + 2*Width = 14
   => 2L + 2W = 14
   => L + W = 14/2
   => L + W = 7 ———— (let us number this equation 1) 
   
2. Area = Length*Width = 12
   => L*W = 12 ———— (let us number this equation 2)
   
3. Length & Width are both Integers

This can be solved very easily with a Simple Quadratic Equation as below. I will also provide a way to solve this problem with Charts without using Quadratic Equation, since I have seen similar questions on 3rd & 4th Grade Math Perennial & Olympiad Tests.

Solving with Simple Quadratic Equation:

From (1): W = (7-L)

Putting this value of W in equation (2):

L * W = 12

=> L * ( 7 – L ) = 12

=> 7L  –  L2 = 12

=> L2 – 7L -12 = 0

=> (L – 3)(L-4) = 0

=> L = { 3, 4}

So the 2 possible values of Length L are 3 & 4.

If L = 3, from equation (2), 

W = 12/L

=> W = 12/4 

=> W = 4

Let us look into the second value of L which is 4.

If L = 4, from equation (2),

W = 12/L

=> W = 12/4

=> W = 3

So the possible pair combinations of Length & Width that will create a rectangle of perimeter 14 sq. meters & perimeter of 12 meters are:

When Length = 3, Width = 4

When Length = 4, Width = 3

Answer: The longest side, the Length will be 4 meters.

Solving with Charts

Now let us solve this problem without using quadratic equations so it will be understood by 3rd & 4th graders.

1. Perimeter = 2*Length + 2*Width = 14
   => 2L + 2W = 14
   => L + W = 14/2
   => L + W = 7 ———— (let us number this equation 1) 
   
2. Area = Length*Width = 12
   => L*W = 12 ———— (let us number this equation 2)
   
3. Length & Width are both Integers

From equation (2) let us factorize 12:

So the factors of 12 are: 1, 2, 3, 4, 6, 12

Let us assume that the Length could be any of these values (1, 2, 3, 4, 6, 12). Let us create a table to find out the possible values of Width for any of these Lengths.

From equation (2), Width = (12/Length) – we will use the various possible values of Length & this equation to find the possible values of Width.

So we see that the Perimeter is 14 when either Length = 3 & Width = 4 OR when Length = 4 & Width = 3. In both these cases the perimeter will be 14 & area will be 12.

Since Length is the longer of the 2 sides, the Length of the rectangle will be 4 meters.

Answer: 4 meters.