Analyzing and Integrating Mathematical Information

Analyzing and Integrating Mathematical Information in order to construct a solution

Solving mathematical problems often requires us to borrow knowledge from different branches of math, such as geometry, algebra, number systems, probability and statistics. Often times, we make use of this information to convert relevant information into mathematical equations. Therefore, it is imperative that we keep a good understanding of the interrelated branches in math and use them appropriately in constructing solutions for a mathematical problem.

Remember that when constructing solutions, we have to follow these steps:

  • Define the problem
  • Identify relevant information
  • Define the variable
  • Convert relevant information into mathematical equations

Let’s look at the first example:

A rectangle has a perimeter of 20 cm. The length of the rectangle is 2 units longer than the width. Find the dimensions of the rectangle (geometry sense).

Constructing solution:

Define the problem:What are the dimensions (length and width) of a rectangle if its perimeter is 20 cm?

Identify relevant information:

Perimeter of rectangle = 20 cm

Length = width + 2 cm

Define the variable:

We want to find the length and width of the rectangles, so we have two variables here. Let length be equal to l and width be equal to w.

Convert relevant information to mathematical equation:

There are two pieces of relevant information that will help us convert information into a mathematical equation.

  1. The perimeter of the rectangle is equal to 20 cm.
    Note: Perimeter of a rectangle = 2 * (length + width)
    Therefore: 20 = 2 * (l + w)
    Or, 10 = (l + w)
    Or, l + w = 10……..(i)
  2. The length of the rectangle is 2 units longer than its width. Therefore:
    l = w + 2……..(ii)
    Solve mathematical equation:
    There are two equations:
    l + w = 10…..(i)
    l = w + 2……(ii)
    Substitute the value of l in equation (i) from equation (ii)
    (w + 2) + w = 10
    w + 2 + w = 10
    2w + 2 = 10
    2w = 10 – 2
    2w = 8

    3. Dividing both sides by 2:

    w = 4

    Now substitute the value of w in equation (ii)

    l = w + 2 = 4 + 2 = 6

    Hence, the dimensions of the rectangle are 6 (length) and 4 (width) centimeters.

    Keep in mind that if we had not known the appropriate formula for the perimeter of the rectangle, we would not have been able to solve this problem.

    Now, let’s look at an example that uses procedures from number systems for constructing its solution.

The sum of three consecutive even integers is 30. Find the three numbers.

Constructing solution:

Define the problem:

Find three consecutive even integers such that their sum is equal to 30.

Identify relevant information:

The integers are consecutive and even and their sum is equal to 30.

Define the variable:

Remember that even numbers are multiples of 2. So we will need to define our variable such that it is a multiple of 2. Therefore, instead of defining our first number as x, we will define it as 2x.

Now, let our first even integer be 2x. Therefore, our next even integer will be 2x + 2, and our third even integer will be 2x + 4. (Note: The difference between two consecutive even integers is always 2).

Convert relevant information to mathematical equation:

The sum of three consecutive even integers is 30” can be written as:

2x + (2x + 2) + (2x + 4) = 30 ……(i)

Solve mathematical equation:

We will solve equation (i), and find x:

2x + (2x + 2) + (2x + 4) = 30

Or, 2x + 2x + 2 + 2x + 4 = 30

Or, (2x + 2x + 2x) + (2 + 4) = 30 (combining like terms)

Or, 6x + 6 = 30

6x = 30 – 6

6x = 24

Dividing both sides by 6:

x = 4

1st even integer = 2 * x = 2 * 4 = 8.

2nd even integer = (2 * x + 2) = (2 * 4 + 2) = 10

3rd even integer = (2 * x + 4) = (2 * 4 + 4) = 12

Thus, the three consecutive even integers such that their sum is 30 are 8, 10, and 12.

Now try this exercise:

The perimeter of a rectangle is 13 cm and its width is 2 ¾ cm. Find its length.

Answer:

Define the problem: length of a rectangle

Identify relevant information:

perimeter of a rectangle = 13 cm

Width of a rectangle = 2.75 cm

Define the variable:

We want to find the length of a rectangle, so we have one variable here. Let length be equal to x cm.

Convert relevant information into mathematical equations:

The perimeter of the rectangle = 2 ( length + width )

= 2( x + 2 ¾ )

= 2( x + 11/4)

Given that the perimeter = 13 cm

Hence,    13 = 2 ( x + 11/4 )

13/2 = x + 11/4

13/2 - 11/4 = x

13*2 – 11*1 = 26 - 11 = 15 = x
          4                  4           4

Hence, x = 15/ 4 = 3 ¾ cm