# Problem Solving: Use Algebra

The main benefit of algebra in written problems is it converts lengthy writing into short simple statements. We try to translate problems into an equation or equations, considering what arithmetic operations are needed.

The following is an outline of how to do this:

• Carefully read problem and identify what you need to find and what information is given;
• Assign a letter to any unknown quantities (e.g. x = width of rectangle);
• Relate the known and unknown quantities to each other using an equation, formula or ratio;
• Find the solution;

Worked Examples

Let’s read these two examples carefully (the first one we have already seen in an earlier section and will now approach using algebra:

Circle in Square Example

Below is a diagram of a red square with a purple circle drawn through its vertices and a blue square that touches the purple circle on all sides.

What is the relationship between the area of the red square and the blue square and how do you know?

Triangular Numbers Example

The formula for a triangular number is T = ½ n (n+1). Prove that 8T+1 is always a square number.

Solutions to Worked Examples

Circle in Square Example

Below is a diagram of a red square with a purple circle drawn through its vertices and a blue square that touches the purple circle on all sides.

What is the relationship between the area of the red square and the blue square and how do you know?

Solution

We notice that the diameter of the purple circle is the same as the length of the blue square, so if we let d = diameter, then the blue square has area d2.

Looking at the red square we then notice that its diagonals have length d. We can use this to find the area of the red square, by splitting it into two halves, as shown below (note that the b and h on the diagram mean the base and the height of the marked triangle):

So the area of this triangle is ½ x d x d/2 = ¼d2. The red square’s area is twice this, so ½d2.

So we can see algebraically that the area of the red square(½d2) is half the area of the blue square (d2).

Triangular Numbers Example

The formula for a triangular number is T = ½ n (n+1). Prove that 8T+1 is always a square number.

Solution

If T = ½ n (n+1), then 8T+1 = 8(½ n (n+1)) + 1 = 4n(n+1) + 1.

If we expand the brackets we get 4n2+4n+1, which factorises to give (2n+1)2, which will clearly always be a square number.

28 Questions of Increasing Difficulty

Solutions to Question Set