# Solving Word Problems Using Graphs

Graphs are used to solve real world problems.

Example

A new small business bakes and sells biscuits. The fixed costs of the business are \$400. The cost of making each biscuit is \$1. The business plans to sell each biscuit at \$1.5. Using graphical method find out the following:

1. Loss to business if 300 biscuits are sold;
2. The exact number of biscuits that the business needs to sell to cover full cost;
3. Profit made by the business if 1200 biscuits are sold.

Solution

Expenses = Fixed Costs + Cost of Number of Biscuits Made

So total cost y for x biscuits made is the equation

y = 400 + x

Income = 1.5 × Number of Biscuits Sold

Expressed in equation form

y = 1.5x

y represents money in dollars whether it is expenses or income. x represents number of biscuits.

Both equations above are linear and their graphs are straight lines.

Cost of making 500 biscuits = \$400 + \$500 = \$900

Cost of making 1000 biscuits = \$400 + \$1000= \$1400

This provides coordinates for two points on the graph of the expenses equation. Since cost is dependent on (or function of ) number of biscuits made, number of biscuits is taken as the first of the two coordinates. Coordinates of two points obtained are (500, 900) and (1000, 1400).

Income when 500 biscuits are sold = 1.5 × \$500 = \$750

Income when 1000 biscuits are sold = 1.5 × \$1000 = \$1500

So the two points on income equation graph are (500, 750) and (1000, 1500).

All four points are shown as red dots in the graph below. The black solid line represents expenses and the red solid line represents income.

The two lines meet at C where income equals expenses. Number of biscuits at C is 800. This is the number of biscuits that the business needs to produce and sell to cover full cost.

At 300 biscuits, income is lower than expenses by 250. This is shown by the purple dashed lines. So the business has a loss of \$250.

At 1200 biscuits, income is greater than expenses by 200. This is shown by the blue dashed lines. So the business has a profit of \$200.

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