Problem Solving with Equations
Now that we know how to solve equations, including problems that involve fractions we are given, the next step is to create our own equations to solve given a particular situation or problem.
Let's work through an example first and then reflect on the general approach to take.
Worked Examples
Question 1
When a number is added to both the numerator and denominator of $\frac{1}{5}$15, the result is $\frac{3}{7}$37.
Let $n$n represent the number. Solve for $n$n.
Question 2
To manufacture sofas, the manufacturer has a fixed cost of $\$27600$$27600 plus a variable cost of $\$170$$170 per sofa. Find $n$n, the number of sofas that need to be produced so that the average cost per sofa is $\$290$$290.
Question 3
A commercial airplane has a total mass at take off of $51000$51000 kg. The luggage and fuel are $\frac{1}{3}$13 the mass of the unloaded plane, and the crew and passengers are $\frac{1}{4}$14 the mass of the fuel and luggage. Solve for $p$p, the mass of the unloaded plane.
The Overall Approach
So it seems that the following general steps can be taken to solve a problem through equation building:
1) Identify the unknown value you are trying to solve for and let it be represented by a pronumeral (the question may already have given you the pronumeral to use).
2) Identify any equations, concepts or formula that may be relevant to the problem. For example, if the question refers to averages, it may be useful to remember that $average=\frac{\text{sum of scores }}{\text{number of scores }}$average=sum of scores number of scores
The hardest step: Weaving it all together.
3) Try to relate the unknown to the other values given in the problem (either using words or mathematical symbols) to form an equation.
It may be useful to describe the relationship(s) you can see in words before writing them out as mathematical equations, or even to form smaller and more obvious mathematical expressions and see how these expressions relate to one another.
4) Solve the equation.
5) Test your solution!