Problem solving with equations I
Now that we know how to solve equations 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.
Examples
Question 1
Skye, a squash player has won $48$48 out of $63$63 matches in her career. Find $x$x, the number of matches she must win in a row to raise her win percentage to $80%$80%.
Solution:
I wonder what percentage she has won so far?
$\frac{48}{63}=76.19%$4863=76.19%
If she won the next game, what would the percentage change to?
| $\frac{48+1}{63+1}$48+163+1 | $=$= | $\frac{49}{64}$4964 |
| $=$= | $76.56%$76.56% |
We want to add the number of games $x$x, so that
$\frac{48+x}{63+x}=80%$48+x63+x=80%
This is our equation. Now we need to solve it.
| $\frac{48+x}{63+x}$48+x63+x | $=$= | $80%$80% | |
| $48+x$48+x | $=$= | $\frac{80}{100}\left(63+x\right)$80100(63+x) | multiply both sides by $63+x$63+x |
| $48+x$48+x | $=$= | $50.4+\frac{8x}{10}$50.4+8x10 | multiply the $\frac{80}{100}$80100 by both terms inside the parenthesis |
| $x-\frac{8x}{10}$x−8x10 | $=$= | $50.4-48$50.4−48 | combine the variables on the left and the constant terms on the right |
| $\frac{2x}{10}$2x10 | $=$= | $2.4$2.4 | simplify both sides |
| $2x$2x | $=$= | $24$24 | multiply both sides by $10$10 |
| $x$x | $=$= | $12$12 | divide both sides by $2$2 |
| so she must win $12$12 more games in a row |
Question 2
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 3
Sisters Judy and Tara specialise in two different triathlon events. Judy finds that her average cycling speed is $8$8 mph faster than Tara's average running speed.
Judy can cycle $22$22 miles in the same time that it takes Tara to run $11$11 miles.
If Tara's running speed is $n$n miles per hour, solve for $n$n.
Determine Judy's average cycling speed.
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 variable (the question may already have given you the variable 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!