So now that we know what a percentage is, how is it used? Let’s look at some sample percentage problems.

Melbourne’s Silvan Reservoir has a capacity of 40,446 ML(megaliters). Currently, it is 88% full. How much water is in the reservoir? In other words, what is 88% of 40,446?

Whenever you see or interpret a problem where you need to take a percentage “of” something, equate the word “of” with “multiply”. So to take 88% of 40,446, we multiply 40, 446 by 88%. If you do this on a calculator, you need to use the decimal equivalent of 88% which is 0.88 (see my previous post). On some calculators, there is a percent key. On these, you can type 88 which the calculator will interpret as 0.88 when you use that key. Regardless of which calculator you use, when you multiply 40, 446 by 88%, you should get 35592.48 ML.

If you do not live in Melbourne, the above problem does not interest you much. As money is of interest to most everyone, let’s look at some typical money related percentage problems.

Since DavidTheMathsTutor has effectively educated the masses on how to use percentages, calculators with percentage keys are no longer in demand. So a store has discounted the normal $24.95 price of these calculators by 30%. What is the new price?

This is a two-step problem: first find what the amount of the discount is, then subtract it from the original price. The amount of the discount is 30% × 24.95 = 0.3 × 24.95 = $7.48 (rounded to 2 decimals as we are talking about money). So the store is taking $7.48 off each calculator. So the new price is 24.95 – 7.48 = $17.47.

On the other hand, again because of DavidThe MathsTutor, there is a big demand on fancy calculators that do all sorts of mathematical things like graph equations. So the store decides to markup the normal price of these calculators by 25% to make up for the loss of the percentage calculators. The normal price of these are $149.50 each. What is the new price?

Again, a two-step problem, but this time you are find the amount of the price increase, then add it to the original price since this is a markup. The amount of markup is 0.25 × 149.50 = $37.38. So the store is increasing the price by $37.38, so the new price is 149.50 + 37.38 = $186.88.

Many times, you need to calculate the original price. For example, you are looking to buy a car. The sticker says $24,500. The salesperson says that’s a good price because they are only making a 5% profit on it. What is the cost of the car to the store? This is a reverse markup problem: what price plus 5% of that price is $24,500?

In equation form, that is the equivalent maths sentence, this is

*x* + (0.05 × *x*) = 24,500.

If you remember my posts on equations, factoring, and the distributive property, this is solved by factoring the left side and then dividing both sides by the number that results from the factoring:

\[

\begin{array}{l}

{{x}\hspace{0.33em}{+}\hspace{0.33em}{(}{0}{.}{05}\hspace{0.33em}{+}\hspace{0.33em}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}{x}{(}{0}{.}{05}\hspace{0.33em}{+}\hspace{0.33em}{1}{)}\hspace{0.33em}{=}\hspace{0.33em}{1}{.}{05}{x}\hspace{0.33em}{=}\hspace{0.33em}{24500}}\\

{\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{24500}{1.05}\hspace{0.33em}{=}\hspace{0.33em}{\$}{23333}{.}{33}}

\end{array}

\]

If you knew the cost to the store was $23,333.33 and knew that they marked up that cost by 5%, if you calculated the new price as we did before, you would get $24,500.

Sometimes you need to calculate the actual percentage. What if your salary went up from 88,000 to 99,000? What percentage pay rise is this (so you can brag to your friends)?

The amount of pay increased by 99,000 – 88,000 = 11,000, so we want to know what percentage of 88,000 is 11,000. Note that you always work with the original price or amount when working out a percentage. So the equivalent maths sentence is

88,000 × *x*% = 11,000

Dividing both sides by 88,000 gives

\[

{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{11000}{88000}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{125}\hspace{0.33em}{=}\hspace{0.33em}{12}{.}{5}{\%}

\]

You must be very good at your job!