Part of a Whole

Let’s again go over the most important rule from this Math Essential:

Rule

PartofaWhole

When a part is $n\phantom{\rule{0.17em}{0ex}}\text{%}$ of a whole, you have that

 $\text{part}=\frac{n}{100}\cdot \text{whole}=\frac{n\cdot \text{whole}}{100}$

In the activities you have seen so far, you have been given $n$ and the whole most of the time. In other cases, you might be given the part and the whole, or the part and the percentage $n$ instead. In all of these cases, setting up an equation is the easiest way to solve the problem, so we’ll look at how to do that first. After that, we’ll go over a way to solve these problems without using equations.

Solving Problems Using Equations

The rule above can also be used to find the whole when you know $n$ and the part. You do that by calling the whole $x$, then solving the resulting equation. The same method also works if you don’t know $n$.

Example 1

A suit is on sale with a $\text{}\text{}40\text{}$ discount. You are told that the discount is $\text{}20\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of the initial price. Find the initial price.

In this case, the initial price is the whole, which you call the unknown $x$. The part is the discount. You enter the information you have into the rule, which gives you

 $\text{}40=\frac{20}{100}\cdot x=\frac{1}{5}\cdot x$

You get rid of $\frac{1}{5}$ by multiplying by 5 on both sides, which leaves $x$ on its own on the right-hand side:

 $5\cdot \text{}40=x$

That means the initial price $x$ was \$$200$.

The next example also has just one unknown, but the problem is phrased differently.

Example 2

A jacket currently costs $\text{}\text{}60\text{}$, and this new price is $\text{}75\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of the old price. Find the old price.

The old price is the whole and the unknown is $x$. You just enter the information you have into the rule:

 $\frac{75}{100}\cdot x=\frac{3}{4}\cdot x=\text{}60$

You want $x$ alone on the left-hand side, so you multiply both sides by the fraction turned upside down, $\frac{4}{3}$. Then the left-hand side becomes

 $\frac{4}{3}\cdot \frac{3}{4}\cdot x=1\cdot x=x$

and the equation turns into

 $x=\frac{4}{3}\cdot \text{}60=\text{}80$

When you look at prices in a store, they rarely say “this item costs $75$ % of its old price”. They usually tell you the discount, or just the discounted price. It’s mostly in school exercises that you’re given the new price in percentages like this. But it’s still good to know what to do!

You can also use equations when you know the whole and a part, where $n\phantom{\rule{0.17em}{0ex}}\text{%}$ is the unknown.

Solving Problems Without Using Equations

If you really don’t want to solve problems by using equations, you can use this rule:

Rule

FindingtheWholeWithoutUsingEquations

When you know that a part is $n\phantom{\rule{0.17em}{0ex}}\text{%}$ of the whole, you can find the whole like this:

 $\text{Whole}=\frac{100}{n}\cdot \text{part}$

We can solve the same examples as above by using this rule:

Example 3

A suit is on sale with a $\text{}\text{}40\text{}$ discount. You are told that the discount is $\text{}20\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of the initial price. Find the initial price.

The initial price was \$$200$.

Rule

FindingthePercentageWithoutEquations

When you know the part and the whole, you can find the percentage by looking at the fraction

 $\frac{\text{part}}{\text{whole}}$

You find the percentage via decimal numbers by moving the decimal mark two places to the right.

You learned how to convert percentages to decimal numbers earlier in this Math Essential. You should practice doing this if you’re not comfortable with it yet. Here is an example of how to use the rule:

Example 4

A jacket is sold at a $\text{}\text{}10\text{}$ discount. The initial price was $\text{}\text{}25\text{}$. By what percentage was the price of the jacket reduced?

 $\frac{\text{}10}{\text{}25}=\frac{2}{5}=0.4=0.40=40\phantom{\rule{0.17em}{0ex}}\text{%}$

You can also practice the memory aids for moving the decimal mark the correct way from earlier in this Math Essential, and you can use those rules to check whether you put the decimal mark in the correct place.

Drawing a sketch that shows the whole (\$$25$) and the part (the \$$10$ discount) can also be useful. Remember that the initial price should be the whole.

Example 5

After finishing Example 4, you want to check your answer by making an estimate. You can think like this: The jacket is reduced by \$$10$, which is just below half of the whole. That means the percentage discount should be just below $50$ %, so the answer of $40$ % looks reasonable.