# Section 3A

...

## Generalities

Percentage always means some fraction

P% means $$\frac{P}{100}$$

for example, 25%=$$\frac{25}{100}=\frac{1}{4}$$ is a quarter.

however ...

## There are three ways of using percentages:

• Simply as a fraction

• To describe change

• For comparison

We consider these ways one at a time now.

## Percentage as a fraction.

You have to be able to transform back and force:

$$P\% = \frac{P}{100}$$

Specifically:

$$20\%=\frac{20}{100}=\frac{1}{5}=0.2$$

$$0.37=\frac{37}{100}=37\%$$

... see p.121 for Brief Review ...

## Percentage as a fraction.

Example 1 (p.122)

An opinion poll finds that $$64\%$$ of $$1069$$ people surveyed said that the President is doing a good job. How many said the President is doing a good job?

Solution

We need to calculate $$64\%$$ out of $$1069$$. That is

$64\% \times 1069 = 0.64 \times 1069= 684.16 \approx 684$

## Using percentage to describe Change

You have to distinguish between:

Absolute change

... and ...

Relative change

### Using percentage to describe Change

absolute change = new value - reference value

relative change = $$\frac{ \text{new value - reference value}}{\text{reference value}}$$

Example. I had $$1$$ ball, and got $$2$$ more.

While the absolute change is $$2$$ balls,

relative change = $$\frac{3-1}{1} = 2 = 200\%$$

Another example. I had $$2$$ ball, and lost $$1$$ of them.

Now the absolute change is $$1-2=-1$$ balls, and

relative change = $$\frac{1-2}{2} = \frac{-1}{2}= -0.5 = -50\%$$

## Using percentage for Comparison

We do not have any change now , but just two values to compare.

You have to distinguish between:

Absolute difference change

... and ...

Relative difference change

## Using percentage for Comparison

absolute difference = compared value - reference value

relative difference = $$\frac{ \text{compared value - reference value}}{\text{reference value}}$$

Example. I have $$1$$ ball in my hand, while there are $$2$$ balls on the desk.

There are $$2-1=1$$ ball more on the desk than in my hand (absolute difference), and

relative difference = $$\frac{2-1}{1} = 1 = 100\%$$

##### Precise meaning of some words:

Of ... More than (Less than)

P% of value is $$\frac{P \times value}{100}$$

Example. 5% of 700 is $$\frac{5 \times 700}{100} = 35$$

P% more than value is $$\frac{(100+P)\times value}{100}$$

Why so? -- Because P% more than a value is the value itself... plus P% of this value:

$$value + \frac{P \times value}{100}=\frac{(100+P )\times value}{100}$$

Similarly, P% less than value is $$value - \frac{P \times value}{100}=\frac{(100-P )\times value}{100}$$

##### Example

Assume that there are 300 students in this class, while an astronomy class has 20% less students than this one. How many students does the astronomy class have?

Solution:

The astronomy class has $$\frac{(100-20 )\times 300}{100} =240$$ students

Alternatively, 20% of 300 is $$\frac{20 \times 300}{100} =60$$.

Thus the astronomy class has 60 students less than this one, which is 300-60=240.

##### More subtle wording: a change in $$percentage$$ $$points$$ is absolute, not relative!

Example

My mortgage rate is 3.25% while a friend of mine has got a rate of 4%.

The rate of my friend is 4-3.25=0.75 $$percentage$$ $$points$$ higher than mine. This is $$absolute$$ difference.

The rate of my friend is $$\frac{4-3.25}{3.25} \approx 0.23 =23\%$$ higher than mine. This is $$relative$$ difference.

##### Abuses of percentage

Shifting reference value.

Assume that Jane has a compensation rate of $$60\$$ per hour while John earns $$30\$$per hour.

Jane: Your rate is only 50% lower than mine

John: But your rate is 100% higher than mine

Both are correct! They naturally make use of different reference values.

##### Abuses of percentage

Shifting reference value. Another example.

The price of a commodity increases by 20%, and after that decreases by 20%. What happens with the price after all?

Solution. The commodity has become a bit cheaper. Indeed, assume the initial price of $$100\$$. It has become $$120 \$$ after the increase. Now the price decreases by 20% of the new reference value of $$120 \$$, which is $$0.2 \times 120 = 24\$$, and the final price is $$120-24=96\$$. The absolute difference in price after all is $$4\$$. That means a drop in price from the initial one of $$100\$$ by $$\frac{4}{100}=0.04=4\%$$

##### Abuses of percentage

Less than nothing

No positive value can decrease by more than 100% staying positive. Examples comprise prices, energy consumption...

A decrease of the price of a commodity by 100% means that the commodity becomes free. Further decrease will mean that the seller is going to pay you if you take it. In such cases, misunderstanding as a result of abuse of percentage is likely.

However, an increase by more than 100% easily happens.

For example, a flight ticket which was $$300\$$, and is now $$750\$$ has increased in price by $$150\%$$.

##### Abuses of percentage

Never average percentages

Example. Last year, 50% of the nights I was off the island I have spent in a hotel, while on the island I have spent no night, that is 0% in a hotel. However, it is not at all true that I have spent 25% of nights last year in a hotel. In fact, I have been off the island only for two weeks, and have thus spent only 7 night in a hotel which is $$\frac{7}{365} \approx 2\%$$ of nights in a hotel last year.