...
Percentage always means some fraction
P% means \( \frac{P}{100} \)
for example, 25%=\(\frac{25}{100}=\frac{1}{4}\) is a quarter.
however ...Simply as a fraction
To describe change
For comparison
We consider these ways one at a time now.
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 ...
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
\[ 64\% \times 1069 = 0.64 \times 1069= 684.16 \approx 684\]
You have to distinguish between:
Absolute change
... and ...
Relative 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 = \( \frac{1-2}{2} = \frac{-1}{2}= -0.5 = -50\% \)
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
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 = \( \frac{2-1}{1} = 1 = 100\% \)
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 \(value - \frac{P \times value}{100}=\frac{(100-P )\times value}{100} \)
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.
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.
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.
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 \(\frac{4}{100}=0.04=4\%\)
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 \( 150\% \).
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.