Percentages

Fig.1. Now 20% extra discount on prices already 60% off.

A regular price is $20, how much do you have to pay after this discount on discount? And how much if the extra discount would be 40%?

What are percentages?

A percentage is a decimal fraction with denominator 100.

"Per cent" or "per centum" literally means "per hundred" ("in each hundred") in Latin. Percent (per cent) indicates parts per hundred.

A percentage is usually given as the numerator followed by a "%" sign, i.e. the percent sign: 25% pronounced as "twenty-five percent":

25 100 = 0.25 = 25 %

The numerator can be an integer or a fraction itself:

12.15 % = 0.1215

Half of a cake is the same as 50% of a cake or 4/7 of the animals in the shelter are a cat is (approximately) the same as 57.14% of the animals in the shelter are a cat.

1 2 = 1 2 × 100 100 = 50 100 = 50 %


4 7 = 4 7 × 100 100 = 400 7 100 4 7 = 400 7 % 57.14 %

The numerator of a percentage is usually expressed in decimal notation. An expression like 400/7 % is unusual. A fraction with a denominator whose prime factors are only twos and/or fives can be precisely expressed as a decimal fraction and thus can be precisely expressed as a percentage with a numerator being a terminating decimal expansion. Other fractions expressed as percentages have numerators that are non-terminating decimal expansions, which in practice are approximated by decimal fractions.

1 2 = 0.5 = 0.5×100 100 = 50 100 = 50 %


1 8 = 0.125 = 12.5 %


1 3 = 0. 3 = 33. 3 % 33 %


4 7 = 0. 571428 4 7 = 57.1428 % 57.14 %


Learn the next percentages by heart:

fractiondecimalpercentage
1 2 0.5 50%
1 3 ≈ 33.33%
1 4 0.25 25%
1 5 0.2 20%
1 8 0.125 12.5%
1 10 0.1 10%
1 20 0.05 5%
1 25 0.04 4%
1 40 0.025 2.5%
1 50 0.02 2%

Now reduce 40%, 60%, 75%, 80%, 100% and 225% to lowest terms. What percentages are approximately equal to 1 6 and 1 7 ?

Percentages in word problems

Percentages are dimensionless ratios. A ratio is a quantity relative to an other quantity, as we have seen in the previous part. We can use a percentage to calculate an absolute quantity (the actual measurable or countable amount) if we, for example, know the total absolute value. For example, with a given unemployment rate of 5% and a total number of potential workers of 1 million, we can calculate the actual number of unemployed people.

We say "five percent of one million equals fifty thousand unemployed persons". As we've seen earlier with fractions in general, the word "of" needs to be interpreted as "times". "Five percent of one million" means 5% × 1 000 000.

Seventy percent of 200:

70 % × 200 = 70 × 200 100 70 % × 200 = 70 × 2 = 140

This is actually solving a proportion, as we covered in the previous part, i.e., 70 to 100 is as x to 200. We need to convert 70 per 100 to a fraction per 200:

70 100 = x 200 100×x = 70×200 x = 70×200 100 = 140

How to calculate the percentage? "What percentage of 200 is 140?" or "how many percent is 140 of 200?". We can use a little algebra and fill in a variable for percentage or total in the initial equation:

p × 200 = 140 p×200 200 = 140 200 p = 70 100 = 70 %

Or:

p 100 = 140 200 200×p = 140×100 p = 140×100 200 = 70

And now we calculate the total, i.e., "70% of how much is 140?"

70 % × t = 140 t = 140 70 % t = 140×100 70 = 200

Or: 70 to 100 is as 140 to the total, that is, 200.

70 100 = 140 t 70×t = 140×100 t = 140×100 70 = 200

This all is basically nothing more than division being the opposite of multiplication, as we have seen before. Only now we convert to per 100 instead of per 1.

140 200 = 70 % because 70 % × 200 = 140 .

Can you see why the "general formula" of a percentage is sometime presented as:

percentage = part total × 100

When we calculate with percentages we mostly calculate a part of a whole, i.e., of some total. The total corresponds to 100%. The difficulty with percentages (and fractions in general) is often to find the right total. Furthermore, in word problems often the need to add or subtract a percentage from a total is concealed in the posing of the problem.

Another point of special interest is the addition and subtraction of percentages. Percentages can only be added or subtracted correctly if they are relative to the same total.

Examples

Example 1:

A store offers a 20% off sale on coats with a regular price of €40. How much does a coat cost with the 20% discount?

A coat costs:

80 % × 40 = 32

Or:

80 100 = p 40 p = 40×80 100 p = 4×8×100 100 = 32

The total price is 100%. The discount is 20%, so 80% (100% − 20%) of the price needs to be paid. It is of course also possible to first calculate the saving, i.e. 20% of the total price, and then subtract this from the total price (€40 − €8 = €32).

A customer will save:

20 % × 40 = 8


Example 2:

A constructor offers to install a new bathroom for a price of €2000 before 12% value-added tax. What is the total price after tax?

Total price:

112 % × 2000 = 2240

Or:

112 100 = p 2000 p = 112×2000 100 p = 112×20×100 100 = 2240


The 12% tax is of €2000. This €2000 is the total, the 100%. So, 112% (100% + 12%) of €2000 is the price after tax.


Example 3:

A store advertises with "Sale: 15% off". A customer is charged £46.03 at the cash register. What would have been charged without the discount?

Pre-discount amount:

85 % × t = £ 46.03 t = £ 46.03 85 % t = £ 46.03 85 × 100 t = £ 54.15

Or:

85 100 = 46.03 t t = 46.03×100 85 t = 4603 85 = 54.15

The total is 100%. The discount is 15%, so 85% (100% − 15%) of the original amount needs to be paid. The charged amount would have been £54.15 without discount. You can use long division to calculate 4603 ÷ 85 = 54.15 (rounded to cents).

Can you calculate the absolute discount?


Example 4:

Fifty individuals of a community of twelve hundred fifty people are baptised. What is the baptism ratio in this community represented as a percentage?

The baptism ratio is 50 1250 . In percentage:

50 1250 = 50 50×25 50 1250 = 1 25 = 4 100 50 1250 = 4 %

Or use long division: 50/1250 = 0.04.

Or:

50 1250 = p 100 p = 100×50 1250 p = 100 25 = 4

An amount of 50 individuals per a total of 1250 people is equal to 4%.

Do you think it was reasonable to assume that a relationship like this is directly proportional, that is, do you think the baptism rate is the same for any number of (random) individuals from a community?


Example 5:

A store offers a discount on cakes with a regular price of $11. A customer pays $9.35. How much is the relative discount expressed as a percentage?

The discount is the next ratio expressed as a percentage:

$ 1.65 $ 11 = 15 %


1.65 11 = 165 11×100 = 11×15 11×100 = 15 100


Or:

1.65 11 = p 100 p = 100×1.65 11 p = 165 11 = 15

The absolute discount is $1.65 ($11 − $9.35). The relative discount expressed as a percentage is as the absolute discount relative to the regular price of $11 (the total).


Example 6:

The cost of a meal at a restaurant is $23.45. A customer pays $25, including $1.55 tip. What percentage tip did the customer leave?

The tip is:

$ 1.55 $ 23.45 6.61 100 6.61 %

Or:

1.55 23.45 = p 100 p = 100×1.55 23.45 p 6.61

You can use long division along with long multiplication to calculate (155 ÷ 2345) × 100 = 6.61 (rounded to hundredths).


Example 7:

A salesman earns a base salary of $150.00 per week with an additional relative commission on everything he sells. Last week this salesman sold $6050.00 worth of items, and his total pay was $876.00. What is his commission expressed as a percentage?

His absolute commission was $726 ($876 − $150). The relative commission is:

$ 726 $ 6050 = 12 %


726 6050 = 6×121 50×121 = 12 100


Or:

726 6050 = p 100 p = 100×726 6050 p = 12


Example 8:

An employee receives a raise in salary from €3334 to €3410 per month. What is the pay raise in percentages?"

$ 76 $ 3334 2.3 %

So, the pay raise is almost 2.3% (of €3334); the employee earns now almost 102.3% of his initial salary.

Suppose that the employee earns now 200% of his first salary when he started to work at this company. How many percent has his salary increased in all those years at this company?

As said earlier: in word problems the hardest task is often to find the total that corresponds to 100%. Sometimes it can even be ambiguous.

For instance: One employee receives a salary of €3334, the other €3410 per month. What is the percentage salary difference? The absolute difference is $76, but what is the relative difference? Relative to which of the two salaries? Which one is the total? Jenny earns 4% more than Jake. Is that 4% of Jenny's salary or 4% of Jake's salary?


Example 9:

In a country 60% of the population are happy and 10% are wealthy. If 5% of the happy people are also wealthy, what percentage of the wealthy are happy?

The answer is:

both happy and wealthy total wealthy


5%×60%×population 10%×population = 5%×60% 10% = 30 %


5%×60% 10% = 30010 000 10100 = 301000 1001000 = 30 100

5% of the happy people are both happy and wealthy, 30% of the wealthy are both happy and wealthy.

60% of the population are happy, so 3% (5% of 60%) of the population are both happy and wealthy.

7% (10% − 3%) of the population are wealthy but not happy, 57% (60% − 3%) of the population are happy but not wealthy.

What percentage of the population are not happy and not wealthy?

Happy, not wealthy, 57% Wealthy, not happy 7% Not happy, not wealthy Happy and wealthy, 3%

Compound percentages

A compounding percentage is a percentage of a percentage or percentages applied sequentially. Compounding percentages cannot be calculated by simply adding up the percentage values and apply this "sum percentage" to the total. The problem is that different percentages refer to different totals. A 5% increase in price followed by a 5% decrease in price will change the final price. An extra 40% discount on prices that are already 60% off does not mean that you will get the product for free.

Figure 1 shows a discount of 40% on prices already discounted 60%. The 60% refers to the original price and the 40% refers to the already discounted price. If both percentages were calculated of the original price, than indeed the final price would be zero and the total discount would be 100% = 1. The correct calculation of the final price is however: 60% of 40% of the original price.

final price = 60 % × 40 % × initial price final price = 60100 × 40100 × initial price final price = 240010000 × initial price final price = 24 % × initial price


total discount = initial price final price total discount = 100 % 24 % = 76 %


That is 76% of the initial price. Percentages can be subtracted because they refer to the same total (the initial price).

Suppose a price rises 5% and then drops 10%. Then the new price will be 94.5% of the original price:

new price = 90 % × 105 % × initial price new price = 90100 × 105100 × initial price new price = 945010000 × initial price new price = 94.5 % × initial price

In total, the price drops 100% – 94.5% = 5.5% of the initial price, from the original price.

Note that the total price decrease cannot be calculated as 10% × 5% × initial price. This calculates 10% of the initial absolute rise, while 10% refers to the total increased price. The price drop from the interim high (the price after the 5% rise) is 10% × 105% × initial price = 10.5% × initial price.

Percentage points

Suppose an interest rate rises from 5% per annum to 7% per annum. It is popular, yet fallacious, to say that in this case the interest rate increases by 2%. One takes the "sum percentage", 7% – 5% = 2%, and call that the increase.

However, if we take absolute interests and calculate the interest rise in percentages:

7 % × savings 5 % × savings = x100 x = 7 % 5 % × 100 x = 140

So, the interest rises 40% and not 2%! A rise of 2% of the initial rate of 5% results in a new interest rate of 5.1%, instead of 7%:

102 % × 5 % = 102100 × 5100 102 % × 5 % = 51010000 = 5.1 %

So, a rise from 5% per annum to 7% per annum is a rise of 40%:

p × 5 % = 7 % p = 7 % 5 % = 140 100 = 140 %

To avoid confusion, we say that an increase from 5% to 7% is an increase of 40%, or an increase of 2 percentage points, rather than 2%. A percentage point or percent point (pp) is a unit to express the arithmetic difference between two percentages.

Example 10:

An investor owns 41% of all shares of a company. He now owns 3.5% more shares of this company than one year earlier. What percentage of the shares did the investor own one year earlier?

If the 3.5% increase is interpreted as a 3.5 pp increase, he owned 41% – 3.5% = 37.5% of the shares a year earlier. If it is interpreted correctly as a percentage increase, he owned approximately 39.6% of the shares one year earlier:

41 % × all shares x % × all shares = 103.5 %


41 % x % = 103.5 100 x % = 41 103.5 x % 0.3961 x 39.6