Percentages - MathsTips.com

The word ‘percentage’ is very familiar to us as it is used regularly in the media to describe anything from changes in the interest rate, to the number of people taking holidays abroad, to the success rate of the latest medical procedures or exam results. Percentages are a useful way of making comparisons, apart from being used to calculate many taxes that we pay such as VAT, income tax, domestic fuel tax and insurance tax, to name but a few. So percentages are very much part of our lives.

What does percentage actually mean? ‘Per cent’ means ‘out of 100’; and ‘out of’, in mathematical language, means ‘divide by’. So if you score $85\%$ (using the symbol ‘%’ for percentage) on a test then, if there were a possible $100$ marks altogether, you would have achieved $85$ marks.

So, $85\% = \dfrac{85}{100}$

A few more examples:

$75\% = \dfrac{75}{100} = \dfrac{3}{4} = 0.75$

$21\% = \dfrac{21}{100} = 0.21$

$50\% = \dfrac{50}{100} = 12 = 0.50$

Thus, all percentage amounts have their fractional and decimal equivalents.

It is worth noting that 50% can be found by dividing by 2, and that 10% is easily found by dividing by 10.

Now, let us look at writing fractions as percentages.

For example, say you get 18 marks out of 20 in a test. What percentage is this?

First, write the information as a fraction. You gained 18 out of 20 marks, so the fraction is $\dfrac{18}{20}$ Since a percentage requires a denominator of $100$ , we can turn $\dfrac{18}{20}$ into a fraction out of $100$ by multiplying both numerator and denominator by $5$ :

$\dfrac{18}{20}= \dfrac{18 \times 5}{20 \times 5} = \dfrac{90}{100} = 90\%$ .

Since we are multiplying both the numerator and the denominator by $5$ , we are not changing the value of the fraction, merely ﬁnding an equivalent fraction.

In that example it was easy to see that, in order to make the denominator $100$ , we needed to multiply $20$ by $5$ . But if it is not easy to see this, such as with a score of, say, $53$ out of $68$ , then we simply write the amount as a fraction and then multiply by $\dfrac{100}{100}$ :

$\dfrac{53}{68} \times \dfrac{100}{100} = \dfrac{53}{68} \times 100\% = 77.94\%$ which is %latex 78\% $ to the nearest whole number.

Although it is easier to use a calculator for this type of calculation, it is advisable not to use the % button at this stage. We shall look at using the percentage button on a calculator at the end of this lesson.

Thus the key point in this discussion is that : percentage means ‘out of 100’, which means ‘divide by $100$ ’.To change a fraction to a percentage, divide the numerator by the denominator and multiply by $100\%$ .

Example: 7 out of every 10 people questioned who expressed a preference liked a certain brand of cereal. What is this as a percentage?

Solution: 7 out of 10 people liked the brand of cereal. Now, 7 out of 10 expressed as a fraction is $\dfrac{7}{10}$

$\dfrac{7}{10} = \dfrac{7}{10} \times \dfrac{100}{100} = \dfrac{7 \times 100}{10 \times 100} = \dfrac{70}{100} = 70\%$

Find percentages

For many calculations, we need to ﬁnd a certain percentage of a quantity.

For example, it is common in some countries to leave a tip of $10\%$ of the cost of your meal for the waiter. Say a meal costs $Rs \: 25.40$ :

$10\% \hspace{1mm} of \hspace{1mm} Rs \: 25.40 = \dfrac {10}{100} \times Rs \: 25.40 = Rs \: 2.54$ .

As mentioned before, an easy way to ﬁnd $10\%$ is simply to divide by $10$ . However the written method shown above is useful for more complicated calculations, such as the commission a salesman earns if he receives $2\%$ of the value of orders he secures.

For example, in one month he secures $Rs \: 250,000$ worth of orders. How much commission does he receive?

Solutuion: $2\% \hspace{1mm} of \hspace{1mm} Rs \: 250,000 =\dfrac{2}{100} \times Rs \: 250,000 =Rs \: 5,000$

Find original amount before percentage change

Let us look at a situation where we need to ﬁnd an original amount before a percentage increase has taken place.

Example: An insurance company charges a customer $Rs \: 320$ for his car insurance. The price includes government insurance premium tax at $5\%$ . What is the cost before tax was added?

Solution: Here, the $Rs \: 320$ represents $100\% + 5\% \hspace{1mm} or, \hspace{1mm} 105\%$ of the cost, so to calculate the original cost, $100\%$ , we need to calculate $Rs \: (\dfrac{320}{105} \times 100) = Rs \: 304.76$ .

Here is one more similar calculation, but this time there has been a reduction in cost.

Example: A shop has reduced the cost of a coat by $15\%$ in a sale, so that the sale price is $Rs \: 127.50$ . What was the original cost of the coat?

Solution: In this case, $Rs \: 127.50$ represents $85\%$ (i.e., $100\%-15\%$ ) of the original price. So if we write this as a fraction.

We divide by $85$ to ﬁnd $1\%$ and then multiply by $100$ to ﬁnd the original price.

$\therefore$ original price of the coat $=Rs \: (\dfrac{127.50}{85} \times 100) = Rs \: 150$

The key point which has to be noted here is that if you are given a percentage change and the ﬁnal amount, write the ﬁnal amount as $100\%$ plus (or minus) the percentage change, multiplied by the original amount.

Expressing change as percentage

We might wish to calculate the percentage by which something has increased or decreased.

To do this we use the rule: $\dfrac{actual \hspace{1mm} increase/decrease}{original \hspace{1mm} cost} \times 100\%$ .

So we write the amount of change as a fraction of the original amount, and then turn it into a percentage.

Example: Four years ago, a couple paid $Rs \: 180,000$ for their house. It is now valued at $Rs \: 350,000$ . Calculate the percentage increase in the value of the house.

Solution:

$Percentage \hspace{1mm} increase = \dfrac{actual \hspace{1mm} increase}{original \hspace{1mm} cost} \times 100\%$

$=\dfrac{Rs \: 350,000- Rs \: 180,000}{Rs 180,000} \times 100\%$

$= \dfrac{Rs \: 170,000}{Rs \: 180,000} \times 100\%= 94\% (approx.)$

Example: A car costs $Rs \: 12,000$ . After $3$ years it is worth $Rs \: 8,000$ . What is the percentage decrease?

Solution: $Percentage \hspace{1mm} decrease = \dfrac{actual \hspace{1mm} decrease}{original \hspace{1mm} cost} \times 100\%$

$=\dfrac{Rs \: 12,000-Rs \: 8,000}{Rs \: 12,000} \times 100\%$

$= \dfrac{Rs \: 4,000}{Rs \: 12,000} \times 100\%$

$= 33\% (approx.)$

Point to note: To write an increase or decrease as a percentage, use the formula:

$\dfrac{actual \hspace{1mm} increase \hspace{1mm} or \hspace{1mm} decrease}{original \hspace{1mm} cost} \times 100\%$

Understanding percentage on calculator

The $\%$ button on the calculator should be used only when one is aware of the effect that it will bring in the calculation. Using the $\%$ button randomly will not produce any fruitful result. Here is a warning about using the percentage button on a calculator: the result depends on when you press the $\%$ button in your calculation. Sometimes it has no eﬀect, sometimes it seems to divide by $100$ , and at other times it multiplies by $100$ . Here are some examples :

Pressing “ $48 \div 400 \%$ ” gives an answer of $12$ . Now $\dfrac{48}{400} = 0.12$ , so pressing the $\%$ button has had the eﬀect of multiplying by $100$ . This has found 48 as a percentage of 400.
Pressing “ $1 \div 2 \times 300 \%$ ” gives the answer $1.5$ . Now $\dfrac{1}{2} \times 300 = 150$ , so pressing the $\%$ button here has divided by $100$ . This has found $300\%$ of the faction $\dfrac{1}{2}$ .
Pressing “ $400 \times 50 \%$ ” gives an answer of $200$ . Now $400 \times 50 = 20,000$ , so pressing $\%$ here has divided by 100. This has found $50\% \hspace{1mm} of \hspace{1mm} 400$ .
Pressing “ $50\% \times 400$ ” results in $400$ on the display, requiring “ $=$ ” to be pressed to display an answer of $20,000$ . So pressing the $\%$ button here has had no eﬀect.

Exercise

In a test you gained $24$ marks out of $40$ . What percentage is this?
$30$ out of $37$ gambling sites on the Internet failed to recognise the debit card of a child. What is this as a percentage?
At the end of $1999$ you bought shares in a company for $Rs \: 100$ . During $2000$ the shares increased in value by $10\%$ . During $2001$ the shares decreased in value by $10\%$ . How much were the shares worth at the end of $2001$ ?
What is the amount of Value Added Tax or VAT (at a rate of $17\%$ ) which must be paid on an imported computer game costing $Rs \: 16.00$ ?
A batsman scored $110$ runs which included $3$ boundaries and $8$ sixes. What percent of his total score did he make by running between the wickets?

Find percentages

Expressing change as percentage

Understanding percentage on calculator

Exercise

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