Introduction

Percentages are a handy way of expressing a proportion of one quantity relative to another. The term 'percent' means per hundred.

The number 1 is equal to 100%.

Some calculators have special keys for working with percentages. Another way of working with them is to mentally multiply a value by 100, before writing it down as a percentage.

Here are some useful behavioural rules to go by when using percentages:

  • Percentages are expressed using the '%' symbol, e.g. 75%.

  • A percentage is 100 times a ratio. So 75% means 0.75 as a ratio.

  • Some calculators may allow you to add or subtract a percentage of a quantity to itself e.g.
    \(2.4 + 15 \% = 2.4 \times (1 + \frac{15}{100} ) = 2.76 \),
    \(2.4 - 15\% = 2.4 \times (1 - \frac{15}{100} ) = 2.04 \),
    \(7 + 30\% = 7 \times (1 + \frac{30}{100} ) = 9.1 \),
    \(7 - 30\% = 7 \times (1 - \frac{30}{100} ) = 4.9 \).

  • Some calculators may allow you to multiply or divide a percentage of a quantity to itself e.g.
    \(2.4 \times 15\% = 2.4 \times \frac{15}{100} = 0.36 \),
    \(2.4 \div 15\% = 2.4 \div \frac{15}{100} = 2.4 \times \frac{100}{15} = 16 \),
    \(7 \times 30\% = 7 \times \frac{30}{100} = 2.1 \),
    \(7 \div 30\% = 7 \div \frac{30}{100} = 7 \times \frac{100}{30} = \frac{700}{30} = \frac{70}{3} = 23.33 \).

  • Some calculators may allow you to convert a number to a percentage directly. Otherwise, multiply the number by 100. To convert a percentage to a number, divide by 100 e.g.
    \( 30\% \leftrightarrow 0.3 \),
    \( 0.6 \leftrightarrow 60\% \),
    \( 1.591 \leftrightarrow 159.1\% \),
    \( 0.62\% \leftrightarrow 0.0062 \).

Assignment Answers and Commentary

Questions 1 through 4

Items 1 through 4 are the conversion of numerical expressions to fractions. Items 5 through 10 are the conversion of fractions to numerical expressions. All answers are given to 4 sf (significant figures) unless otherwise stated.

item Input
 Percentage
 Comment
1
 \( \frac{3}{8} \)
 37.5%
 \( 3 \div 8 \) = 0.375. This can be multiplied by 100 to get the answer. You could also do \( \frac{3}{8} \times 100 \), which yields \( 37 \frac{1}{2} \), or 37.5.
2
 \( \frac{7}{16} \)
 43.75%
 \( 7 \div 16 \) = 0.4375. This can be multiplied by 100 to get the answer. You could also do \( \frac{7}{16} \times 100 \), which yields \( 43 \frac{3}{4} \), or 43.75.
3
 0.35
 35%
 The input can be multiplied by 100 to get the answer.
4
 0.42
 42%
 The input can be multiplied by 100 to get the answer.


Question 5

A drum of cable has 163 metres of cable on it. 92 metres are then used to wire some lights. What percentage of the original drum is left?

Solution

I will use the symbol \( p \) to describe the proportion of the cable remaining. The amount of cable remaining is 71 metres (163 - 92). The calculation is shown below.

\( p = \frac{163 - 92}{163} = \frac{71}{163} \) = 0.436.

To convert to a percentage, multiply by 100. The percentage remaining is therefore 43.6%.

A table outlining the percentage calculations is shown below.

Description Quantity (m)
 Percentage of
Initial Amount
Initial Amount of Cable
 163 100.0%
Amount used
 92 56.4%
Amount remaining
 71 46.4%


Question 6

You are paid $420 for an electrical job. $219 goes to the wholesaler for the electrical fittings used on the job. What percentage of the total bill is yours?

Solution

I will use the symbol \( p \) to describe the proportion of the money remaining. The amount of money remaining is $201 (420 - 219). The calculation is shown below.

\( p = \frac{420 - 219}{420} = \frac{201}{420} \) = 0.479.

To convert to a percentage, multiply by 100. The percentage remaining is therefore 47.9%.

A table outlining the percentage calculations is shown below.

Description Price (m)
 Percentage of
Initial Amount
Job Payment
 $420 100.0%
Costs to Wholesaler
 $219 52.1%
Amount remaining
 $201 47.9%



Last modified: Friday, 8 May 2020, 6:24 PM