Numbers
4. Percentages
From the Latin per centum, meaning ‘by a hundred’, a percentage is a value expressed as a fraction of 100, using the percentage symbol (%).
An understanding of percentages is particularly useful for conveying proportions, and making comparisons.
If 100% represents the whole, or total value, of something, whatever that amounts to (the membership of an organisation, for example, or the population of a town), then half of that whole is 50% (100 divided by 2) and a quarter is 25% (100 divided by 4).
Values expressed as percentages are often easier to picture than fractions. Let's look at an example.
Shopping around for a new mobile phone, you find an online store offering to sell your preferred model for only 6/8 (six eighths) of the usual price, while another store is advertising the same phone for 7/10 (seven tenths) of the usual price. Which one appears most attractive?
A further company is offering this model for 75% of the usual price and yet another store is selling for 70% of the usual price. Which of these appears most attractive?
As 70 is less than 75, the 70% price is likely to be the best deal, although you would need to check the usual (100%) price at each company.
Calculate a percentage of a given value
Divide the percentage by 100, and multiply by the given value.
For example:
15% of ticket sales on the first night of an upcoming production by Portimadie Amateur Dramatics Society (PADS) will be donated to the local playgroup.
If ticket sales amount to £800, what will the playgroup receive?
15 ÷ 100 = 0.15
0.15 x 800 = 120
15% of 800 is 120.
Convert a fraction to a percentage
Divide the numerator by the denominator, and multiply by 100.
For example:
90 out of 120 (90/120) members of Portimadie's bowling club participated in a recent competition.
What percentage of the membership was represented?
90 ÷ 120 = 0.75
0.75 x 100 = 75
75% of the membership was represented.
Calculators: Most mobile phones include a 'Calculator' app, as do Windows and Mac OS operating systems on computers and laptops. There are also online calculators you might find useful, for example Calculator.net.
Click the headings below for more examples of percentages in use.
Food packaging must display nutritional information for 100 grammes of that food product. This helps us to make informed choices about what we purchase and consume, particularly when items are sold in different sizes or weights.
You could be looking at two very different foodstuffs, for example a small packet of biscuits and a large bar of chocolate, but if typical values are provided for 100g of that product, you can quickly make a comparison about which product contains the most sugar or fat.
Chocolate cake
by OpenClipart-Vectors at Pixabay
Remembering that a percentage is a value expressed as a fraction of 100, if 39g out of a 100g of a chocolate cake, as shown in the nutrition information label above, is sugar (39/100), then we can also say that 39% of the chocolate cake is sugar.
So, how do we check how much sugar there is in a large, 120g portion of cake?
Method 1
If 39% of the cake is sugar, our 120g portion also contains 39% sugar. To find 39% of a 120g portion, multiply 120 by 39/100 (39 divided by 100).
120 x 39/100 = 120 x 0.39 = 46.8
Which tells us that a 120g portion of this cake contains 46.8g of sugar.
Method 2
Multiply 120 by 39 (120 x 39 = 4680), and then divide that answer by 100 (4680/100 = 46.80).
Dividing whole numbers by 100
To divide a whole number by 100, imagine there is a decimal point to the right of it and then move this decimal point two places to the left.
For example: 457 ÷ 100 = 4.57
If all the digits after a decimal point are zeros, you can ignore them (and the point).
For example: 3200 ÷ 100 = 32
Similarly, to divide a number by 10, move the decimal point one place to the left.
For example: 237 ÷ 10 = 23.7
Try it out
How much sugar is there in a 75g portion of this cake?
There are 29.25g of sugar in a 75g portion of the cake.
75 x 39/100 = 75 x 0.39 = 29.25
A store is offering a 20% discount on everything. If the item you are purchasing normally costs £37.00, how much will you pay?
Original price: £37.00
Discount: 20%
There are several ways you can work out how much you will pay for the item.
Calculate the 20% discount and subtract this from the original price:
- Convert the percentage discount to a decimal: 20% = 20/100 (20 divided by 100) = 0.2
- Multiply the decimal by the original price: 0.2 x 37.00 = 7.4, so a saving of £7.40 (two decimal places are used with money)
- Subtract this from the original price: 37.00 - 7.40 = 29.60
Discounted price: £29.60
Calculate the 80% discounted price:
Knowing that 20% means 20 out of a 100, then subtracting 20 from 100 tells us that we will pay 80% of the original price (100 - 20 = 80).
- Convert 80% to a decimal: 80% = 80/100 (80 divided by 100) = 0.8
- Multiply this by the original price: 0.8 x 37.00 = 29.60
Discounted price: £29.60
Calculate 10% first and work from this:
To find 10% you simply move the decimal point in the original price, one place to the left.
This method, once you have the hang of it, can be quicker to do in your head. Once you have 10%, then to find 20% you simply double it (10 x 2 = 20). Similarly if you were looking at a 5% discount, this would be half of 10% (10 ÷ 2 = 5).
- Calculate 10% of the price: 37.00 divided by 10 (move the decimal point one position to the left) = 3.70
- Multiply by 2 (double it) to find 20%: 3.70 x 2 = 7.40
- Subtract from the original price: 37.00 - 7.40 = 29.60
Alternatively, multiply 10% of the original price by 8 to find 80%.
- Calculate 10% of the price: 37.00 divided by 10 (move the decimal point one position to the left) = 3.70
- Multiply by 8 to find 80%: 3.70 x 8 = 29.60
Discounted price: £29.60
Try it out
If you book early, you can save 30% on the price of a concert ticket.
Original price: £80.00
Discount: 30%
Calculate 10% of the full price and the use this to work out how much you will pay.
10% of £80.00 is £8.00 (80 ÷ 10 = 8).
Applying a 30% discount leaves 70% to pay (100 - 30 = 70).
70% of £80.00 is £56.00 (£8 x 7 = 56).
£56.00 is the discounted price for early purchase.
When managing your household budget, it is important to set money aside for essential expenditure, such as rent or mortgage, utility bills, such as electricity and gas, and food shopping.
Once your essential expenditure is covered, you can then work out what you have left for other expenditure and savings.
The 50-30-20 rule
The 50-30-20 rule is a budgeting method in which a fixed percentage of household income is allocated to three categories of expenditure as follows:
| % of income | Category | Example expenditure |
|---|---|---|
| 50% | Needs | Essential expenses, such as rent or mortgage payments, utility bills, food and transport. |
| 30% | Wants | Non-essentials such as subscriptions, gym membership, entertainment, eating out. |
| 20% | Savings | Putting money into savings or repaying debts such as credit cards or student loans. |
How helpful this method might be will depend on individual circumstances and income, but let’s look at an example of how it could be applied.
Mairead receives £1700 per month, after tax. She would really like join her friends on a holiday abroad next year but rarely has any money left in her account by the end of the month. One of her friends has suggested she try the 50-30-20 rule to manage her money.
Needs
Mairead notes her essential (Needs) expenditure.
Does the expenditure she has noted under the ‘Needs’ category meet the 50% allowed under the 50-30-20 rule?
| Rent | 350 |
|---|---|
| Travel | 45 |
| Supermarket / weekly shopping | 250 |
| Electricity | 95 |
| Council tax | 80 |
| TV licence | 14 |
| Phone | 16 |
| Total | 850 |
Yes, Mairead’s essential costs do (just) add up to 50% of her income.
50% = ½, so half of Mairead’s income is £1700 ÷ 2 = £850
Wants
Next Mairead looks at her non-essential (Wants) expenditure. Her gym membership and streaming TV services are fixed monthly amounts, but she never keeps an eye on what she spends going out and shopping for clothes.
If she can stick to the rule (30% of income to Wants), this is where she can make some savings.
How much of Mairead’s salary should she allocate to this category?
| Gym | 45 |
|---|---|
| TV streaming subscriptions | 18 |
| Miscellaneous | ? |
Under the rule, Mairead can allocate 30% of her £1700 income to this category.
10% of £1700 is £170. Multiply this by 3 to get 30%, which is £510. Taking into account the fixed expenses, this would leave 447 for all other Wants expenditure.
Savings
The remainder of Mairead’s income can be allocated to the Savings (20%) category.
Mairead currently pays her credit card £120 per month. How much can she allocate to her holiday savings account once she has paid her credit card, under the 50-30-20 rule?
10% of £1700 is £170. Multiply this by 2 to get 20%, which is £340. Subtract the £120 credit card payment which leaves £220 which she can put into a savings account.
If she sticks to her budget, Mairead will have £2640 (220 x 12) in the holiday account in a year's time!
Multiplication and division are covered in more detail in the Operations and calculations book.