# Fractions, Decimals and Percentages

Pizzas are very useful, mathematically speaking. However much we hate fractions, we all know what half a pizza looks like, and that’s the point. Numbers don’t have any intrinsic meaning, and we can’t picture them unless they relate to something in the real world, so pizzas are just a useful way of illustrating fractions, decimals and percentages.

They all do the same job of showing what share of something you have, and a common question involves converting from one to another, so here are a few tips…

## Fractions to Decimals

### Calculator

• Simply divide the numerator by the denominator, eg 3/4 = 3 ÷ 4 = 0.75.

### Non-calculator

You can always use the standard ‘bus stop’ method to divide the numerator by the denominator on paper (or in your head), but the numbers may be easy enough for you to use a shortcut.

• If the denominator is a power of 10 (eg 10 or 100), write the numerator down straight away as a decimal. You just have to make sure you end up with the digits in the right columns, eg a fraction involving hundredths needs to end in the second column after the decimal point, so 29/100 = 0.29.
• If the denominator ends in zero, you may be able to simplify the fraction into tenths first and then convert it into a decimal, eg 16/20 = 8/10 = 0.8.
• If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert, eg 36/45 = 4/5 = 0.8. Just make sure that you learn all these by heart, especially the eighths!

#### Quiz

1. What is 5/10 as a decimal?
2. What is 8/40 as a decimal?
3. What is 36/60 as a decimal?
4. What is 27/36 as a decimal?
5. What is 77/88 as a decimal?

## Fractions to Percentages

### Calculator

• Simply divide the numerator by the denominator, multiply by 100 and add the ‘%’ sign, eg 3/4 = 3 ÷ 4 x 100 = 0.75 x 100 = 75%.

### Non-calculator

You can always convert the fraction into a decimal (see above) and then multiply by 100 and add the ‘%’ sign. Otherwise, try these short cuts in order.

• If the denominator is a factor of 100 (eg 10, 20, 25 or 50), multiply the numerator by whatever number will turn the denominator into 100 and add the ‘%’ sign, eg 18/25 = 18 x 4 = 72%.
• If the denominator is a multiple of 10 (eg 30, 40 or 70), divide the numerator by the first digit(s) of the denominator to turn the fraction into tenths, multiply the numerator by 10 and add the ‘%’ sign, eg 32/80 = 32 ÷ 8 x 10 = 4 x 10 = 40%.
• If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert from memory, eg 8/64 = 1/8 = 12.5%.

#### Quiz

1. What is 4/10 as a percentage?
2. What is 6/20 as a percentage?
3. What is 24/40 as a percentage?
4. What is 14/70 as a percentage?
5. What is 40/64 as a percentage?

## Decimals to Fractions

Every decimal is really a fraction in disguise, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Check the final column of the decimal (eg tenths or hundredths) and place all the digits over the relevant power of 10 (eg 100 or 1000) before simplifying if necessary, eg 0.625 = 625/1000 = 5/8.

#### Quiz

1. What is 0.4 as a fraction?
2. What is 0.25 as a fraction?
3. What is 0.24 as a fraction?
4. What is 0.875 as a fraction?
5. What is 0.375 as a fraction?

## Decimals to Percentages

Again, this is an easy one, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Multiply by 100 and add the ‘%’ sign, eg 0.375 x 100 = 37.5%.

#### Quiz

1. What is 0.27 as a percentage?
2. What is 0.1 as a percentage?
3. What is 0.55 as a percentage?
4. What is 0.001 as a percentage?
5. What is 1.5 as a percentage?

## Percentages to Fractions

You can think of a percentage as simply a fraction over 100, so the method is easy enough whether you’re allowed a calculator or not.

### Calculator/non-calculator

• If the percentage is a whole number, remove the ‘%’ sign, place the percentage over 100 and simplify if necessary, eg 75% = 75/100 = 3/4.
• If not, turn the fraction into a whole number as you go by multiplying the numerator and denominator by whatever number you need to (usually 2, 3 or 4), eg 37.5% = (37.5 x 2) / (100 x 2) = 75/200 = 3/8.

#### Quiz

1. What is 22% as a fraction?
2. What is 15% as a fraction?
3. What is 37.5% as a fraction?
4. What is 87.5% as a fraction?
5. What is 6.25% as a fraction?

## Percentages to Decimals

This is easy enough, so the method is the same whether you’re allowed a calculator or not.

### Calculator/non-calculator

• Remove the ‘%’ sign and divide by 100, eg 70% ÷ 100 = 0.7.

#### Quiz

1. What is 40% as a decimal?
2. What is 70% as a decimal?
3. What is 35% as a decimal?
4. What is 45.5% as a decimal?
5. What is 62.1% as a decimal?

## Ordering Fractions, Decimals and Percentages

A common question in the 11+ or 13+ involves putting a list of fractions, decimals and/or percentages in size order—either from largest to smallest or smallest to largest.

There are a number of ways of doing this, and it depends what kind of numbers are involved. However, a good first step is to start with the first two numbers and ask yourself if one is ‘obviously’ bigger than another. For instance, it might be quite difficult to compare 1/17 and 18/19 by converting them to fractions with the same denominator, but you don’t have to because 1/17 is clearly smaller!

After that, you can look at each number one by one and work out where it fits in your list. To keep track of everything, it’s a good idea to put numbers in pencil next to each value. Once you have the final order, you can write them all down on the answer line.

One simple question you can always ask yourself is whether the two fractions, decimals or percentages are smaller or larger than a half. If one is smaller but the other is larger, then the answer’s obvious.

If that doesn’t work, here are a few more ways to do it.

### Ordering Fractions

If two fractions have the same denominator, the larger one will be the one with the larger numerator, eg 2/3 is bigger than 1/3.

If the fractions have different denominators, turn them into fractions with the same denominator and then compare the numerators, eg 5/6 and 7/8 are the same as 40/48 and 42/48, so 7/8 must be larger.

#### Quiz

1. Put these numbers in order from largest to smallest: 1/2, 1/4, 2/5, 4/7, 5/8
2. Put these numbers in order from largest to smallest: 3/4, 1/8, 5/6, 4/9, 3/8
3. Put these numbers in order from largest to smallest: 4/5, 1/9, 3/4, 7/8, 1/4
4. Put these numbers in order from largest to smallest: 1/3, 3/4, 2/3, 1/8, 5/6
5. Put these numbers in order from largest to smallest: 2/5, 1/2, 2/3, 4/5, 3/4

### Ordering Decimals

Decimals are easy to sort. It’s a bit like putting words in alphabetical order:

• Start with the first digit after the decimal point, which is the number of tenths. The number with the bigger first digit is bigger overall, eg 0.2 is bigger than 0.1.
• If the numbers have the same number of tenths, compare the hundredths, eg 0.12 is bigger than 0.11.
• Repeat until you find the first digit that’s different. Just remember that if one number ends before you get a different number, it will always be smaller, eg 0.45 is smaller than 0.456.

#### Quiz

1. Put these numbers in order from smallest to largest: 0.2, 0.3, 0.11, 0.2, 0.33
2. Put these numbers in order from smallest to largest: 0.8, 0.6, 0.55, 0.5, 0.555
3. Put these numbers in order from smallest to largest: 0.9, 0.4, 0.8, 0.11, 0.1
4. Put these numbers in order from smallest to largest: 0.13, 0.103, 0.301, 0.013
5. Put these numbers in order from smallest to largest: 0.4444, 0.44444, 0.444, 0.44, 0.4

### Ordering Percentages

Percentages are also easy to sort as they’re just values that you can put in numerical order, eg 35% is bigger than 17% because 35 is bigger than 17.

#### Quiz

1. Put these numbers in order from largest to smallest: 25%, 12%, 80%, 100%, 4%
2. Put these numbers in order from largest to smallest: 13%, 103%, 31%, 30%, 30.1%
3. Put these numbers in order from largest to smallest: 2%, 222%, 22%, 2.2%, 2.22%
4. Put these numbers in order from largest to smallest: 24%, 4%, 4.4%, 80%, 42%
5. Put these numbers in order from largest to smallest: 14%, 71%, 3.5%, 5.3%, 4%

### Ordering a Mixture

This is where it gets tricky. There’s no single way of comparing fractions, decimals and percentages, so once you’ve numbered the values that are ‘obviously’ bigger and smaller, you’ll have to convert the others into the most common form, eg if there are three fractions, two decimals and a percentage, turn them all into fractions.

This usually saves time, but look out for ‘awkward’ numbers that you can’t easily turn into a different format, eg 0.618 is impossible to turn into a common fraction, and the number π is an ‘irrational number’ that can’t be converted into anything else!

#### Quiz

1. Put these numbers in order from smallest to largest: 0.2, 11%, 25%, 1/4, 3/8
2. Put these numbers in order from smallest to largest: 99.9%, 0.9, 7/8, 8/9, 0.99
3. Put these numbers in order from smallest to largest: 0.8, 4/5, 5/6, 81%, 90%
4. Put these numbers in order from smallest to largest: 0.5, 55%, 4/5, 7/8, 77%
5. Put these numbers in order from smallest to largest: 77%, 0.7, 3/4, 2/3, π

# Shortcuts It’s just through here…

There is always more than one way of solving a Maths problem. That can be confusing, but it can also be an opportunity – if only you can find the right trade-off between speed and accuracy.

I’ve taught a lot of QTS numeracy candidates recently, and the Maths itself isn’t particularly difficult, particularly in the mental arithmetic section.

The trick is to be familiar with all the possible short cuts and capable of using the right one at the right time. It may mean having to do more sums, but it will be much simpler and quicker in the long run. You don’t have to use all of these all the time, but it is useful to know what they are just in case you need them.

• Multiplying and dividing by 5
The most useful short cut I’ve come across is very simple. To multiply by 5, try multiplying by 10 and then dividing by 2 (or vice versa), eg
13 x 5
= 13 x 10 ÷ 2
= 130 ÷ 2
= 65
You have to do two sums rather than one, but the point is that you should be able to save time and improve the chances of getting the answer right by doing both in your head rather than having to work out a more difficult sum on paper.
You can do divide by 5 in a similar way by multiplying by 2 and dividing by 10 (or vice versa), eg
65 ÷ 5
= 65 x 2 ÷ 10
= 130 ÷ 10
= 13
You can do a similar trick with 50, 500 etc simply by multiplying or dividing by a higher power of ten.
• Chunking
If you have to multiply by a two-digit number outside your times tables, chunking is an easy way to do the sum in your head. Instead of writing it down on paper and using long multiplication (which would take a long time and is easy to get wrong!), try multiplying by the tens and the units separately and adding up the results, eg 16 x 15 = 10 x 15 + 6 x 15 = 150 + 90 = 240. The numbers might still be too tricky to do it comfortably, but it’s often worth a try.
• Rounding
To avoid sums with ‘tricky’ numbers, try rounding them up to the nearest ‘easy’ figure and adjusting at the end. This is particularly useful when working out start and end times, eg
‘The morning session in a school began at 09:25. There were three lessons of 50
minutes each and one break of 20 minutes. At what time did the morning session end? Give your answer using the 24-hour clock.’
If you assume the lessons last an hour, you can add three hours to 09:25 to get 12:25. You would normally then knock off 3 x 10 = 30 minutes, but the 20-minute break means you only need to subtract 10 minutes, which means the session ended at 12:15.
• Money problems
There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it – 1 x 2 x 3 is just the same as 3 x 2 x 1. The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning or turn multiplication by a fraction of a pound into a division sum, eg
‘All 30 pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get an average of 18 spellings correct each. The average amount of sponsorship was 20 pence for each correct spelling. How many pounds would the class expect to raise for charity?’
The basic sum is 30 x 18 x 20p, and there are a couple of ways you could do this:
1) Knock off the zeroes in two of the numbers, change the order of the numbers to make it easier and double and halve the last pair to give yourself a sum in your times tables, ie
30 x 18 x 20p
= 3 x 18 x 2
= 3 x 2 x 18
= 6 x 18
= 12 x 9
= £108
2) Convert pence into pounds, turn it into a fraction, change the order of the numbers, divide by the denominator and, again, double and halve the last pair to give yourself a sum in your times tables, ie
30 x 18 x 20p
= 30 x 18 x £0.20
= 30 x 18 x ⅕
= 30 x 18 ÷ 5
= 30 ÷ 5 x 18
= 6 x 18
= 12 x 9
= £108
• Percentages
Many students get intimidated by percentages, fractions and decimals, but they are all just different ways of showing what share you have of something. You will often by asked to add or subtract a certain percentage. The percentage will usually end in zero (eg 20%, 30% or 40%), so the easiest way is probably to find 10% first. That just means dividing by 10, which means moving the decimal point one place to the left or, if you can, knocking off a zero. Once you know what 10% is, you can simply multiply by 2, 3 or 4 etc and add or subtract that number to find the answer, eg
‘As part of the numeracy work in a lesson, pupils were asked to stretch a spring to extend its length by 40 per cent. The original length of the spring was 45 centimetres. What should be the length of the extended spring? Give your answer in centimetres.’
You need to find 40% of 45cm, so you can start by finding 10%, which is 45 ÷ 10 or 4.5cm. You can then multiply it by 4 to find 40%, which is best done by doubling twice, ie 4.5 x 2 x 2 = 9 x 2 = 18. Finally, you just add 18cm to the original length of the spring to find the answer, which is 45 + 18 = 63cm.
• Common fractions
An awful lot of questions involve converting between fractions, percentages and decimals. There is a proper technique for doing any of those, but it’s very useful if you learn the most common fractions and their decimal and percentage equivalents by heart, eg
½ = 0.5 = 50%
¼ = 0.25 = 25%
¾ = 0.75 = 75%
⅕ = 0.2 = 20%
⅖ = 0.4 = 40%
⅗ = 0.6 = 60%
⅘ = 0.8 = 80%
⅛ = 0.125 = 12.5%
⅜ = 0.375 = 37.5%
⅝ = 0.625 = 62.5%
⅞ = 0.875 = 87.5%
• Times tables
3 x 24
= (3 x 2) x (24 ÷ 2)
= 6 x 12
= 72
Alternatively, you can halve just one of the numbers and double the result, eg
24 x 9
= 12 x 9 x 2
= 108 x 2
= 216
• Multiplying by 4
If you have to multiply by 4 and the number is not in your times tables, a simple way to do it is to double it twice, eg
26 x 4
= 26 x 2 x 2
= 52 x 2
= 104
• Multiplying by a multiple of 10
If you have to multiply by a multiple of 10 such as 20 or 30, try knocking the zero off and adding it in again afterwards. That way, you don’t have to do any long multiplication and, with any luck, the sum will be in your times tables, eg
12 x 30
= 12 x 3 x 10
= 36 x 10
= 360
• Multiplying decimals
This can be a bit confusing, so the best way of doing it is probably to ignore any decimal points, multiply the numbers together and then add back the decimal point to the answer so that you end up with the same number of decimal places as you had in the beginning, eg
0.5 x 0.5
= 5 x 5 ÷ 100
= 25 ÷ 100
= 0.25
• Using the online calculator
The second section of the QTS numeracy test consists of on-screen questions that can be answered using an online calculator. This obviously makes working out the answer a lot easier, and short cuts are therefore less useful. However, just because the calculator’s there doesn’t mean you have to use it, particularly for multiple-choice questions. If you have to add up a column of cash values, for example, and compare it with a number of options, you could simply tot up the number of pence and pick the option with the right amount. Alternatively, the level of accuracy needed in the answer may give you a helping hand if it rules out all but one of the possible answers, eg 6 ÷ 21 to one decimal place is always going to be 0.3. Why? Well, it’s a bit less than 7 ÷ 21, which would be a third or 0.3 recurring. An answer of 0.4 would be more than that, and 0.2 would be a fifth, which is far too small, so it must be 0.3.
• Don’t do more than you have to!
There are several types of question that could tempt you into doing more work than you need to do. If you’re trying to work out how many tables you need at a wedding reception for a given number of guests, the answer is always going to need rounding up to the next whole number, so you don’t need to spend any time working out the exact answer to one or two decimal places. Equally, some numbers are so close to being an ‘easy’ number that you don’t need to add or subtract anything after rounding up or down to make the basic sum easier, eg
‘For a science experiment a teacher needed 95 cubic centimetres of vinegar for each pupil. There were 20 pupils in the class. Vinegar comes in 1000 cubic centimetre
bottles. How many bottles of vinegar were needed?
If you round 95cc to 100cc, the answer is 20 x 100 ÷ 1000 or 2 bottles, and the remainder consisting of 20 lots of 5cc of vinegar can safely be ignored.