Student Support Services
Author: Lynn Ireland, revised by Dave Longstaff
A fraction is the ratio of two integers (whole numbers) eg ,
The number at the top is called the numerator, the number at the bottom is called the denominator.
has a numerator 2 and denominator 5. It is spoken ‘two fifths’.
has a numerator 3 and denominator 7.
It is spoken ‘three sevenths’.
Equivalent Fractions
Here is shaded
Here is shaded
Here is shaded
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In all three diagrams the shaded regions represent the same quantity so that
= =
The fractions are said to be equivalent.
Fractions are equivalent if you can convert one into the other by multiplying (or dividing) the numerator and the denominator by the same number. e.g. is equivalent to since →
is equivalent to since →
is equivalent to since
→
Equivalent fractions are needed when we come to addition and subtraction of fractions.
Mostly the preferred form for a fraction is the simplest i.e. when numerator and denominator have no common factors so that we cannot divide any further. eg can be made simpler by dividing numerator and denominator by 7 to get and we cannot make any simpler.
This simplification is called cancelling.
Cancelling fractions makes them easier to work with particularly when we multiply and divide fractions.
The four operations we need to look at are , +, ­ and ÷ and in that order:-
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Multiplication
We can get a simple rule for this operation as follows:
↑
4
↓
← 3 →
The shaded area in the diagram
Represents 3 x 4 = 12 and if we use the same idea but the sides of the diagram now represent 1 unit we can illustrate
↑
↓
← →
The shaded area in the diagram represents x =
(= after cancelling)
The rule is:- multiply numerators, multiply denominators e.g. i) ii) iii) =
Algebraically =
(= after cancelling)
Addition and subtraction + -
Type I
If two (or more) fractions have the same denominator, then we just add the numerators e.g.
It is easy to see this is the correct method by looking at the following diagram
←
The shaded region is
→ + ← 2/8 →
3

Similarly
Type II
If the fractions have different denominators then it is not possible to use the above method. We do, however, have a technique for changing fractions to equivalent fractions and we use this to convert two fractions with different denominators into two fractions with the same denominator and then just use the Type I method. e.g. i)
If we multiply the numerator and denominator of by 4 i.e.
→ and if we multiply the numerator and denominator of by 5 i.e.
→ we have produced two equivalent fractions whose denominators are the same.
We can simply add:- = ii) +
Multiply numerator and denominator of by 4 and multiply numerator and denominator of by 3 to produce two equivalent fractions:-
→ , and + =
Choosing what we multiply by is not difficult. It is (usually) the denominator of the other fraction. e.g. +
Multiply numerator and denominator of by 5 (the denominator of ) to get
Multiply numerator and denominator of by 7 (the denominator of to get =
Then
Do not forget to cancel your fractions, if possible, to produce the simplest form of your answer. e.g.
4

So,
= and
, which equals , which equals (simplest form)
Subtraction
This is done in exactly the same way except instead of adding we subtract! e.g. (Same denominators, so Type I subtraction)
- (Different denominators so produce equivalent fractions)
→ and → so
Algebraically, + =
2 nd term
This looks a little awkward, but if we look at the answer we can, very quickly, produce it as follows
+
A C
+
B D 3 rd term
1 st term
3
1 1
+
3 4
3 4
12
4
4 + 3
12
7
=
12
5

5
2 1 4 + 5 = 9
5 2 10 10 10
4
Division
To find a method for this operation we proceed as follows:
(Just rewriting the sum)
= (Multiply numerator and denominator by the same number to produce an
equivalent fraction)
x 12
= 2 x 12 4
1
3 (Cancelling)
3 x 12 3
1
4
= = (simplifying)
Before working this out (using the method from multiplying fractions) look at the number we have obtained.
has become i.
e. The first fraction has remained the same, the divide sign ÷ has become times x, and the second fraction has turned upside down.
This happens in general and gives us a simple method for dividing fractions. e.g. i)
ii)
iii)
=
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Algebraically
Summarising i) = ii) iii)
It is best NOT to remember these as formulas but as methods.
Mixed Numbers
If we have a, so called, mixed number to deal with, i.e. a fraction and a whole number e.g. 2 , 4 , it is best to convert the whole number to a top heavy fraction or improper fraction, perform the operation using the above rules and then convert back to a mixed number if necessary. e.g. i) 2 3
2 → and 3 →
So 2 ii) 3
3
(cancelling)
and 2 →
So 3 =
To convert a mixed number to a fraction we multiply the whole number by the denominator of the fraction and then add on the numerator.
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Schematically
A
Again do NOT remember this as a formula but as a method.
Exercise 1
1. For each group of fractions, state which fractions are equivalent: a) 1
2
,
1
4
,
2
4
,
3
4
b)
3
8
,
2
7
,
6
21
,
4
15
c) 4
5
,
9
10
,
2
3
,
12
15
2. Cancel the following fractions down to their simplest form: a)
5
25
b)
36
108
c)
20
64
3. For each of the following pairs of fractions, state which one is the larger: a) 3
4
,
7
8
b) 5
8
,
6
7
c) 12
15
,
3
5
4. Convert the following mixed fractions into improper fractions: a) 5
7
8
b) 6
1
8
c) 2
5
16
5. Convert the following improper fractions into mixed fractions: a) 18
5
b) 26
7
c) 19
3
6. Work out the following (simplify your answer if possible): a) 4× 1
5 b) 5× 8
9 c) 6× 3
12
7. Work out the following (simplify your answer if possible): a) 1
4
× 1
5
b) 3
8
× 1
2
c) 4
5
× 2
3
8. Work out the following divisions (simplify your answer if possible): a)
8
9
÷ 2
3
b)
3
7
÷ 1
2
c) 4
5
÷ 1
5
9. Work out the following divisions (simplify your answer if possible): a)
2
3
÷4 b) 1
2
÷8 c) 5
8
÷6
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For extra help with Fractions consult Mathematics leaflet ‘Fractions’ available on the web at www.hull.ac.uk/studyadvice
1.5, 2.7, 1.333, 12.6 are all decimals.
The decimal point ( .
) is used to distinguish the parts of the number.
Numbers to the left of the decimal point are the normal counting numbers.
Numbers to the right of the decimal point are parts of numbers.
Example
123.456. Here we have 123 and a bit. The bit is 0.456.
Place Value
The value of a number is dependent upon its position.
This is called place value .
Thousands Tens Units
• Tenths Hundredths Thousandths
1
5
6
1 • 0
2
•
5
7 • 9
0
•
0
1
0 4
The table above shows how place value works for decimals.
1.01 has one unit and one hundredth.
2.5 has two units and 5 tenths.
57.9 has five tens, seven units and 9 tenths
160.004 has one hundred, 6 tens, and 4 thousandths
Decimal-Speak
It is usual to say the numbers after the decimal point as individual numbers. For example 4.93 would be said as ‘four point nine three’ not ‘four point ninety three’
Notice that where a number does not have a value for a column, a nought is used.
This preserves the value of the following numbers. In this way 0.2 is different from
0.02 in the same way that 20 is different from 2.
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As with numbers in front of the decimal point, noughts not contained within a number are not usually written. i.e. 5.1 is really 5.1000000000000000000000000… but we can just assume that the following noughts are there.
Multiplying and Dividing by 10
If you multiply a number by ten, its digits will remain the same, but they will move in relation to the decimal point.
Example
12×10 = 120. As 12 is the same as 12.0, all we have done is to move the decimal point one place to the right, so 12.0 becomes 120.
Alternatively you can think of this as the number moving one place to the left.
Whichever you prefer, the end result is the same.
This system works for numbers with decimal places too.
Examples
1.64×10 = 16.4 2.85×10 = 28.5 6.2×10 = 62
0.5×10 = 5 0.67×10 = 6.7 2.05×10 = 20.5
Notice that in the last example, the nought is treated in the same way as all other digits.
When you are on a ward or in a clinic, you may be asked to measure doses of medication. For these calculations, a sound grasp of place value is essential, as 0.1 grams is 100 times the amount of a medicine that 0.001 grams would be.
Multiplying by 100, 1000 etc is performed in a similar way to multiplying by 10.
Example
We have seen that 2.85×10 = 28.5
Multiplying by 10 again gives 28.5×10 = 285
As 100 = 10×10, multiplying by 100 is exactly the same as multiplying by 10, then multiplying the result by 10.
10

Each time we multiply by 10 we move the decimal point one place to the right.
Multiplying by 100 moves the decimal point one place to the right twice, so the overall effect is to move the decimal point two places to the right.
So, looking at the example again:
2.85×10 = 28.5 28.5×10 =285
2.85×100 = 285
More examples
5.6×100 = 560 4.35×100 = 435 3.509×100=350.9
The most common multiplication of this type you will have to do will be multiplication by 1000.
As 1000 = 10×10×10, we can look at multiplying by 1000 as multiplying by 10 three times in succession.
Looking at our example:
2.85×10 = 28.5 28.5×10 = 285 285×10 = 2850
2.85×1000 = 2850
The overall effect of multiplying by 1000 is to move the decimal point three places to the right.
Division
As division by 10 is the inverse process to multiplication by 10, we simply apply the same processes but in reverse.
To multiply by 10, we move the decimal point one place to the right.
To divide by 10 we move the decimal point one place to the left.
Examples
12

10= 1.2 143

10= 14.3 2.85

10= 0.285
In the same way we can divide by 100 and 1000.
Examples
12

100 =0.12 143

100= 1.43 2.85

100= 0.0285
11

Note- when dividing by 10, 100, 1000 etc, it may be useful to write some noughts in front of you r number so that you don’t lose track. i.e. 5.1

1000 = 0005.1

1000
0005.1

10 = 000.51
000.51

10 = 00.051
00.051

10 = 0.0051, so
5.1

1000 = 0.0051
Dividing by numbers smaller than 1
Sometimes you may be asked to divide by numbers smaller than one.
Example
Evaluate 12÷0.1. Essentially this is asking us how many 0.1s are in 12.
The first thing that we do is note that 0.1 is one tenth.
We know from our work on fractions that there are ten tenths in a unit.
We have 12 of these units.
Hence our question can be changed to:
Evaluate 12×10=120
When we divide a value by a number less than 1, our answer will be larger than the value we started with.
Exercise 2
1. Express the following in terms of hundreds, tens, units, tenths etc: a) 125.9 b) 87.03 c) 102.065
2. Write these numbers in figures: a) One unit, six tenths and one thousandth b) Five tens and five tenths c) Three hundreds, six units, nine hundredths and one thousandth
3. Evaluate the following: a) 18 × 10 b) 1.4 × 10 c) 0.02 × 10 d) 26.8 × 100 e) 2.09 × 100 f) 3.94 × 100
12

g) 2.1 × 1000 h) 12.9 × 1000 i) 1.08 × 1000
4. Evaluate the following: a) 18

10 b) 1.4

10 c) 0.02

10 d) 26.8

100 e) 2.09

100 f) 3.94

100 g) 2.1

1000 h) 12.9

1000 i) 1.08

1000
5. Copy the procedure below to answer the following questions:
The question asks for 6.3÷0.01. I am dividing by 0.01 .
0.01 is one hundredth, so there are 100 of them in 1 unit.
I have 6.3 units, so I must have 6.3 × 100 hundredths.
The question
6.3÷0.01 is equivalent to 6.3 × 100
6.3 × 100 = 630, so 6.3÷0.01 = 630 a) 2.9 ÷ 0.1 b) 32 ÷ 0.001 c) 0.48 ÷ 0.01
For extra help with this section consult the Mathematics leaflet ‘Powers of 10…’ available on the web at www.hull.ac.uk/studyadvice
Answers to exercises
Exercise 1
1.
2.
a) a) 1
5
1
2
,
2
4 b) 2
7
,
6
21 c) 4
5
,
12
15 b) 1
3 c)
5
16
3.
4.
a) a)
7
8
47
8
5.
a) 3
3
5 b) b) 3
6
7 b) 49
8
5
7 c) 12
15 c) 37
16 c) 6
1
3
6 .
7.
8.
9. a) 4
5 a)
1
20 a) a)
4
3
1
6 b) b) b) b)
6
7
40
9
3
16
1
16 c) c) c) c)
3
2
8
15
4
5
48
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Exercise 2
1. a) one hundred, two tens, five units and nine tenths
b) eight tens, seven units, and three hundredths
c) one hundred, two units, six hundredths and five
thousandths.
2. a) 1.601
3. a) 180 d) 2680 g) 2100
4. a) 1.8 d) 0.268 g) 0.0021 b)50.5 b) 14 e) 209 h) 12900 b) 0.14 e) 0.0209 h) 0.0129
5. a) 2.9 ÷ 0.1= 2.9 x 10 = 29 b) 32 ÷ 0.001 = 32 x 1000 = 32 000 c
) 0.48 ÷ 0.01 = 0.48 x 100 = 48 c) 306.091 c) 0.2 f) 394 i) 1080 c) 0.002 f) 0.0394 i) 0.00108
The Department of Nursing and Midwifery
Dr Bunnell provides support for students with their mathematics.
To contact Dr Bunnell email T.bunnell@hull.ac.uk
Disclaimer
Please note that the author of this document has no nursing or medical experience.
The topics in this leaflet are dealt with in a mathematical context rather than a medical one.
© 04/2008
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