Study Advice Service
Student Support Services
Mathematics Practice For
Nursing and Midwifery
This leaflet contains theory, examples and exercises on the topics:
• Fractions
• Decimals
• Ratio
• Percentage
• Using Formulae
• Unit Conversion
• Dosage Calculation
There are often different ways of doing things in Mathematics and the methods suggested in the
worksheets may not be the ones you were taught. If you are successful and happy with the
methods you use it may not be necessary for you to change them. If you have problems or need
help in any part of the work then there are a number of ways you can get help.
For students at the University of Hull
• Ask your lecturers
•
•
•
You can contact a Mathematics Tutor from the Study Advice Service on the ground floor of
the Brynmor Jones Library if you are in Hull or in the Keith Donaldson Library if you are in
Scarborough; you can also contact us by email
Come to a Drop-In session organised for your department
Look at one of the many textbooks in the library.
For others
• Ask your lecturers
• Access your Study Advice or Maths Help Service
• Use any other facilities that may be available.
If you do find anything you may think is incorrect (in the text or answers) or want further help please
contact us by email.
Tel:
01482 466199
Web: www.hull.ac.uk/studyadvice
Email: [email protected]
Where can you find help for your mathematics?
There are many sources of mathematics help available to you.
The Study Advice Services desk is located in the Brynmor Jones Library on the Hull
campus. Turn left after entering via the turnstiles and you will find our desk. In Scarborough we
are based in room C17b, and appointments are made via the Keith Donaldson Library. Help is
available from the Study Advice Services in two main ways:
• A large number of leaflets containing theory, examples and exercises (with answers) on
different mathematical topics. These can be collected from the desk or downloaded from
the internet via the Study Advice Services website.
• Individual appointments with the Mathematics Tutor. These last 30 minutes, and provide
help with your specific difficulty. If your colleagues also have the same problem as you, it
may be possible to book a room in order to hold a workshop on that topic.
For Scarborough students, mathematics support is available by email or by a pre-arranged
telephone link with the Hull tutor. Mathematics leaflets are available in the corridor opposite the
library entrance, in the Library Quiet Room and on the Study Advice Services website.
The Internet
There are many useful sites out there to explain and help you practise mathematics.
Some of the best are:
• S-cool www.s-cool.co.uk/
This site offers revision on all school subjects up to and including A level.
• BBC AS-Guru www.bbc.co.uk/schools/
This links to GCSE bite-size and AS-Guru.
• Nursing Standard www.nursing-standard.co.uk
This site contains help for drugs calculations as part of its Resources section.
The School of Nursing
Dr Bunnell provides support for students with their mathematics.
To contact Dr Bunnell, email on [email protected]
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.
Version 1 of this leaflet produced Sept-Nov 2002, by L.Towse.
If you find any errors in this document, please tell me via email.
A4 version adapted from Version 1, Jan 03.
Updated June 2004 by L. Ireland
2
1. Fractions
1
1
is a fraction of a whole. The bottom number, the denominator tells us how many equal
2
2
parts the whole has been split into.
1
The top number, the numerator, tells us how many of these parts we have.
2
In this diagram, the whole is split into 8 equal parts. 4 of these parts are
4
shaded. This represents .
8
In this diagram, the whole has been split into 4 equal parts. 2 of these are
2
shaded. This represents .
4
The greater the number of pieces a whole is cut into, the smaller each piece is.
1
1
1
is smaller than , but is larger than
9
8
10
e.g.
Fraction-speak - 12 is one ‘half’, 32 is three ‘halves’, 1 is one ‘third’, 1 is one ‘quarter’, 1 is one
3
4
5
‘fifth’
Equivalent Fractions
When two fractions represent the same quantity, they are said to be
equivalent fractions.
(4 )
In the diagram above, you can see that, 4 is the same as 2 . This is because a quarter, 1 , is
(8 )
8
4
twice as big as an eighth 1 .
So 2 is equivalent to 4 .
4
8
Two fractions are equivalent if you can produce one from the other by multiplying the top and
bottom by the same number.
Example
×2
×2
×2
×2
⎯→ 4 ⎯⎯
⎯→ 8
2 ⎯⎯
4 ⎯⎯
⎯→ 8 ⎯⎯
⎯→ 16
×3
⎯→ 3
1 ⎯⎯
3 ⎯⎯
⎯→ 9
×3
In general, the preferred form for a fraction is the simplest, where the numerator and
denominator have no common factor other than 1. Dividing the top and bottom of a fraction by
the same number is called cancelling down.
3
Example
Cancel down 8 .
24
Notice that both the top and bottom numbers are divisible by 8. Dividing top and bottom by 8 we
get 8 = 1 .
24
3
Note we could have divided through by 4 instead of 8, this would have given us 8 = 2 . This is
24
6
valid, but 2 and 6 have a common factor of 2. So we need to repeat the process, dividing
through by 2 this time.
So, 8 = 2 = 1 as before.
24
6
3
Cancelling down fractions makes them easier to work with-for example it is simpler to work out
1 of a quantity than 8 .
3
24
Adding Fractions
1
1
is a different type of creature to .
3
4
You cannot add fractions with different denominators together as they stand.
However, if you convert them into fractions with the same denominator, you can.
The key to adding fractions is to notice that
Examples
1.What is 1 + 1 ?
3
4
We need to get both fractions over the same denominator.
We do this by finding the smallest common multiple of the two denominators, ie the smallest
nubmer that they both divide into exactly.
In this case, we have 3 and 4 and their smallest common multiple is 12.
We need these fractions to be put into the form A , where A is a value we have to find.
12
To make 3 into 12 we multiply by 4. To ensure that the fraction stays the same size, any
multiplication of it must happen to both the numerator and the denominator.
Hence we must multiply both top and bottom by 4.
4 × 1 = 4 (We are performing the opposite operation to ‘cancelling down’.)
4
3
12
To make 4 into 12 we multiply by 3. Again, we must also multiply the numerator by the same
amount.
3×1 = 3
3
4
12
So we now have as our question
What is 4 + 3 ? Our answer is 4 + 3 = 7
12
12
12
12
12
2. What is 2 + 4 ?
3
5
The lowest common multiple of 3 and 5 is 15.
So we calculate 5 × 2 = 10 and 3 × 4 = 12
5
10
15
+
12
15
=
3
15
3
5
15
22
15
4
3. What is 1 + 3 ?
10
2
Here we have denominators of 10 and 2. As 2 is a factor of 10, we can leave the first fraction as
it stands and only alter the second one, converting it into tenths.
5
3
15
15
16
8
1
5 × 2 = 10 . So we have 10 + 10 = 10 = 5
Note: Subtracting fractions works in exactly the same way, except that the last step is a
subtraction, not an addition.
Relative Size of Fractions
We have seen that the larger the denominator the smaller the pieces are that the whole has
been cut into. So we know that a quarter is smaller than a third. But if the numerator is not equal
to 1, it is harder to tell the difference in size.
To get around this problem, we put the fractions we are comparing over the same
denominator, like we do for adding fractions.
Then the difference should be clear.
Examples
1. Which is larger 2 or 4 ? It is not immediately obvious.
3
5
In example 2 above, we found that 2 = 10 and 4 = 12 .
3
15
5
15
Since the second fraction has more fifteenths than the first, it is the larger.
2. Which is larger 3 or 4 ?
8
9
The smallest common multiple of 8 and 9 is 8×9=72.
So 9 × 3 = 27 and 8 × 4 = 32 , and we can see that 4 is larger than 3
9
8
72
8
9
72
9
8
Mixed Fractions
3
Mixed fractions are a combination of a whole number and a fraction, such as 2 .
4
This represents 2 wholes and 3 quarters of a whole.
As each whole can be split into 4 equal pieces, we have a total of 11
quarters.
In other words:
4 ⎞ 3 8 3 11
3 ⎛
2 = ⎜2× ⎟ + = + =
4⎠ 4 4 4 4
4 ⎝
(Equivalently the process is as follows: multiply the whole number by the denominator and then
add this result to the numerator)
11
4
is called an improper or top-heavy fraction, because the numerator is greater in value than
the denominator.
5
We can also convert improper fractions into mixed fractions by the following method:
We want to convert 26 .
12
As 26=12+12+2 we can split the fraction up: 26 = 12 + 12 + 2 .
12
12
12
We know that
26
12
12
12
12
= 1 , so we have
= 1+1+ 2 = 2 + 2 = 2 2 .
12
12
12
As 2 and 12 have a common factor of 2, we can cancel this fraction down to 1 , leaving us with
6
21
6
(You could also work this out by dividing the numerator by the denominator- the whole number
result becomes the number of wholes and the remainder becomes the fractional part)
Multiplying fractions
To multiply a fraction by a whole number, you just multiply the numerator by the whole number
and leave the denominator alone.
Example
What is 3× 2 ?
5
We multiply the numerator 3×2=6 and leave the denominator alone.
Our result is 6 .
5
(If we multiplied both numerator and denominator by 3 we would get 6 which is equivalent to
15
2
5
and so is not 3 times the size of 2 )
5
To multiply a fraction by another fraction, we multiply the numerators together and the
denominators together.
Example
Multiply 2 by 3
5
7
We perform two calculations. 2×3=6 and 5×7=35
Our result is 6
35
Dividing fractions
How would we work out 2 ÷ 3 ?
5
7
To work this out we use a trick. Using the fact that dividing by a number (such as 3) is the same
as multiplying by 1 over that number (such as 1 ), we can turn our division into a multiplication:
3
2
5
÷ 3 is the same as 2 × 7
7
5
3
We can then continue as before:
2×7=14, 5×3=15
So 2 ÷ 3 = 14
5
7
15
6
In general the procedure is change the division sign to a multiplication sign, turn the second
fraction upside down and then multiply.
This method does not work immediately for mixed fractions, which need to be converted back
into improper fractions first.
Note that this procedure works every time you want to divide by a fraction, whether or not the
first value is in the form of a fraction, a decimal or a whole number.
To divide a fraction by a whole number, we can follow the same procedure as above, only this
time our second fraction will consist of our whole number over one.
For example to divide by 3 is the same as to divide by 3 .
1
Example
3
÷4 = 3 ÷ 4 = 3× 1 = 3
7
7
1
7
4
28
Exercise 1
1. For each group of fractions, state which fractions are equivalent.
b) 3 , 2 , 6 , 4
c) 4 , 9 , 2 , 12
a) 1 , 1 , 2 , 3
2 4 4 4
8 7 21 15
5 10 3 15
2. Cancel the following fractions down to their simplest form:
b) 36
c) 20
a) 5
25
108
64
3. For each of the following pairs of fractions, state which one is the larger:
b) 5 , 6
c) 12 , 3
a) 3 , 7
4 8
15 5
8 7
4. Convert the following mixed fractions into improper fractions:
b) 6 1
c) 2 5
a) 5 7
8
16
8
5. Convert the following improper fractions into mixed fractions:
b) 26
c) 19
a) 18
5
7
3
6. Work out the following (simplify your answer if possible):
a) 4× 1
b) 5× 8
c) 6× 3
5
9
12
7. Work out the following (simplify your answer if possible):
b) 3 × 1
c) 4 × 2
a) 1 × 1
4
5
8
2
5
3
8. Work out the following divisions (simplify your answer if possible):
b) 3 ÷ 1
c) 4 ÷ 1
a) 8 ÷ 2
9
3
7
2
5
5
9. Work out the following divisions (simplify your answer if possible):
b) 1 ÷8
c) 5 ÷6
a) 2 ÷4
3
2
8
7
For extra help with Fractions please consult Mathematics leaflet ‘Fractions’
• 2. Decimals
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
•
0
1
2
•
5
5
7
•
9
1
6
0
•
0
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.
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 5.10000000000000000000… but we can just assume that the following noughts
are there.
Multiplying by 10, 100, 1000
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
8
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.
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
2.85×1000 = 2850
285×10 = 2850
The overall effect of multiplying by 1000 is to move the decimal point three places to the right.
Division by 10, 100, 1000
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
Note- when dividing by 10, 100, 1000 etc, it may be useful to write some noughts in front of your
number so that you don’t lose track.
9
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 you divide by a number less than 1, your 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
d) 26.8 × 100
e) 2.09 × 100
g) 2.1 × 1000
h) 12.9 × 1000
4. Evaluate the following:
a) 18 ÷ 10
b) 1.4 ÷ 10
d) 26.8 ÷ 100
e) 2.09 ÷ 100
g) 2.1 ÷ 1000
h) 12.9 ÷ 1000
c) 0.02 ×10
f) 3.94 × 100
i) 1.08 × 1000
c) 0.02 ÷ 10
f) 3.94 ÷ 100
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 please consult Mathematics leaflet ‘Powers of 10…’
10
• 3. Ratio
Ratio describes the relationship between two quantities.
Here we have 3 grey squares and 2 white squares.
We can say that the ratio of grey squares to white squares is 3 to 2.
This is usually written 3:2 where the colon replaces the ‘to’.
3:2 means that for every 3 items of the first type we have 2 items of the second.
Similarly the ratio of white squares to grey squares is 2:3.
In this diagram, we have 16 grey squares and 8 white squares.
The ratio of grey squares to white squares is 16:8.
However, as can be seen from the diagram, in each row we have 4 grey
squares for every 2 white squares.
This means that a ratio of 16:8 is the same as a ratio of 4:2.
We have cancelled down the ratio by dividing both sides by a common factor (in this case 4).
Looking at the ratio 4:2, we can see that 4 and 2 have a common factor of 2. This means that
the ratio can be cancelled down further (as we did with fractions earlier).
So ‘for every 16 grey we have 8 white’ becomes:
The ratio of grey to white is 16:8
This is the same as 4:2
Which is the same as 2:1.
So the ratio of grey to white, is 2:1.
Using Ratios
Examples
1. A chocolate cake recipe requires the ratio of cocoa to flour to be 1:3. You have measured out
2 ounces of cocoa.
How much flour do you need?
Here we have that 1 part of cocoa is 2 ounces.
We need 3 parts of flour.
So we need to measure out 3 × 2 = 6
6 ounces of flour.
2. Solution X is made from the contents of bottles A and B at a ratio of 3:2. We have already
measured out 600ml of A.
How many mls of B are required to make up X?
3:2 means that for every 3 parts of A we need 2 parts of B.
We have 600ml of A. This is the same as 3 parts of 200ml each.
To make up the solution we need 2 parts of B. So we need 2 x 200ml = 400ml.
11
Ratios can also be linked to fractions.
Examples
1. The ratio of drug A to water in a solution is 1:4.
This means that for every part of A we need four parts of water.
Alternatively, it means that for every 5 parts of the solution, 1 is A and 4 are water.
So, 1 of the solution is A.
5
2. The ratio of A to B in a solution is 3:4.
This means that for every 3 parts of A there are 4 parts of B.
It also means that out of every 7 parts, 3 are A and 4 are B.
So, 3 of the solution is A and 4 is B.
7
7
Note Some drugs may be labelled by ratios of milligrams to millilitres; in these situations the
units are not the same on both sides. Always check labels carefully.
Also 10mg per ml may be written 10mg/ml.
Exercise 3
1. For the following diagrams, state i) the ratio of shaded to unshaded; ii) the ratio of unshaded
to shaded:
a)
b)
c)
d)
If possible cancel the ratios down to their simplest form.
2. Draw diagrams to represent the following ratios:
a) 1:3
b) 3:5
c) 6:7
3. Write the following ratios in their simplest forms
a) 12:8
b) 5:15
c) 28:7
4. The ratio on ward X of male patients to female patients is 2:5.
a) If there are 6 male patients, how many female patients are there?
b) If there are 20 female patients, how many male patients are there?
5. Medication Q is made up of solutions A, B and C.
To make 50 mg of the medication you need
10mls of A
20mls of B
5mls of C
a) What is the ratio of: i) A to B? ii) B to C? iii) C to A?
b) If you needed to produce 100mg of Q how many mls of A, B and C would you need?
c) There are 40mls of A left.
i) What is the maximum dosage of Q that you can produce?
ii) What quantities of B and C are needed to produce this dose?
6. For the following ratios of A:B, state what fraction of the solution is A and what fraction of the
solution is B. Cancel down where possible.
a) 2:6
b) 1:8
c)12:3
d) 2:3
12
• 4. Percentage
‘Per cent’ literally means ‘per hundred’, so percentage is concerned with parts of a hundred.
The symbol % is used to denote percentages.
Some commonly used percentages are:
100% of something means the whole amount. (Literally 100 per 100)
50% of something means that you are looking at half of it, as 50 is half of 100.
10% of something means that you are looking at a tenth of it as 10 is a tenth of 100.
We can work out percentages in many different ways. The best method to use is the one that
you find easiest. Two of the methods are detailed below.
Method 1-Use Fractions
As percentages are closely linked to fractions, we can use this fact to help with our calculations.
We know that 50% means ‘50 out of a hundred’, so we can write this as 50 in the same way as
100
we know that 1out of 2 can be written
as 1
2
.
The following table shows the fraction form of some common percentages:
Percentage Fraction
Simplified
Fraction
100%
50%
25%
10%
5%
1%
100
100
50
100
25
100
10
100
5
100
1
100
1
1
2
1
4
1
10
1
20
1
100
You may wish to perform the cancelling down yourself to check the final column.
The general procedure for converting a percentage (say 15%) into a fraction is:
• Write the percentage as a fraction of 100 i.e. 15
100
•
•
Cancel the fraction down to its lowest terms. In this case we can divide top and bottom by
the common factor, 5.
When the fraction is in its lowest terms, the job is done. 15%= 3
20
Cancelling the fraction down means that any subsequent calculation we perform uses the
smallest possible numbers and is thus easier to work out.
When we have converted our percentage to a fraction it is quite simple to use.
13
Example
Find 10% of 50.
10% is the same as 1 (from the table).
10
So 10% of 50 = 1 ×50
10
= 50×1 as we first multiply by the numerator.
=
10
50 = 5
10
1
as 50 and 10 have a common factor of 10
=5
Example
Find 30% of 25.
30%= 30 = 3
100
10
30% of 25 =25× 3 = 25×3 = 75
10
10
10
As 75 and 10 have a common factor of 5, we can cancel the fraction down
75 = 15
10
2
This is an improper fraction, so we convert it into a mixed fraction.
= 14 + 1
15
2
=7 +
2
2
1 =7 1
2
2
Method 2 - Use Decimals
As the number 1 is used to represent a whole, we can also use it to represent 100%.
We know that 50% is half of 100%, so 50% of 1 must be half of 1, which as a decimal is 0.5.
The following table shows the decimal form of some common percentages:
Percentage
Decimal
100%
1
50%
0.5
25%
0.25
10%
0.1
5%
0.05
1%
0.01
The general procedure for converting a percentage (say 15%) into a decimal is:
• Take the numerical value of the percentage, in this case 15, and divide it by 100.
• So 15% = 0.15. That’s all there is to it.
Example
Find 10% of 50
10÷100=0.1
so 10% of 50 = 0.1×50=5
Notice that this result is the same as the one we found earlier, using fractions.
Both methods will give the same answer for any percentage problem.
Note In calculating medicines, it is vital that your calculations are accurate.
A nought in the wrong place can make a large difference to a dose.
For this reason it is always a good idea to check your results, preferably by performing the
calculation again using a different method, or by performing it in reverse.
14
More Examples
John weighs 120lbs and is 6ft 1.
He is in hospital and cannot leave until he has increased his weight by 25%. How much must he
weigh when he is allowed to leave?
The question asks for the total weight after the gain. To start off we need to know how much he
needs to gain.
He is currently 120lbs.
We need to find 25% of 120
Method 1 - Fractions
25% =
1
4
25
100
=
5
20
=
1
4
by cancelling
Method 2 - Decimals
25% = 25 = 0.25
100
× 120=30 so 25% of 120 is 30 0.25×120=30
His total weight will be
120+30=150
150 lbs
His total weight will be
120+30=150
150 lbs
An alternative method is to notice that his total weight will be 100% of his original weight + 25%
of his original weight. So his eventual weight will be 125% of his original weight.
This means that we can shorten the above calculations:
125% = 125 = 25 = 5 by cancelling
125% = 125 = 1.25
5
4
1.25×120=150
100
20
4
× 120=150
His total weight will be 150 lbs
100
His total weight will be 150 lbs
Decreasing by a percentage
Extra care needs to be taken when decreasing by a percentage.
Example
An item costing £30 is reduced by 20% in the sale, what is the new price?
We can tackle this problem in two different ways.
Method 1
We find out what 20% of the item is and take that value away from the original cost.
20% = 20 = 1 = 0.2
100
5
0.2×30=6
30-6=24
The final cost is £24
Method 2
We notice that if we take away 20% of an item, we have 80% left. So we can work out what
80% is in one calculation.
80% = 80 = 8 = 0.8
100
10
0.8×30=24
The final cost is £24
15
As a rule, the fraction method is best if working on paper and the decimal method is best when
using a calculator.
Always check that your answer makes sense. A good check is to perform your calculation in
reverse, so if you’ve found 25% of something, multiply it by 4 and you should have your original
quantity back.
Exercise 4
1. Express as i) a fraction (simplify if possible), ii) a decimal
a) 20%
b) 30%
c) 45%
e) 9%
f) 12%
g) 84%
2. Using the method of your choice, evaluate the following:
a) 20% of 15
b) 30% of 10
c) 45% of 200
e) 9% of 300
f) 12% of 50
g) 84% of 25
d) 95%
h) 29%
d) 95% of 100
h) 29% of 300
3. A baby’s weight has increased since birth by 10%. When it was born it weighed 3kg. What is
its new weight?
4. An item costs £50. There is a price increase of 10%, followed by a decrease of 10% in a sale.
What is the sale price of this item?
For extra help with Percentages please consult Mathematics leaflets ‘Linking Fractions,
Decimals and Percentages’ and ‘Percentages’.
• 5. Using Formulae
Algebra is widely used in mathematics and is often used as a form of shorthand.
In algebra, letters are used to denote numbers.
We do this if the number the letter represents is still to be worked out, or if this number could
change.
You may use formulae in order to work out medicine doses which are dependent on the patients
weight.
Example
We could use p as the price in pounds of an item.
Then 3×p (written 3p) would represent the total cost of 3 items at £p each.
In order to make our work look neater/save time, we often omit the multiplication sign when
using algebra. To do this we simply write the items being multiplied next to each other:
3 × x = 3x
4 × g = 4g
f × g = fg
Formulae
Formulae is the plural of formula. A formula is an algebraic expression or ‘rule’.
Examples
Volume of a box: Length × Width × Height = l × w × h = lwh
Converting Temperatures: F = ( 95 C ) + 32 where F is Fahrenheit and C is Celsius
Substitution
Putting a known value into a formula in order to work out an unknown value.
16
Example
The relationship between quantities X and Z is such that
X=3Z+2
What is the value of X when Z=4?
Solution
The meaning of this formula in words is ‘multiply Z by 3, then add 2 and you get the value of X.
We have been given the value of Z so the only unknown value is X
We substitute Z=4 into the equation
X=3Z+2
X=(3×4)+2 (as 3Z means 3×Z)
X=12+2=14
Exercise 5
1. For the following formulae
i) explain in words what the formula means
ii) substitute in the given value of Z
iii) work out the value of X
a)X=5Z, Z=3
b)X=4Z+3, Z=2
c)X=12Z-6, z=5
2. Find the value of X in the following formulae when Z=4
a) X = 2Z
b) X = 3Z + 6
c) X=3Z+2× ( 8Z )
4
2 Z −3
Z +4
(You are unlikely to meet anything more complicated than the previous formulae)
For extra help with Algebra please consult Mathematics leaflet ‘Basic Algebra 1’
• 6. Unit Conversion
In your chosen field you are likely to need to convert weights and volumes from one unit to
another.
Metric Measurements of Weight
Name
Kilogram
Gram
Milligram
Microgram
Nanogram
Abbreviation
kg
g
mg
mcg
ng
Notes
Approximate weight of a litre of water
One thousand grams to a kilogram
One thousand mg to the gram
One million mcg to the gram
One thousand ng to the mcg
17
Conversion Chart
Number of
Kilograms
×1000
÷1000
Number of Grams
×1000
÷1000
Number of
Milligrams
×1000
÷1000
Number of
Micrograms
×1000
÷1000
Number of
Nanograms
To move up one stage we divide by 1000 and to go down one stage we multiply by 1000. If we
want to move up two stages we divide by 1000 two times (i.e. divide by 1000000= 1 million)
Example
Convert 3 kilograms into grams.
As we can see from the table, there are 1000 grams in a kilogram.
We have 3 kilograms, so 3×1000=3000
3kg=3000g.
From the conversion chart, the arrow from kilograms to grams carries the instruction ‘×1000’.
So 3×1000=3000. We have 3000g.
Note There is a greater chance of serious error when using abbreviations of measures.
For example mg, ng, and mcg may be hard to distinguish if written by hand. To avoid this, it is
always best to write out the whole name of the measure.
18
Metric Measurements of Liquids
Name
Litre
Millilitre
Abbreviation
Notes
l
Abbreviation is a lower-case L
ml
One thousand millilitres to a litre
Conversion Chart
Number of
litres
×1000
÷1000
Number of
Millilitres
There is also the Centilitre (cl), so named as there are a hundred of them in a litre.
A single Centilitre is equivalent to 10ml. Centilitres are normally used to measure wine.
Examples
1. Convert 575 millilitres into litres.
From the diagram, we see that to convert from millilitres to litres, we divide the number of
millilitres by 1000.
So we have 575÷1000=0.575 litres
2. Convert 2.67 litres into millilitres.
To convert from litres to millilitres we multiply the number of litres by 1000.
So we have 2.67×1000=2670 millilitres
Estimation
Always look at the answers you produce to check they are sensible. A good way to do this is to
estimate.
In Example 1 above we can use our knowledge of litres and millilitres to estimate the result. We
have 575 millilitres. If we had 1000 millilitres we would have a litre. Half a litre would be 500
millilitres, so our result will be a little over half a litre.
Exercise 6
1. Copy and complete the following, using the tables and diagrams
a) 1 kilogram = ____ grams
b) 1 gram = ____ milligrams
c) 1 gram = ____ micrograms
d) 1 microgram = ____ nanograms
e) 1 litre = ____ millilitres
2. Convert the following into milligrams
a) 6 grams
b) 26.8 grams
c) 3.924 grams
d) 405 grams
3. Convert the following into grams
a) 1200mg
b) 650mg
c) 6749mg
4. Convert the following into milligrams
a) 120 micrograms b) 1001 micrograms
c) 2675 micrograms d) 12034 mcg
19
d) 3554mg
5. Convert the following: (you may find it easier to work out the answers in two stages):
a) 1.67grams into micrograms
b) 0.85grams into micrograms
c) 125 micrograms into grams
d) 6784 micrograms into grams
e) 48.9 milligrams into nanograms f) 3084 nanograms into milligrams
6. Convert the following into litres
a) 10 millilitres
b) 132 millilitres
c) 2389 millilitres
d) 123.4 ml
7. Convert the following into millilitres
a) 4 litres
b) 6.2 litres
c) 0.94 litres
d) 12.27 litres
8. A patient needs a dose of 0.5 g of medicine A. They have already had 360mg.
a) How many more mg do they need?
b) What is this value in grams?
c) A dose of 1400 mcg has been prepared. Will this be enough?
For extra help with Units please consult Mathematics leaflet ‘Powers of 10…’
• 7. Dosage Calculations
Dosage calculations vary depending on whether you are dealing with liquid or solid medications,
or if the dose is to be given over a period of time. In this section I will go over each of these
situations in turn.
It is very important that you know how drug dosages are worked out, because it is good practise
to always check calculations before giving medication, no matter who worked out the original
amount. It is far better to point out a mistake on paper than overdose a patient.
Tablets
Working out dosage from tablets is simple.
Formula for dosage:
Total dosage required = Number of tablets required
Dosage per tablet
Note-If your answer involves small fractions of tablets, it would be more sensible to try to find
tablets of a different strength rather than try to make 2 of a tablet for example.
3
Examples
1. A patient needs 500mg of X per day. X comes in 125mg tablets. How many tablets per day
does he need to take?
Total dosage required is 500mg,
Dosage per Tablet is 125mg
So our calculation is 500 =4
125
He needs 4 tablets a day
20
Liquid Medicines
Liquid medicines are a little trickier to deal with as they will contain a certain dose within a
certain amount of liquid, such as 250mg in 50ml.
To work out the dosage, we will use the formula:
What you want × What it’s in
What you’ve got
Note: In order to use this formula, the units of measurement must be the same for ‘What you
want’ and ‘What you’ve got’; i.e. both mg or both mcg etc.
Examples
2. We need a dose of 500mg of Y. Y is available in a solution of 250mg per 50ml.
In this case,
What we’ve got= 250
What we want= 500
What it’s in= 50
So our calculation is 500 × 50 =100
250
We need 100ml of solution.
3. We need a dose of 250mg of Z. Z is available in a solution of 400mg per 200ml.
In this case,
What we’ve got= 400
What we want = 250
What it’s in= 200
So our calculation is 250 × 200 = 125
400
We need 125ml of solution.
Medicine over Time
1. Tablets/liquids
This differs from the normal calculations in that we have to split our answer for the total dosage
into 2 or more smaller doses.
Look at Example 1 again. If the patient needed the 500mg dose to last the day, and tablets
were taken four times a day, then our total of 4 tablets would have to be split over 4 doses.
Total amount of liquid/tablets for day = Amount to be given per dose
Number of doses per day
We would perform the calculation: 4÷4=1. So he would need 1 tablet 4 times a day
2. Drugs delivered via infusion
For calculations involving infusion, we need the following information:
The total dosage required.
The period of time over which medication is to be given.
How much medication there is in the solution.
21
Example
4. A patient is receiving 500mg of medicine X over a 20 hour period.
X is delivered in a solution of 10mg per 50ml.
What rate should the infusion be set to?
Here our total dosage required is 500mg
Period of time is 20 hours
There are 10mg of X per 50ml of solution
Firstly we need to know the total volume of solution that the patient is to receive.
Using the formula for liquid dosage we have:
500 ×50=2500 So the patient needs to receive 2500mls.
10
We now divide the amount to be given by the time to be taken: 2500 =125
20
The patient needs 2500mls to be given at a rate of 125mls per hour
Note: Working out medicines over time can appear daunting, but all you have to do is to work
out how much medicine is needed in total, and then divide it by the amount of hours/doses
needed
Drugs labelled as a percentage
Some drugs may be labelled in different ways to those used earlier.
V/V and W/V
Some drugs may have V/V or W/V on the label.
V/V means that the percentage on the bottle corresponds to volume of drug per volume of
solution. i.e 15% V/V means for every 100ml of solution, 15ml is the drug.
W/V means that the percentage on the bottle corresponds to the weight of drug per volume of
solution. Normally this is of the form ‘number of grams per number of millilitres’. So in this case
15% W/V means that for every 100ml of solution there are 15 grams of the drug.
If we are converting between solution strengths, such as diluting a 20% solution to make it a
10% solution, we do not need to know whether the solution is V/V or W/V.
Examples
5. We need to make up 1 litre of a 5% solution of A. We have stock solution of 10%.
How much of the stock solution do we need? How much water do we need?
We can adapt the formula for liquid medicines here:
What we want× What we want it to be in
What we’ve got
We want a 5% solution. This is the same as 5 or 1 .
100
We’ve got a 10% solution. This is the same as
10
100
20
or 1 .
10
We want our finished solution to have a volume of 1000ml.
Our formula becomes
1
20
1
10
=1
2
×1000 = 1 × 10 ×1000 (using the rule for dividing fractions)
20
1
×1000=500 . We need 500mls of the A solution.
Which means we need 1000-500=500mls of water.
22
6. You have a 20% V/V solution of drug F. The patient requires 30ml of the drug. How much of
the solution is required?
20% V/V means that for every 100ml of solution we have 20ml of drug F.
Using our formula:
What you want × What it’s in
What you’ve got
This becomes 30 ×100=150
20
We need 150mls of solution.
7. Drug G comes in a W/V solution of 5%. The patient requires 15 grams of G. How many mls
of solution are needed?
5% W/V means that for every 100mls of solution, there are 5 grams of G.
Using the formula gives us
15 ×100=300
5
300mls of solution are required.
Note In very rare cases, a drug may be labelled with a ratio. If this is the case, refer to the Drug
Information Sheet for the specific medication in order to be completely sure how the solution is
made up.
Exercise 7
1. How many 30mg tablets of drug B are required to produce a dosage of:
a) 60mg
b) 120mg
c) 15mg
d) 75mg
2. Medicine A is available in a solution of 10mg per 50ml. How many mls are needed to produce
a dose of:
a) 30mg
b) 5mg
c) 200mg
d) 85mg
3. Medicine C is available in a solution of 15micrograms per 100ml. How many mls are needed
to produce a dose of:
a)150mcg
b) 45mcg
c)30mcg
d) 75mcg
4. Medicine D comes in 20mg tablets. How many tablets are required in each dose for the
following situations:
a) total dosage 120mg , 3 doses b) total dosage 60mg, 2 doses c) total dosage 100mg, 5
doses
d) total dosage 30mg, 3 doses
5. At what rate per hour should the following infusions be set?
a) Total dosage 300mg, solution of 25mg per 100mls, over 12 hours
b) Total dosage 750mg, solution of 10mg per 30mls, over 20 hours
c) Total dosage 450mg, solution of 90mg per 100mls, over 10 hours
23
6. Drug B comes in a 20% V/V stock solution.
i) How much of the solution is needed to provide:
a) 50ml of B
b) 10ml of B
c) 200ml of B
ii) How would you make up the following solutions from the stock solution?
a) Strength 20% volume 1 litre
b) Strength 10% volume 750ml
iii) What strength are the following solutions?
a) Volume 1 litre, made up of 600ml stock solution, 400ml water
b) Volume 600ml, made up of 300ml stock solution, 300ml water
7. Drug C comes in a 15% W/V stock solution.
i) How much of the solution is needed to provide:
a) 30g of C
b) 27.5gof C
c) 90g of C
ii) How would you make up the following solutions from the stock solution?
a) Strength 5% volume 900ml
b) Strength 10% volume 750ml
iii) How many grams of C are in the following solutions?
a) Volume 1 litre, made up of 400ml stock solution, 600ml water
b) Volume 800mls, made up of 450ml stock solution, 350ml water
For further help on this topic there are several books available in the Brynmor Jones Library.
Note, however, that books on the subject by non-British authors may use notation or measures
not commonly used in the UK.
Suggested Reading
Drug Calculations for Nurses-A Step By Step Approach
Robert Lapham and Heather Agar
BJL 2nd Floor East RS57L3
ISBN 0-340-60479-4
Nursing Calculations Fifth Edition
J.D. Gatford and R.E.Anderson
BJL 2nd Floor East RT68G2
ISBN 0-443-05966-7
24
Answers to exercises
Exercise 1
1. a) 1 , 2 b) 2 , 6
c) 4 , 12
2 4
a) 1
5
a) 7
8
a) 47
8
a) 3 3
5
2.
3.
4.
5.
7 21
b) 1
3
b) 6
7
b) 49
8
b) 3 5
7
5 15
c) 5
16
c) 12
15
c) 37
16
c) 6 1
3
6.
a) 4
5
a) 1
20
a) 4
3
a) 1
6
7.
8.
9.
3
2
c) 8
15
c) 4
b) 40
9
b) 3
16
6
b)
7
b) 1
16
c)
c) 5
48
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
b)50.5
c) 306.091
3. a) 180
d) 2680
g) 2100
b) 14
e) 209
h) 12900
c) 0.2
f) 394
i) 1080
4. a) 1.8
d) 0.268
b) 0.14
e) 0.0209
c) 0.002
f) 0.0394
g) 0.0021
h) 0.0129
i) 0.00108
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
Exercise 3
1. a) i) 5:2
c) i) 3:3 = 1:1
2. a)
ii) 2:5
b) i) 1:4
ii) 4:1
ii) 3:3 = 1:1
d) i) 3:2
ii) 2:3
b)
c)
3. a) 3:2
b) 1:3
c) 4:1
4. a) 15 women
b) 8 men
5. a) i) 10:20 = 1:2
b) 20mls of A
c) i) 200mg
ii) 20:5 = 4:1
40mls of B
ii) 80mls of B
iii) 5:10 = 1:2
10mls of C
20mls of C
6. a) A. 2 = 1
B. 6 = 3
b) A. 1 , B. 8
B.
d) A.
c) A.
8
12
15
4
= 4
5
8
3
15
4
=1
5
9
2
5
25
, B.
9
3
5
Exercise 4
1. a) 20 = 2 = 1 = 0.2
e)
100
9
100
10
b) 30 = 3 = 0.3
5
= 0.09
100
f)
12
100
=
c) 45 = 9 = 0.45
10
6
50
=
3
25
= 0.12
100
84
100
g)
=
20
42
50
2. a) 3
b) 3
c) 90 d) 95
e) 27
f) 6
g) 21
3. New weight is 3.3kg=3300g
4. After increase price is £55. The sale price is £49.50.
=
21
25
d) 95 = 19 = 0.95
= 0.84
h)
100
29
100
20
= 0.29
h) 87
Exercise 5
1. a) i) Multiply Z by 5 to find the value of X
ii)X=5×3 iii) X=15
b) i) Multiply Z by 4 then add on 3 to find the value of X
ii) X=(4×2)+3 iii) X=11
c) i) Multiply Z by 12 then subtract 6 to find the value of X
ii) X=(12×5)-6 iii) X=54
2.
X = 2× 4 = 8 = 2
a) X = 2Z
b) X =
4
3Z + 6
2 Z −3
4
4
( 3×4 )+ 6
X =
= 18
( 2×4 )−3
5
c) X = 3Z + 2( 8Z )
Z +4
X = (3 × 4) + 2( 48×+44 ) = 12 + 2( 328 ) = 12 + 8 = 20
Exercise 6
1. a) 1kg=1000g
e) 1 litre=1000ml
b) 1g=1000mg
c) 1g= 1000000mcg
d) 1mcg=1000ng
2.
a) 6g=6000mg
b) 268g=26800mg
c) 3.924g=3924mg
d) 405g=405000mg
3.
a) 1200mg=1.2g
b) 650mg=0.65g
c) 6749mg=6.749g
d) 3554mg=3.554g
4.
a)120mcg=0.12mg b) 1001mcg=1.001mg
d) 12034mcg=12.034mg
5.
a) 1.67g=1670000mcg
d) 6784mcg=0.006784g
b) 0.85g=850000mcg
e) 48.9mg=48900000ng
c) 125 mcg= 0.000125g
f) 3084ng=0.003084mg
6.
a) 10ml=0.01litres
d) 123.4ml=0.1234 litres
b) 132ml=0.132litres
c) 2389ml=2.389litres
7.
a) 4litres=4000ml
d) 12.27litres=12270ml
b) 6.2litres=6200ml
c) 0.94litres=940ml
8.
a) 140 milligrams
b) 0.14 grams
c)2675 mcg= 2.675mg
c) no, the correct dose would be 140000mcg
Exercise 7
1. a) 2 tablets
b) 4 tablets
c) 1 tablet
d) 2 1 tablets
2. a) 150ml
3. a) 1000ml
4. a) 2 tablets
b) 25ml
b) 300ml
b) 1 1 tablets
c) 1000ml
c) 200ml
c) 1 tablet
d) 425ml
d) 500ml
d) 1 tablet
2
2
5. a) 100ml per hour
b) 112.5 ml per hour
2
2
c) 50ml per hour
26
6. i)a) 250ml
b) 50ml
c) 1 litre
ii) a) 1 litre stock, no water
b) 375ml stock, 375ml water
iii) a) 600ml stock contains 120ml B b) 300ml stock contains 60ml B
So 120ml in 1000ml= 120 =12%
So 60ml in 600ml= 60 =10%
600
1000
7. i) a) 200ml
b) 150ml
c) 600ml
ii) a) 300ml stock, 600ml water b) 500ml stock, 250ml water
iii) a) 60g
b) 67.5g
We would appreciate your comments on this worksheet, especially if you’ve found any
errors, so that we can improve it for future use. Please contact the Maths tutor by email
at [email protected].
th
updated 16 June 2004
The information in this leaflet can be made available in an
alternative format on request. Telephone 01482 466199
27