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Web: www.hull.ac.uk/skills

Email:[email protected]

Maths for Healthcare ProfessionalsMathematics Skills Guide

All the information that you require to enable you to pass the mathematics

tests within your course is contained within.

Contents

Fractions Page 2

Decimals Page 11

Ratio Page 17

Percentages Page 21

BMI Page 29

Averages Page 30

Unit

Conversion

Page 31

Dosage

Calculations

Page 36
http://www.hull.ac.uk/skillshttp://www.hull.ac.uk/skillsmailto:[email protected]:[email protected]:[email protected]:[email protected]://www.hull.ac.uk/skills


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2

Fractions

A fraction is the ratio of two integers (whole numbers)

Example ,

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

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.

Examples

is equivalent to

since

is equivalent to


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since

is equivalent to

since

Equivalent fractions are needed when we come to addition and subtraction of

fractions.

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.

For example 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 multiplying

and dividing them.

The four operations we need to look at are , +, and and in that order:-

Multiplication

We can get a simple rule for this operation as follows:

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)

4

3


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The rule is:- multiply numerators, multiply denominators

Examples i)

ii)

iii) = (= after cancelling)

Algebraically =

Addition +

Type I

If two (or more) fractions have the same denominator, then we just add the

numerators

Examples

It is easy to see this is the correct method by looking at the following diagram

+ 2/8

The shaded region is

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.

Examples i)

If we multiply the numerator and denominator of by 4 i.e.


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5

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 now 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.

Example +

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.

Example

= and

So, , which equals , which equals (simplest form)

Subtraction -

This is done in exactly the same way except instead of adding we subtract!

Example (same denominators, so Type I subtraction)


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- (different denominators so produce equivalent fractions)

and

so

Algebraically, + = and - =

so - = =

This looks a little awkward, but if we look at the answer we can, very quickly,

produce it as follows

+

5

2 1 = 4 5 = 9

5 2 1 0 10 10

4

Division

To find a method for this operation we proceed as follows:

A C

B D

2ndterm

3rdterm

1stterm

1 1

3 4

3 4

3

12

4

4 3

12

7

=

12


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(just rewriting the sum)

= (multiply numerator and denominator by the same number to produce an

equivalent fraction)

x 12

= 2 x 124

31

3 x 123

41

= = (simplifying)

Before working this out (using the method from multiplying fractions) look at the

expression 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.

Examples i)

ii) =

iii)

Algebraically

Summarising

i) =


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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 Examples 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.

Examples

i) 2 3

2 and 3

So 2 (cancelling)

ii) 3

3 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.

Schematically

A Again do NOT remember this as a formula but as a method.


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Fractions Exercise

1. For each group of fractions, state which fractions are equivalent:

a)4

3

4

2

4

1

2

1 ,,, b)15

4

21

6

7

2

8

3 ,,, c)15

12

3

2

10

9

5

4 ,,,

2. Cancel the following fractions down to their simplest form:

a)25

5 b)108

36 c)64

20

3. For each of the following pairs of fractions, state which one is the larger:

a)8

7

4

3 , b)7

6

8

5 , c)5

3

15

12 ,

4. Convert the following mixed fractions into improper fractions:

a)8

75 b)816 c)

16

52

5. Convert the following improper fractions into mixed fractions:

a)518 b)

726 c)

319

6. Work out the following (simplify your answer if possible):

a) 45

1 b) 59

8 c) 612

3

Note: Any whole number can be written as a fraction with denominator 1, ie. 3 = ,

7 = , and we just use the usual rules so 3 x = x =

7. Work out the following (simplify your answer if possible):

a)4

1 5

1 b)8

3 2

1 c)5

4 3

2

8. Work out the following divisions (simplify your answer if possible):

a)9

8 3

2 b)7

3 2

1 c)5

4 5

1

9. Work out the following divisions (simplify your answer if possible):

a)3

2 4 b)2

1 8 c)8

5 6


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Fractions Exercise - Answers

1. a)4

2

2

1 , b)21

6

7

2 , c)15

12

5

4, 6. a)

5

4 b)9

40 c)2

3

2. a)5

1 b)3

1 c)16

5 7. a)20

1 b)16

3 c)15

8

3. a)8

7 b)7

6 c)15

12 8. a)3

4 b)7

6 c)4

4. a) 847

b) 849

c) 1637

9. a) 61

b) 161

c) 485

5. a)5

33 b)7

53 c)3

16


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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 pointare the normal counting numbers.

Numbers to the right of the decimal pointare 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 its 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 means one unit and one hundredth.

2.5 means two units and 5 tenths.

57.9 means five tens, seven units and 9 tenths

160.004 means 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 numbers following. 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 5.1000000000000000000000000 but we can just assume that the

following noughts are there.


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Multiplying and dividing by 10/100/1000 etc.

When we multiply a number by ten its digits remain the same but the decimal pointmoves one place to the right.

Examples 12.4 x 10 = 124

231.47 x 10 = 2314.7

14 x 10 = 140

0.03 x 10 = 0.3

When we multiply a number by one hundred its digits remain the same but thedecimal point moves two places to the right.

Examples 1.24 x 100 = 124

433.62 x 100 = 43362

8.4124 x 100 = 841.24

When we multiply a number by one thousand its digits remain the same but the

decimal point moves three places to the right.

Examples 2.316 x 1000 = 2316

81.42 x 1000 = 81420

0.0031 x 1000 = 3.1

When multiplying we just move the decimal point as many places as there are

noughts to the right.

Division is the inverse process to multiplication so that when dividing by 10/100/1000

we simply reverse the above process.

Examples 12 10 = 1.2

21 100 = 0.21

274 1000 = 0.274

ie. we move the decimal one, two or three places but this time to the left.


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Multiplying Decimals

We get a method for this by noting that 32.4 = and 2.42 = using

the ideas from the previous section.

So 32.4 x 2.42 = x

We perform the usual long multiplication

324

242

64800

12960

648

78408

And now divide by 10 and 100, ie. move the decimal point

three places,1 place from 32.4 and 2 places from 2.42 to get

78.408

This is why the method works. We can abbreviate it to:

Just perform a long multiplication, ignoring the decimal point

Now put the decimal point back in the answerit will have just as many

decimal places as the two original numbers combined.

Example 21.4 x 2.31

214

231

42800

6420

214

49434

We have 1 dp from 21.4

and 2 dp from 2.31

giving a total of 3 dp

Putting the decimal point in gives 49.434

Example 0.03 x 1.1

11

3

33


and 1 dp from 1.1




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You should always perform a rough check to make sure your answer is of the correct

order.

Example 20.42 x 3.12

2042

312

612600

20420

4084

637104


and 2 dp from 3.12



As a rough check 20.42 is approximately 20 and 3.12 is approximately 3

So our answer should be about 60, which it is.

Dividing decimals

To get a method for this operation we use the idea of equivalent fractions.

(Remember we can produce a fraction equivalent to a given fraction by multiplying

(or dividing) the numerator and denominator by the same number)

Example 5.39 1.1 = 5.39

1.1

Now multiply numerator and denominator by 10 to produce 53.9 and we can now

perform the usual long division 11

4.9

11 53.9

So that 5.39 = 4.9

1.1

As with multiplication, do a rough check to make sure your answer is of the correct

order.

Example 0.0325 0.013

ie. 0.0325

0.013


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Multiply numerator and denominator by 1000 to produce 32.5 and then perform the

long division. 13

2.5

13 32.5

Once again a rough check shows the answer is of the correct order.

As with multiplication of decimals we can abbreviate the method to:

Move the decimal point in the denominator to produce a whole number

Move the decimal point in the numerator the same number of places

Perform the long division

Example

Find 0.275

0.25

This becomes 27.5 (two places in the numerator and denominator)

25

1.1

25 27.5

A rough check shows the answer is of the correct order

Find 2.405

0.37

This becomes 240.5 (two places in the numerator and denominator)

37

6.5

37 240.5

Once again a rough check shows the answer is of the correct order


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Decimals Exercise

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

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. Find:

a) 2.54 0.2 b) 34.56 0.9 c) 18.48 1.2

d) 30.72 2.4 e) 0.085 0.025

Decimals Exercise - Answers

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 b) 14 c) 0.2 d) 2680 e) 209f) 394 g) 2100 h) 12900 i) 1080


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4. a) 1.8 b) 0.14 c) 0.002 d) 0.268 e) 0.0209f) 0.0394 g) 0.0021 h) 0.0129 i) 0.00108

5. a) 12.7 b) 38.4 c) 15.4 d) 12.8 e) 3.4

Ratio

Ratiodescribes the relationship between two quantities.

Here we have 3 grey squares and 2 white squares. We can say

that the

ratioof grey squares to white squares is 3 to 2.

This is usually written 3:2where 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 the top part of 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 bottom part of the diagram, in each row we have

4 grey squares for each 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 in the

fractions section).

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.


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Using Ratios

Examples

1. The small intestine is about 20 feet in

length. If the ratio of the small intestine to

the large intestine is 4:1 how long is the large intestine?

We have small:large 4:1 meaning to every 4 feet of small intestine we have

1 foot of large. So that for 20 feet of small intestine we must have 5 feet of

large. The large intestine is about 5 feet long.

2. Solution X is made from the contents of bottles A and B in the ratio of 3:2. We

have already measured out 600mL of A.

How many mL 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.

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,5

1 of the solution is A.

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,7

3 of the solution is A and7

4 is B.


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NoteSome 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.

Ratio Exercise

1. For the following diagrams, state i) the ratio of grey to white; ii) the ratio of white to

grey:

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

10mL of A

20mL of B

5mL 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 mL of A, B and C would you

need?

c) There are 40mL 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


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Ratio Exercise Answers

a) i) 5:2 ii) 2:5 b) i) 1:4 ii) 4:1c) i) 3:3 = 1:1 ii) 3:3 = 1:1 d) i) 3:2 ii) 2:3

2. a) b) c)

3. a) 3:2 b) 1:3 c) 4:14. a) 15 women b) 8 men5. a) i) 10:20 = 1:2 ii) 20:5 = 4:1 iii) 5:10 = 1:2

b) 20mL of A 40mL of B 10mL of Cc) i) 200mg ii) 80mL of B 20mL of C

6. a) A.4

1

8

2 B.

4

3

8

6 b) A.

9

1 B.9

8

c) A.5

4

15

12 B.

5

1

15

3 d) A.

5

2 B.5

3


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Percentages

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. Two of the methods are

detailed below.

Method 1- Using 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

100

50 in the same way as we know that 1out of 2 can be written as2

1 .

The following table shows the fraction form of some common percentages:

Percentage Fraction Simplified

Fraction100%

100

100 1

50%100

50 2

1

25%100

25 4

1

10%100

10 10

1

5%100

5 20

1

1%1001

1001

You may wish to perform the cancelling down yourself to check the final column.


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The general procedure for converting a percentage (say 15%) into a fraction is:

Write the percentage as a fraction of 100 i.e.100

15

Cancel the fraction down to its simplest form. In this case we can divide topand bottom by the common factor, 5.

When the fraction is in its simplest form, we are done. 15%= 203

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.

Example

Find 10% of 50.

10% is the same as10

1 (from the table).

So 10% of 50 =10

1 50

10

150 as we first multiply by the numerator.

10

50

1

5 as 50 and 10 have a common factor of 10

= 5

Example

Find 30% of 25.

30%=10

3

100

30

30% of 25 =2510

75

10

325

10

3

As 75 and 10 have a common factor of 5, we can cancel the fraction down

2

15

10

75

This is an improper fraction or top heavy fraction, so we convert it into a mixed

number.

= 7

Method 2 - Using DecimalsAs 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


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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 by100.

So 15% = 0.15, 17% = 0.17, 37% = 0.37Example

Find 10% of 50.

10100 = 0.1

so 10% of 50 = 0.150 = 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 decimal point 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.

More Examples

John weighs 120lbs and is 6ft 1in

He is in hospital and cannot leave until he has increased his weight by 25%. How

much must he weigh before 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 currently weighs 120lbs.

We need to find 25% of 120

Method 1 - Fractions Method 2 - Decimals

25%4

1

20

5

100

25 by cancelling

4

1 120=30 so 25% of 120 is 30

His total weight will be120+30=150150 lbs

25% 100

25 0.25

0.25120=30

His total weight will be120+30=150150 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%4

5

20

25

100

125 by 125%

100

125 1.25


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cancelling

4

5 120=150

His total weight will be 150 lbs

1.25120=150

His total weight will be 150 lbs

Increasing by a percentage

Example

A patient weights 150 kg. They have a 12% weight gain. What is his new weight?

Method 1

Find out what 12% of 150 is and add that to the original weight.

12% = = 0.12

And 0.12 x 150 = 18

So that 150 + 18 = 168

The patients new weight is 168 kg

Method 2

Notice that if we add 12% to the original

We now have 112%

1.12 x 150 = 168

As above the patients new weight is 168 kg

Decreasing by a percentage

Example

The dose of a drug given to a patient is to be reduced by 15%. If the patient had

been originally prescribed 300 mg of drug A what is the new dosage in mg?

Method 1

Find out what 15% of 300 is and subtract that from the original dose

15% = = 0.15

And 0.15 x 300 = 45


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So that 30045 = 255

The new dosage is 255 mg

Method 2

Notice that if we reduce the dosage by 15%

We have 85% left

85% = 0.85

0.85 x 300 = 255

As above, the new dosage is 255 mg

Always check that your answer makes sense. A good check is to perform your

calculation in reverse, so if youve found 25% of something, multiply it by 4 and you

should have your original quantity back.


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Percentages Exercise 1

1. Express as i) a fraction (simplify if possible), ii) a decimal

a) 20% b) 30% c) 45% d) 95%

e) 9% f) 12% g) 84% h) 29%

2.Using the method of your choice, evaluate the following:

a) 20% of 15 b) 30% of 10 c) 45% of 200 d) 95% of 100

e) 9% of 300 f) 12% of 50 g) 84% of 25 h) 29% of 300

3. A babys weight has increased since

birth by 10%. When it was born it weighed 3kg. What is its new weight?

4. A young adultsheight was measured

and found to be 1.3m. They grow by 10% over the next year. What is their new

height?

5. A patient loses 7% of their body weight

after surgery. If they originally weighed 195 kg what is their new weight?

For extra help with Percentages consult Mathematics leaflets Fractions, Decimals

and Percentages: how to link them and Percentagesavailable on the web at

www.hull.ac.uk/studyadvice

Percentages Exercise 1 - Answers

1. a) 2.05

1

10

2

100

20 b) 3.0

10

3

100

30 c) 45.0

20

9

100

45 d) 95.0

20

19

100

95

e) 09.0100

9 f)

12.025

3

50

6

100

12

g)

84.025

21

50

42

100

84

h) 29.0100

29

2. a) 3 b) 3 c) 90 d) 95e) 27 f) 6 g) 21 h) 87

3. New weight is 3.3kg or 3300g4. 1.43m


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5. 181.35 kg

Percentage increase/decrease

We often need to find the percentage increase or decrease in a patients weight. Todo this we use the formula:

Change in Weight x 100

Original Weight

Example:

A patient who originally weighed 50kg loses 2 kg. What is her percentage weightloss?

Her change in weight is 2 kg and her original weight is 50 kg.

So we have 2_ x 100 = 4

50

This represents a 4% weight loss.

A patient who originally weighed 125 kg now weighs 135 kg. What is his percentageweight gain?

Here the change in weight is 10 kg and the original weight is 125 kg.

So that we have 10__ x 100 = 8

125

This represents an 8% weight gain.


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Percentages Exercise 2

Find the weight gain/loss of the following patients.

a) Weight originally 80 kg, final weight

92 kg

d) Weight originally 200 kg, final weight

195 kgb) Weight originally 60 kg, final weight

63 kg

e) Weight originally 250 kg, final weight

245 kg

c) Weight originally125 kg, final weight

120 kg

Percentages Exercise 2 Answers

a)15% weight increase d) 2.5% weight decrease

b) 5% weight increase e) 2% weight decrease

c) 4% weight decrease


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Body Mass Index (BMI)

The BMI provides a simple numeric measure of a persons fatness or thinnesswhich allows healthcare professionals to discuss over and underweight problems

objectively.

The current settings are:

BMI < 20Underweight

20 < BMI < 25Optimum weight

25 < BMI < 30Overweight

BMI > 30Obese

BMI > 40Morbidly obese

BMI is calculated by dividing the persons weight (in Kg) by their height 2 (in metres).

The formula is written W

H2

Example

Find the BMI of a patient who weighs 75 kg and is 1.42 m tall

BMI = 75 /(1.42 x 1.42) = 34.72

To the nearest whole number this is 35, therefore this patient is in the obese range.

Find the BMI of a patient who weighs 93 kg and is 1.95m tall

BMI = 93 / (1.95 x 1.95) = 24.46

To the nearest whole number this is 24, therefore this patient is in the optimum

weight range.

Body Mass Index Exercise with answers

Find the BMI of the following patients to the nearest whole number

a) Weight 80 kg, height 1.83m (24)

b) Weight 115 kg, height 2.00m (29)c) Weight 78 kg, height 1.54m (33)

d) Weight 62 kg, height 1.8m (19)

e) Weight 132 kg, height 1.64m (49)


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Averages

The average or the mean of a set of numbers is just the value you get after adding

the set of numbers up and dividing by how many numbers you have.

Examples

1. Find the average of 2, 6, 4, 8, 5

2+6+4+8+5 = 25

= 5

The average is 5

2. If a patients oral fluid intake on successive days is 120 mL, 200 mL 140 mL

and 260 mL, what was the average intake over 4 days?

120+200+140+260 = 720

= 180

The average intake is 180 mL

Averages Exercise

1. A patients pulse was taken every 30 minutes over 2 hours

It was found to be 110, 105, 95 and 90

What is the average pulse rate over the 2 hours?

2. A patients temperature was taken every 30 minutes over 4 hours.

It was 38C, 38C, 38.5C, 39.1C, 38.4C, 38.1C, 37.4C, and 42.1C

What is the average temperature over:

a) The first two hours

b) The second two hours

Averages Exercise

1. 100

2. a) 38.4C b) 39.0C


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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 Abbreviation Notes

Kilogram kg Approx. the weight of a litre of water

Gram g One thousand grams to a kilogram

Milligram mg One thousand mg to the gram

Microgram mcg One million mcg to the gram

Nanogram ng One thousand ng to the mcg

Conversion Chart

Number of

Kilograms

Number of

Grams

Number of

Milligrams

Number of

Micrograms

Number of

Nanograms

x 10001000

1000

1000

1000

x 1000

x 1000

x 1000

As we move down the

diagram the arrows are

on the right and we

move the decimal point

three places to the right.As we move up the

diagram the arrows are

on the left and we move

the decimal point three

places to the left.


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Metric Measurements of Liquids

Name Abbreviation Notes

Litre L An upper case L

Millilitre mL One thousand millilitres to alitre

Conversion Chart

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.

DO NOT USE A LOWER CASE L AS AN ABBREVIATION FOR LITRES. There is a

chance of misreading 3l as thirty one (31) when it should be 3L. Always use L even

in mL!

Examples

1. Convert 575 millilitres into litres.

From the diagram, we see that to convert millilitres to litres, we divide the number of

millilitres by 1000.

So we have 5751000=0.575 litres

2. Convert 2.67 litres into millilitres.

To convert litres to millilitres we multiply the number of litres by 1000.

So we have 2.671000=2670 millilitres

Estimation

Always look at the answers you produce to check they are sensible. A good way to

do this is to estimate the answer.

Number of

Litres

Number of

Millilitres

x 10001000


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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.

Conversions of lbs kg, kg lbs

It is sometimes necessary to change from imperial units to metric units and vice

versa. The method is shown below:

Weights in kg x 2.2 = weights in pounds.

A patient weighs 124 kg, what is this in pounds (lbs)?

124 x 2.2 = 272.8 lbs

Weights in pounds 2.2 = weights in kg.

A patient weighs 212 lbs, what is this in kg?

212 2.2 = 96.37 (2dp) kg


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Unit Conversion Exercise 1

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 d) 3554mg

4. Convert the following into milligrams

a) 120 micrograms b) 1001 micrograms c) 2675 micrograms

d) 12034 mcg

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 millilitres

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?


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Unit Conversion Exercise 1 - Answers

1 a) 1kg=1000g b) 1g=1000mg c) 1g=1000000mcgd) 1 mcg=1000ng e) 1 litre=1000Ml

2 a) 6g=6000mg b) 268g=26.800mg c) 3.924g=3924mgd) 405g=405000mg

3 a) 1200mg=1.2g b) 650mg=0.65g c) 6749mg=6.749gd) 3554mg=3.554g

4 a)120mcg=0.12mg b) 1001mcg=1.001mg c)2675 mcg= 2.675mgd) 12034mcg=12.034mg

5 a) 1.67g=1670000mcg b) 0.85g=850000mcg

c) 125 mcg=0.000125g d) 6784mcg=0.006784ge) 48.9mg=48900000ng f) 3084ng=0.003084mg

6 a) 10mL=0.01litres b) 132mL=0.132litres c) 2389mL=2.389litresd) 123.4mL=0.1234 litres

7 a) 4litres=4000mL b) 6.2litres=6200mL c) 0.94litres=940mL

d) 12.27litres=12270mL

8 a) 140 milligrams b) 0.14 gramsc) no, the correct dose would be 140000mcg


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Dosage Calculations

Working out a dosage in either tablets or liquids is straightforward. The formula used

is always the same.

What you want x What its in

What youve got

When working with tablets what its in is always one tablet.

To calculate a dosage you must write down 3 numbers.

They are:

What you wantthis is what is prescribed/ordered/required/needed by the patient.

What you have gotthis is what is available.

What its in this is either 1 when we are working with tablets or in mL when working

with liquids.

The order in which you write these down is not difficult to remember, if you think The

patient always comes first ie. What you want.

Examples

1. A patient needs 500mg of drug X per day. X is available in 125mg tablets. How

many tablets per day does he need to take?

What you want = 500mg } The units are both the same

What youve got = 125mg }

What it is in = one tablet

So our calculation is x 1 = 4

The patient needs 4 tablets a day.

2. We need a dose of 500mg of Y. Y is available in a solution of 250mg per 50mL.

In this case,What you want = 500mg } both in mg

Note: In order to use this formula, the units of What you want and What youve

got must be the same, ie. both in mcg, or both in mg, or both in g.


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What youvegot = 250mg }

What its in= 50mL

So our calculation is250

500 50 =100

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 you want = 250mg } both in mg

What youve got= 400mg }

What its in= 200mL

So our calculation is400

250 200 = 125


4. A patient is prescribed 250mg of erythromycin IV.

Stock on hand contains 1g in 10mL once diluted.

What you want = 250mg

What youve got = 1g

What its in = 10mL

The units of What you want are mg and the units of What youve got are g. They

must be the same units.

Both in mg1g = 1000mg

So:

What you want = 250mg

What youve got = 1000mg

What its in = 10mL

Our calculation is

250 x 10 = 2.5

1000

We need 2.5mL

Both in g250mg = 0.25g

= 0.25g

=1g

=10mL

Our calculation is

0.25 x 10 = 2.5

1

We need 2.5mL

Medicine over Time

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.

ExampleA child weighing 12.5kg is prescribed a drug which is to be given in four

equally divided doses. The dosage the child requires is 100mg/kg body weight.The child requires 12.5 x 100mg = 1250mg of the drug.


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So for four equally divided doses

1250 = 312.5

4

They need 312.5mg four times a day.

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

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:

10

500 50=2500 so the patient needs to receive 2500mL.

We now divide the amount to be given by the time to be taken:20

2500=125

The patient needs 2500mL to be given at a rate of 125mL per hour

Note: Working out medicines over time can appear daunting, but all you need to do

is work out how much medicine is needed in total, and then divide it by the amount of

doses needed or the time over which it is to be given.

Drugs labelled as a percentage

Some drugs may be labelled in different ways from 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

volumeof 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.


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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

1. 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 weve got

We want a 5% solution. This is the same as100

5 or20

1 .

Weve got a 10% solution. This is the same as100

10 or10

1 .

We want our finished solution to have a volume of 1000mL.

Our formula becomes

10

1

20

1

1000

=20

1 1

10 1000 (using the rule for dividing fractions)

=2

1 1000=500 . We need 500mL of the A solution.

Which means we need 1000-500=500mL of water.

(Alternatively you can use the fact that a 5% solution is half the strength of a 10%

solution to see that you need 500mL of solution and 500mL of water)

2. 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 its in

What youve got

This becomes20

30 100=150


3. Drug G comes in a W/V solution of 5%. The patient requires 15 grams of G. How

many mL of solution are needed?


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5% W/V means that for every 100mL of solution, there are 5 grams of G.

Using the formula gives us

5

15 100=300

300mL of solution are required.

NoteIn 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.

Dosage Calculations Exercise 1

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 mL are needed

to produce a dose of:

a) 30mg b) 5mg c) 200mg d) 85mg

3. Medicine C is available in a solution of 15 micrograms per 100mL. How many mL

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 100mL, over 12 hours

b) Total dosage 750mg, solution of 10mg per 30mL, over 20 hours

c) Total dosage 450mg, solution of 90mg per 100mL, over 10 hours

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:


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a) 30g of C b) 22.5g of 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 waterb) Volume 800mL, made up of 450mL stock solution, 350mL water

Dosage Calculations Exercise 1 Answers

1. a) 2 tablets b) 4 tablets c)2

1 tablet d)

22

1 tablets

2. a) 150mL b) 25mL c) 1000mL d) 425mL

3. a) 1000mL b) 300mL c) 200mL d) 500mL

4. a) 2 tablets b) 12

1 tablets c) 1 tablet d)2

1 tablet

5. a) 100mL per hour b) 112.5 mL per hour c) 50mL per hour

6. i) a) 250mL b) 50mL c) 1 litreii) a) 1 litre stock, no water b) 375mL stock, 375mL wateriii) a) 600mL stock contains 120mL B

So 120mL in 1000mL=1000

120 =12%

b) 300mL stock contains 60mL B

So 60mL in 600mL=600

60 =10%

7. i) a) 200mL b) 150mL c) 600mLii) a) 300mL stock, 600mL water b) 500mL stock, 250mL wateriii) a) 60g b) 67.5g


A drug is available in 1 mg, 2 mg, 5 mg and 10 mg tablets.

What is the best combination of these (i.e. the smallest number of tablets) to give the

following dosages?

Dosage Tablets required Number of tablets

1 3 mg

2 7 mg

3 8 mg

4 10mg

5 11 mg


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6 14 mg


Tablets

required

Number of

tablets

Tablets

required

Number of

tablets

1 1 mg & 2

mg

2 2 2 mg & 5

mg

2

3 1 mg, 2mg & 5

mg

3 4 5 mg & 5mg

2

5 5 mg, 5

mg & 1

mg

3 6 5 mg, 5

mg, 2 mg

& 2 mg

4


1.A solution contains furosemide (frusemide) 10 mg/mL. How many milligrams of

frusemide are in

a 2 mL b 3 mL c 5 mL of the solution?

2.A solution contains morphine hydrochloride 2 mg/mL. How many milligrams of

morphine hydrochloride are in


3.Another solution contains morphine hydrochloride 40 mg/mL. How many

milligrams of morphine hydrochloride are in

a 2 mL b 5 mL c 10 mL of this solution?

4.A suspension contains phenytoin 125 mg/5 mL. How many milligrams of

phenytoin are in

a 20 mL b 30 mL c 40 mL of the suspension?


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5.A solution contains fluoxetine 20 mg/5 mL. How many milligrams of fluoxetine

are in


6.A suspension contains erythromycin 250 mg/5 mL. How many milligrams of

erythromycin are in

a 10 mL b 20 mL c 30 mL of the suspension?

7.A syrup contains chlorpromazine 25 mg/5 mL. How many milligrams of

chlorpromazine are in

a 10 mL b 30 mL c 50 mL of the syrup?

8.A mixture contains penicillin 250 mg/5 mL. How many milligrams of penicillin are

in

a 15 mL b 25 mL c 35 mL of the mixture?


All answers are in mg

1 a) 20 b) 30 c) 50

2 a) 6 b) 10 c) 14

3 a) 800 b) 200 c) 400

4 a) 500 b) 750 c) 1000

5 a) 40 b) 100 c) 160

6 a) 500 b) 1000 c) 1500

7 a) 50 b) 150 c) 250

8 a) 750 b) 1250 c) 1750


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In each example, you are given the prescribed dosage and the strength of stock on

hand. Calculate the volume to be given:

1. Ordered: penicillin 500 mg On hand: syrup 125 mg/5 mL

2. Ordered: furosemide (frusemide) 40 mg On hand: solution 10 mg/mL

3. Ordered: morphine hydrochloride 100 mg On hand: solution 40 mg/mL

4. Ordered: paracetamol 180 mg On hand: suspension 120 mg/5 mL

5. Ordered: phenytoin 150 mg On hand: suspension 125 mg/5 mL

6. Ordered: erythromycin 1250 mg On hand: suspension 250 mg/5 mL

7. Ordered: fluoxetine 30 mg On hand: solution 20 mg/5 mL

8. Ordered: penicillin 1000 mg On hand: mixture 250 mg/5 mL

9. Ordered: chlorpromazine 35 mg On hand: syrup 25 mg/5 mL

10. Ordered: penicillin 1200 mg On hand: mixture 250 mg/5 mL

11. Ordered: erythromycin 800 mg On hand: mixture 125 mg/5 mL

Dosage Calculations Exercise 4 - Answers

All answers are in mL

1. 20 5. 6 9. 7

2. 4 6. 25 10. 24

3. 2.5 7. 7.5 11. 32

4. 7.5 8. 20


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Dosages of oral medications

1. A patient is ordered paracetamol 1 g, orally. Stock on hand is 500 mg tablets.

Calculate the number of tablets required.

2. Ordered: codeine 15 mg, orally. Stock on hand: codeine tablets, 30 mg. How

many tablets should the patient take?

3. A patient is ordered furosemide (frusemide) 60 mg, orally. In the ward are 40

mg tablets. How many tablets should be given?

4. How many 30 mg tablets of codeine are needed for a dose of 0.06 gram?

5. 750 mg of ciprofloxacin is required. On hand are tablets of strength 500 mg.

How many tablets should be given?

6. A patient is prescribed 150 mg of soluble aspirin. On hand we have 300 mg

tablets. What number should be given?

7. 450 mg of soluble aspirin is ordered. Stock on hand is 300 mg tablets. How

many tablets should the patient receive?

8. 25 mg of captopril is prescribed. How many 50 mg tablets should be given?

9. The stock on hand of diazepam is 5 mg tablets. How many tablets are to be

administered if the order is diazepam 12.5 mg?

10. Digoxin 125 mcg is ordered. Tablets available are 0.25 mg. How many

tablets should be given?

Check that you have used the same unit of weightthroughout a calculation.

Are bothweights in milligrams (mg)? Or are bothweights in micrograms

(mcg)?


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Dosage Calculations Exercise 5 - Anwers

All answers are in tablets

1. 2 5. 1 9.

2. 1 6. or0.5 10. or0.5

3. 1 7. 1

4. 2 8. or0.5


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Dosage Calculation Exercise 6

Calculate the volume of stock required. Give answers greater than 1 mL correct to

one decimal place; answers less than 1 mL correct to two decimal places.

Ordered Stock ampoule

1. Morphine 12 mg 15 mg/mL

2. Calciparine 7000 units 25 000 units in 1 mL

3. Benzylpenicillin 1500 mg 1.2 g in 10 mL

4. Heparin 3000 units 5000 units/mL

5. Phenobarbitone 70 mg 200 mg/mL

6. Pethidine 80 mg 100 mg/2 mL

7. Buscopan 0.24 mg 0.4 mg/2mL

8. Digoxin 200 mcg 500 mcg in 2 mL

9. Furosemide (frusemide) 150 mg 250 mg in 5 mL

10. Ondansetron 5 mg 4 mg in 2 mL

11. Capreomycin 800 mg 1 g in 5 mL

12. Tramadol 120 mg 100 mg in 2 mL

13. Gentamicin 70 mg 80 mg in 2 mL

14. Vancomycin 800 mg 1 g in 5 mL

15. Morphine 7.5 mg 10 mg in 1 mL

16. Ceftriaxone 1250 mg 1 g/3 mL

17. Buscopan 25 mg 20 mg in 1 mL

18. Dexamethasone 3 mg 4 mg/mL

19. Vancomycin 1.2 g 1000 mg/5 mL

20. Naloxone 0.5 mg 0.4 mg/mL


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1. 0.8 6. 1.6 11. 4 16. 3.8

2. 0.28 7. 1.2 12. 2.4 17. 1.3

3. 12.5 8. 0.8 13. 1.8 18. 0.75

4. 0.6 9. 3 14. 4 19. 6

5. 0.35 10. 2.5 15. 0.75 20. 1.3


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Calculate the volume of stock to be drawn up for injection.

1. Pethidine 60 mg is ordered. Stock ampoules contain 100 mg in 2 mL.

2. An adult is ordered metoclopramide 15 mg, for nausea. On hand are ampoules

containing 10 mg/mL.

3. A patient is prescribed erythromycin 250 mg, I.V. Stock on hand contains 1 g in

10 mL, once diluted.

4. Tramadol hydrochloride 80 mg is required. Available stock contains 100 mg in 2mL.

5. A patient is ordered benzylpenicillin 800 mg. On hand is benzylpenicillin 1.2 g in

6 mL.

6. An adult patient with TB is to be given 500 mg of capreomycin every second day,

I.M.I. Stock on hand contains 1 g in 3 mL.

7. Digoxin ampoules on hand contain 500 mcg in 2 mL. Digoxin 150 mcg is

ordered.

8. Stock Calciparine contains 25 000 units in 1 mL. 15 000 units of Calciparine are

ordered.

9. Penicillin 450 mg is ordered. Stock ampoules contain 600 mg in 5 mL.



1. 1.2 4. 1.6 7. 0.6

2. 1.5 5. 4 8. 0.6

3. 2.5 6. 1.5 9. 3.75 (3.8 to 1dp)


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1. An injection of morphine 8 mg is required. Ampoules on hand contain 10 mg in 1

mL. What volume is drawn up for injection?

2. Digoxin ampoules on hand contain 500 mcg in 2 mL. What volume is needed to

give 350 mcg?

3. A child is ordered 9 mg of gentamicin by I.M.I. Stock ampoules contain 20 mg in

2 mL. What volume is needed for the injection?

4. A patient is to be given flucloxacillin 250 mg by injection. Stock vials contain 1 g

in 10 mL, after dilution. Calculate the required volume.

5. Stock heparin has a strength of 5000 units per mL. What volume must be drawnup to give 6500 units?

6. Pethidine 85 mg is to be given I.M. Stock ampoules contain pethidine 100 mg in 2

mL. Calculate the volume of stock required.

7. A patient is to receive an injection of gentamicin 60 mg, I.M. Ampoules on hand

contain 80 mg/2 mL. Calculate the volume required.

8. A patient is prescribed naloxone 0.6 mg, I.V. Stock ampoules contain 0.4 mg/2

mL. What volume should be drawn up for injection?

Think about each answer. Does it make sense? Is it ridiculously large?



1. 0.8 4. 2.5 7. 1.5

2. 1.4 5. 1.3 8. 3.0

3. 0.9 6. 1.7


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Suggested Reading

Drug Calculations for Nurses-A Step By Step Approach

Robert Lapham and Heather Agar

ISBN 0-340-60479-4

Nursing Calculations Fifth Edition

J.D. Gatford and R.E.Anderson

ISBN 0-443-05966-7

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.

We would appreciate your comments on this worksheet, especially if

youve found any errors, so that we can improve it for future use. Please

contact the Maths Skills Adviser by email at [email protected]

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