Maths for Healthcare Professionals

Maths for Healthcare Professionals Mathematics 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 Ratio Page 7 Percentages Page 2 BMI Page 29 Averages Page 30 Unit Conversion Dosage Calculations Page 3 Page 36 Web: www.hull.ac.uk/skills Email: skills@hull.ac.uk

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 2

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: 4 3 The shaded area in the diagram Represents 3 x 4 = 2 and if we use the same idea but the sides of the diagram now represent unit we can illustrate The shaded area in the diagram represents x = (= after cancelling) 3

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. 4

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) 5

- (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 + 2 nd term A B + C D 3 rd term st term 3 + 3 4 2 4 + 3 2 = 7 2 3 4 4 5 2 + = 4 + 5 = 9 5 2 0 0 0 4 Division To find a method for this operation we proceed as follows: 6

(just rewriting the sum) = (multiply numerator and denominator by the same number to produce an x 2 equivalent fraction) = 2 x 2 4 3 3 x 2 3 4 = = (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) = 7

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. 8

Fractions Exercise. For each group of fractions, state which fractions are equivalent: a), 2, 3 2 4 4 4, b) 3 2 6,, 4 8 7 2 5, c) 4 9,, 2 2 5 0 3, 5 2. Cancel the following fractions down to their simplest form: a) 5 25 b) 36 08 c) 20 64 3. For each of the following pairs of fractions, state which one is the larger: a) 3, b) 4 7 8 5, c) 8 6 7 2 3, 5 5 4. Convert the following mixed fractions into improper fractions: a) 5 7 b) 8 6 c) 8 2 5 6 5. Convert the following improper fractions into mixed fractions: a) 8 5 b) 26 7 c) 9 3 6. Work out the following (simplify your answer if possible): a) 4 5 b) 5 8 9 c) 6 3 2 Note: Any whole number can be written as a fraction with denominator, 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 5 b) 3 8 2 c) 4 5 2 3 8. Work out the following divisions (simplify your answer if possible): a) 8 9 2 3 b) 3 7 2 c) 4 5 5 9. Work out the following divisions (simplify your answer if possible): a) 2 3 4 b) 2 8 c) 5 8 6 9

Fractions Exercise - Answers. a), 2 b) 2 6, c) 4, 2 6. a) 4 5 b) 40 9 c) 3 2 2 4 7 2 5 5 2. a) 5 b) 3 c) 5 7. 6 a) 20 b) 3 6 c) 8 5 3. a) 7 8 b) 6 7 c) 2 8. 5 a) 4 3 b) 6 c) 4 7 4. a) 47 8 b) 49 8 c) 37 9. 6 a) 6 b) 6 c) 5 48 5. a) 3 b) 3 c) 6 3 5 5 7 3 0

Decimals.5, 2.7,.333, 2.6 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 23.456. Here we have 23 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 0 2 5 5 7 9 6 0 0 0 4 The table above shows how place value works for decimals..0 means one unit and one hundredth. 2.5 means two units and 5 tenths. 57.9 means five tens, seven units and 9 tenths 60.004 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. is really 5.000000000000000000000000 but we can just assume that the following noughts are there.

Multiplying and dividing by 0/00/000 etc. When we multiply a number by ten its digits remain the same but the decimal point moves one place to the right. Examples 2.4 x 0 = 24 23.47 x 0 = 234.7 4 x 0 = 40 0.03 x 0 = 0.3 When we multiply a number by one hundred its digits remain the same but the decimal point moves two places to the right. Examples.24 x 00 = 24 433.62 x 00 = 43362 8.424 x 00 = 84.24 When we multiply a number by one thousand its digits remain the same but the decimal point moves three places to the right. Examples 2.36 x 000 = 236 8.42 x 000 = 8420 0.003 x 000 = 3. 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 0/00/000 we simply reverse the above process. Examples 2 0 =.2 2 00 = 0.2 274 000 = 0.274 ie. we move the decimal one, two or three places but this time to the left. 2

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 And now divide by 0 and 00, ie. move the decimal point three places, place from 32.4 and 2 places from 2.42 to get 78.408 2960 648 78408 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 answer it will have just as many decimal places as the two original numbers combined. Example 2.4 x 2.3 24 23 42800 6420 24 49434 We have dp from 2.4 and 2 dp from 2.3 giving a total of 3 dp Putting the decimal point in gives 49.434 Example 0.03 x. 3 33 We have 2 dp from 0.03 and dp from. giving a total of 3 dp Putting the decimal point in gives 0.033 3

You should always perform a rough check to make sure your answer is of the correct order. Example 20.42 x 3.2 2042 32 62600 20420 4084 63704 We have 2 dp from 20.42 and 2 dp from 3.2 giving a total of 4 dp Putting the decimal point in gives 63.704 As a rough check 20.42 is approximately 20 and 3.2 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. = 5.39. Now multiply numerator and denominator by 0 to produce 53.9 and we can now perform the usual long division 4.9 53.9 So that 5.39 = 4.9. As with multiplication, do a rough check to make sure your answer is of the correct order. Example 0.0325 0.03 ie. 0.0325 0.03 4

Multiply numerator and denominator by 000 to produce 32.5 and then perform the long division. 3 2.5 3 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. 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 5

Decimals Exercise. Express the following in terms of hundreds, tens, units, tenths etc: a) 25.9 b) 87.03 c) 02.065 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) 8 0 b).4 0 c) 0.02 0 d) 26.8 00 e) 2.09 00 f) 3.94 00 g) 2. 000 h) 2.9 000 i).08 000 4. Evaluate the following: a) 8 0 b).4 0 c) 0.02 0 d) 26.8 00 e) 2.09 00 f) 3.94 00 g) 2. 000 h) 2.9 000 i).08 000 5. Find: a) 2.54 0.2 b) 34.56 0.9 c) 8.48.2 d) 30.72 2.4 e) 0.085 0.025 Decimals Exercise - Answers. 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).60 b) 50.5 c) 306.09 3. a) 80 b) 4 c) 0.2 d) 2680 e) 209 f) 394 g) 200 h) 2900 i) 080 6

4. a).8 b) 0.4 c) 0.002 d) 0.268 e) 0.0209 f) 0.0394 g) 0.002 h) 0.029 i) 0.0008 5. a) 2.7 b) 38.4 c) 5.4 d) 2.8 e) 3.4 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 the top part of this diagram, we have 6 grey squares and 8 white squares. The ratio of grey squares to white squares is 6:8. 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 6:8 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 6 grey we have 8 white becomes: The ratio of grey to white is 6:8 This is the same as 4:2 Which is the same as 2:. So the ratio of grey to white, is 2:. 7

Using Ratios Examples. The small intestine is about 20 feet in length. If the ratio of the small intestine to the large intestine is 4: how long is the large intestine? We have small:large 4: meaning to every 4 feet of small intestine we have 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. The ratio of drug A to water in a solution is :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, is A and 4 are water. So, 5 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, 73 of the solution is A and 74 is B. 8

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 0mg per ml may be written 0mg/mL. Ratio Exercise. 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) :3 b) 3:5 c) 6:7 3. Write the following ratios in their simplest forms a) 2:8 b) 5:5 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 0mL 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 00mg 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) :8 c)2:3 d) 2:3 9

Ratio Exercise Answers a) i) 5:2 ii) 2:5 b) i) :4 ii) 4: c) i) 3:3 = : ii) 3:3 = : d) i) 3:2 ii) 2:3 2. a) b) c) 3. a) 3:2 b) :3 c) 4: 4. a) 5 women b) 8 men 5. a) i) 0:20 = :2 ii) 20:5 = 4: iii) 5:0 = :2 b) 20mL of A 40mL of B 0mL of C c) i) 200mg ii) 80mL of B 20mL of C 6. a) A. 2 8 B. 6 3 4 8 4 b) A. 9 8 B. 9 2 4 c) A. 5 5 B. 3 5 5 2 d) A. 5 3 B. 5 20

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: 00% of something means the whole amount. (Literally 00 per 00) 50% of something means that you are looking at half of it, as 50 is half of 00. 0% of something means that you are looking at a tenth of it as 0 is a tenth of 00. We can work out percentages in many different ways. Two of the methods are detailed below. Method - 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 50 in the same way as we know that out of 2 can be written as 2. 00 The following table shows the fraction form of some common percentages: Percentage Fraction Simplified Fraction 00 00% 00 50% 50 00 25% 25 00 0% 0 00 5% 5 00 % 00 You may wish to perform the cancelling down yourself to check the final column. 2 2 4 0 20 00

The general procedure for converting a percentage (say 5%) into a fraction is: Write the percentage as a fraction of 00 i.e. 5 00 Cancel the fraction down to its simplest form. In this case we can divide top and bottom by the common factor, 5. When the fraction is in its simplest form, we are done. 5%= 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. Example Find 0% of 50. 0% is the same as 0 (from the table). So 0% of 50 = 0 50 50 as we first multiply by the numerator. 0 50 0 = 5 5 as 50 and 0 have a common factor of 0 Example Find 30% of 25. 30%= 30 00 3 0 30% of 25 =25 3 25 3 0 0 75 0 As 75 and 0 have a common factor of 5, we can cancel the fraction down 75 5 0 2 This is an improper fraction or top heavy fraction, so we convert it into a mixed number. = 7 Method 2 - Using Decimals As the number is used to represent a whole, we can also use it to represent 00%. We know that 50% is half of 00%, so 50% of must be half of, which as a decimal is 0.5. The following table shows the decimal form of some common percentages: Percentage Decimal 00% 50% 0.5 25% 0.25 22

0% 0. 5% 0.05 % 0.0 The general procedure for converting a percentage (say 5%) into a decimal is: Take the numerical value of the percentage, in this case 5, and divide it by 00. So 5% = 0.5, 7% = 0.7, 37% = 0.37 Example Find 0% of 50. 0 00 = 0. so 0% of 50 = 0. 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 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 20lbs and is 6ft in 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 20lbs. We need to find 25% of 20 Method - Fractions 25 25% 5 00 20 4 by cancelling 20=30 so 25% of 20 is 30 4 His total weight will be 20+30=50 50 lbs Method 2 - Decimals 25 25% 00 0.25 0.25 20=30 His total weight will be 20+30=50 50 lbs An alternative method is to notice that his total weight will be 00% of his original weight + 25% of his original weight. So his eventual weight will be 25% of his original weight. This means that we can shorten the above calculations: 00 25 5 25% 25 25 00 20 4 by 25%.25 23

cancelling 5 4 20=50 His total weight will be 50 lbs.25 20=50 His total weight will be 50 lbs Increasing by a percentage Example A patient weights 50 kg. They have a 2% weight gain. What is his new weight? Method Find out what 2% of 50 is and add that to the original weight. 2% = = 0.2 And 0.2 x 50 = 8 So that 50 + 8 = 68 The patient s new weight is 68 kg Method 2 Notice that if we add 2% to the original We now have 2%.2 x 50 = 68 As above the patient s new weight is 68 kg Decreasing by a percentage Example The dose of a drug given to a patient is to be reduced by 5%. If the patient had been originally prescribed 300 mg of drug A what is the new dosage in mg? Method Find out what 5% of 300 is and subtract that from the original dose 5% = = 0.5 And 0.5 x 300 = 45 24

So that 300 45 = 255 The new dosage is 255 mg Method 2 Notice that if we reduce the dosage by 5% 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 you ve found 25% of something, multiply it by 4 and you should have your original quantity back. 25

Percentages Exercise. Express as i) a fraction (simplify if possible), ii) a decimal a) 20% b) 30% c) 45% d) 95% e) 9% f) 2% g) 84% h) 29% 2. Using the method of your choice, evaluate the following: a) 20% of 5 b) 30% of 0 c) 45% of 200 d) 95% of 00 e) 9% of 300 f) 2% of 50 g) 84% of 25 h) 29% of 300 3. A baby s weight has increased since birth by 0%. When it was born it weighed 3kg. What is its new weight? 4. A young adult s height was measured and found to be.3m. They grow by 0% over the next year. What is their new height? 5. A patient loses 7% of their body weight after surgery. If they originally weighed 95 kg what is their new weight? For extra help with Percentages consult Mathematics leaflets Fractions, Decimals and Percentages: how to link them and Percentages available on the web at www.hull.ac.uk/studyadvice Percentages Exercise - Answers. 20 2 30 3 45 9 95 9 a) 00 0 5 0. 2 b) 00 0 0. 3 c) 00 20 0. 45 d) 00 20 0. 95 e) 00 0. 09 g) 29 h) 2 6 3 2 00 50 25 0.2 42 00 0. 29 00 50 25 0.84 2. a) 3 b) 3 c) 90 d) 95 e) 27 f) 6 g) 2 h) 87 3. New weight is 3.3kg or 3300g 4..43m 26

5. 8.35 kg Percentage increase/decrease We often need to find the percentage increase or decrease in a patient s weight. To do this we use the formula: Change in Weight x 00 Original Weight Example: A patient who originally weighed 50kg loses 2 kg. What is her percentage weight loss? Her change in weight is 2 kg and her original weight is 50 kg. So we have 2_ x 00 = 4 50 This represents a 4% weight loss. A patient who originally weighed 25 kg now weighs 35 kg. What is his percentage weight gain? Here the change in weight is 0 kg and the original weight is 25 kg. So that we have 0 x 00 = 8 25 This represents an 8% weight gain. 27

Percentages Exercise 2 Find the weight gain/loss of the following patients. a) Weight originally 80 kg, final weight 92 kg b) Weight originally 60 kg, final weight 63 kg c) Weight originally25 kg, final weight 20 kg d) Weight originally 200 kg, final weight 95 kg e) Weight originally 250 kg, final weight 245 kg Percentages Exercise 2 Answers a)5% weight increase d) 2.5% weight decrease b) 5% weight increase e) 2% weight decrease c) 4% weight decrease 28

Body Mass Index (BMI) The BMI provides a simple numeric measure of a person s fatness or thinness which allows healthcare professionals to discuss over and underweight problems objectively. The current settings are: BMI < 20 Underweight 20 < BMI < 25 Optimum weight 25 < BMI < 30 Overweight BMI > 30 Obese BMI > 40 Morbidly obese BMI is calculated by dividing the person s weight (in Kg) by their height 2 (in metres). The formula is written W Example H 2 Find the BMI of a patient who weighs 75 kg and is.42 m tall BMI = 75 / (.42 x.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.95m tall BMI = 93 / (.95 x.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.83m (24) b) Weight 5 kg, height 2.00m (29) c) Weight 78 kg, height.54m (33) d) Weight 62 kg, height.8m (9) e) Weight 32 kg, height.64m (49) 29

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. Find the average of 2, 6, 4, 8, 5 2+6+4+8+5 = 25 = 5 The average is 5 2. If a patient s oral fluid intake on successive days is 20 ml, 200 ml 40 ml and 260 ml, what was the average intake over 4 days? 20+200+40+260 = 720 = 80 The average intake is 80 ml Averages Exercise. A patient s pulse was taken every 30 minutes over 2 hours It was found to be 0, 05, 95 and 90 What is the average pulse rate over the 2 hours? 2. A patient s temperature was taken every 30 minutes over 4 hours. It was 38 C, 38 C, 38.5 C, 39. C, 38.4 C, 38.C, 37.4 C, and 42. C What is the average temperature over: a) The first two hours b) The second two hours Averages Exercise. 00 2. a) 38.4 C b) 39.0 C 30

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 000 000 Number of Kilograms Number of Grams Number of Milligrams x 000 x 000 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. 000 x 000 Number of Micrograms 000 Number of Nanograms x 000 3

Metric Measurements of Liquids Name Abbreviation Notes Litre L An upper case L Millilitre ml One thousand millilitres to a litre Conversion Chart Number of Litres 000 x 000 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 0mL. 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 (3) when it should be 3L. Always use L even in ml! Examples. Convert 575 millilitres into litres. From the diagram, we see that to convert millilitres to litres, we divide the number of millilitres by 000. So we have 575 000=0.575 litres 2. Convert 2.67 litres into millilitres. To convert litres to millilitres we multiply the number of litres by 000. So we have 2.67 000=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. 32

In Example above we can use our knowledge of litres and millilitres to estimate the result. We have 575 millilitres. If we had 000 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 24 kg, what is this in pounds (lbs)? 24 x 2.2 = 272.8 lbs Weights in pounds 2.2 = weights in kg. A patient weighs 22 lbs, what is this in kg? 22 2.2 = 96.37 (2dp) kg 33

Unit Conversion Exercise. Copy and complete the following, using the tables and diagrams a) kilogram = grams b) gram = milligrams c) gram = micrograms d) microgram = nanograms e) 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) 200mg b) 650mg c) 6749mg d) 3554mg 4. Convert the following into milligrams a) 20 micrograms b) 00 micrograms c) 2675 micrograms d) 2034 mcg 5. Convert the following: (you may find it easier to work out the answers in two stages): a).67grams into micrograms b) 0.85grams into micrograms c) 25 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) 0 millilitres b) 32 millilitres c) 2389 millilitres d) 23.4 millilitres 7. Convert the following into millilitres a) 4 litres b) 6.2 litres c) 0.94 litres d) 2.27 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 400 mcg has been prepared. Will this be enough? 34

Unit Conversion Exercise - Answers a) kg=000g b) g=000mg c) g=000000mcg d) mcg=000ng e) litre=000ml 2 a) 6g=6000mg b) 268g=26.800mg c) 3.924g=3924mg d) 405g=405000mg 3 a) 200mg=.2g b) 650mg=0.65g c) 6749mg=6.749g d) 3554mg=3.554g 4 a)20mcg=0.2mg b) 00mcg=.00mg c)2675 mcg= 2.675mg d) 2034mcg=2.034mg 5 a).67g=670000mcg b) 0.85g=850000mcg c) 25 mcg=0.00025g d) 6784mcg=0.006784g e) 48.9mg=48900000ng f) 3084ng=0.003084mg 6 a) 0mL=0.0litres b) 32mL=0.32litres c) 2389mL=2.389litres d) 23.4mL=0.234 litres 7 a) 4litres=4000mL b) 6.2litres=6200mL c) 0.94litres=940mL d) 2.27litres=2270mL 8 a) 40 milligrams b) 0.4 grams c) no, the correct dose would be 40000mcg 35

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 it s in What you ve got When working with tablets what it s in is always one tablet. To calculate a dosage you must write down 3 numbers. They are: What you want this is what is prescribed/ordered/required/needed by the patient. What you have got this is what is available. What it s in this is either 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. Note: In order to use this formula, the units of What you want and What you ve got must be the same, ie. both in mcg, or both in mg, or both in g. Examples. A patient needs 500mg of drug X per day. X is available in 25mg tablets. How many tablets per day does he need to take? What you want = 500mg } The units are both the same What you ve got = 25mg } What it is in = one tablet So our calculation is x = 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 36

What you ve got = 250mg } What it s in = 50mL So our calculation is 500 250 50 =00 We need 00mL 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 you ve got = 400mg } What it s in = 200mL So our calculation is 250 400 200 = 25 We need 25mL of solution. 4. A patient is prescribed 250mg of erythromycin IV. Stock on hand contains g in 0mL once diluted. What you want = 250mg What you ve got = g What it s in = 0mL The units of What you want are mg and the units of What you ve got are g. They must be the same units. Both in mg g = 000mg So: What you want = 250mg What you ve got = 000mg What it s in = 0mL Our calculation is 250 x 0 = 2.5 000 We need 2.5mL Both in g 250mg = 0.25g = 0.25g =g =0mL Our calculation is 0.25 x 0 = 2.5 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. Example A child weighing 2.5kg is prescribed a drug which is to be given in four equally divided doses. The dosage the child requires is 00mg/kg body weight. The child requires 2.5 x 00mg = 250mg of the drug. 37

So for four equally divided doses 250 = 32.5 4 They need 32.5mg 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 0mg 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 0mg 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 2500mL. 0 2500 We now divide the amount to be given by the time to be taken: 20 =25 The patient needs 2500mL to be given at a rate of 25mL 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 5% V/V means for every 00mL of solution, 5mL 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 5% W/V means that for every 00mL of solution there are 5 grams of the drug. 38

If we are converting between solution strengths, such as diluting a 20% solution to make it a 0% solution, we do not need to know whether the solution is V/V or W/V. Examples. We need to make up litre of a 5% solution of A. We have stock solution of 0%. 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 5 We want a 5% solution. This is the same as 00 or 20. 0 We ve got a 0% solution. This is the same as 00 or 0. We want our finished solution to have a volume of 000mL. Our formula becomes 20 000 0 = 0 20 000 (using the rule for dividing fractions) = 2 000=500. We need 500mL of the A solution. Which means we need 000-500=500mL of water. (Alternatively you can use the fact that a 5% solution is half the strength of a 0% 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 00mL of solution we have 20mL of drug F. Using our formula: What you want What you ve got What it s in 30 This becomes 20 00=50 We need 50mL of solution. 3. Drug G comes in a W/V solution of 5%. The patient requires 5 grams of G. How many ml of solution are needed? 39

5% W/V means that for every 00mL of solution, there are 5 grams of G. Using the formula gives us 5 00=300 5 300mL 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. Dosage Calculations Exercise. How many 30mg tablets of drug B are required to produce a dosage of: a) 60mg b) 20mg c) 5mg d) 75mg 2. Medicine A is available in a solution of 0mg 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 5 micrograms per 00mL. How many ml are needed to produce a dose of: a)50mcg 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 20mg, 3 doses b) total dosage 60mg, 2 doses c) total dosage 00mg, 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 00mL, over 2 hours b) Total dosage 750mg, solution of 0mg per 30mL, over 20 hours c) Total dosage 450mg, solution of 90mg per 00mL, over 0 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) 0mL of B c) 200mL of B ii) How would you make up the following solutions from the stock solution? a) Strength 20% volume litre b) Strength 0% volume 750mL iii) What strength are the following solutions? a) Volume 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 5% W/V stock solution. i) How much of the solution is needed to provide: 40

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 0% volume 750mL iii) How many grams of C are in the following solutions? a) Volume litre, made up of 400mL stock solution, 600mL water b) Volume 800mL, made up of 450mL stock solution, 350mL water Dosage Calculations Exercise Answers. a) 2 tablets b) 4 tablets c) 2 tablet d) 2 2 tablets 2. a) 50mL b) 25mL c) 000mL d) 425mL 3. a) 000mL b) 300mL c) 200mL d) 500mL 4. a) 2 tablets b) 2 tablets c) tablet d) 2 tablet 5. a) 00mL per hour b) 2.5 ml per hour c) 50mL per hour 6. i) a) 250mL b) 50mL c) litre ii) a) litre stock, no water b) 375mL stock, 375mL water iii) a) 600mL stock contains 20mL B 20 So 20mL in 000mL= 000 =2% b) 300mL stock contains 60mL B 60 So 60mL in 600mL= 600 =0% 7. i) a) 200mL b) 50mL c) 600mL ii) a) 300mL stock, 600mL water b) 500mL stock, 250mL water iii) a) 60g b) 67.5g Dosage Calculations Exercise 2 A drug is available in mg, 2 mg, 5 mg and 0 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 3 mg 2 7 mg 3 8 mg 4 0mg 5 mg 4

6 4 mg Dosage Calculations Exercise 2 Answers Tablets required Number of tablets Tablets required Number of tablets mg & 2 mg 3 mg, 2 mg & 5 mg 5 5 mg, 5 mg & mg 2 2 2 mg & 5 mg 3 4 5 mg & 5 mg 3 6 5 mg, 5 mg, 2 mg & 2 mg 2 2 4 Dosage Calculations Exercise 3. A solution contains furosemide (frusemide) 0 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 a 3 ml b 5 ml c 7 ml of the solution? 3. Another solution contains morphine hydrochloride 40 mg/ml. How many milligrams of morphine hydrochloride are in a 2 ml b 5 ml c 0 ml of this solution? 4. A suspension contains phenytoin 25 mg/5 ml. How many milligrams of phenytoin are in a 20 ml b 30 ml c 40 ml of the suspension? 42

5. A solution contains fluoxetine 20 mg/5 ml. How many milligrams of fluoxetine are in a 0 ml b 25 ml c 40 ml of the solution? 6. A suspension contains erythromycin 250 mg/5 ml. How many milligrams of erythromycin are in a 0 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 0 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 5 ml b 25 ml c 35 ml of the mixture? Dosage Calculations Exercise 3 Answers All answers are in mg a) 20 b) 30 c) 50 2 a) 6 b) 0 c) 4 3 a) 800 b) 200 c) 400 4 a) 500 b) 750 c) 000 5 a) 40 b) 00 c) 60 6 a) 500 b) 000 c) 500 7 a) 50 b) 50 c) 250 8 a) 750 b) 250 c) 750 43

Dosage Calculations Exercise 4 In each example, you are given the prescribed dosage and the strength of stock on hand. Calculate the volume to be given:. Ordered: penicillin 500 mg On hand: syrup 25 mg/5 ml 2. Ordered: furosemide (frusemide) 40 mg On hand: solution 0 mg/ml 3. Ordered: morphine hydrochloride 00 mg On hand: solution 40 mg/ml 4. Ordered: paracetamol 80 mg On hand: suspension 20 mg/5 ml 5. Ordered: phenytoin 50 mg On hand: suspension 25 mg/5 ml 6. Ordered: erythromycin 250 mg On hand: suspension 250 mg/5 ml 7. Ordered: fluoxetine 30 mg On hand: solution 20 mg/5 ml 8. Ordered: penicillin 000 mg On hand: mixture 250 mg/5 ml 9. Ordered: chlorpromazine 35 mg On hand: syrup 25 mg/5 ml 0. Ordered: penicillin 200 mg On hand: mixture 250 mg/5 ml. Ordered: erythromycin 800 mg On hand: mixture 25 mg/5 ml Dosage Calculations Exercise 4 - Answers All answers are in ml. 20 5. 6 9. 7 2. 4 6. 25 0. 24 3. 2.5 7. 7.5. 32 4. 7.5 8. 20 44

Dosage Calculations Exercise 5 Dosages of oral medications. A patient is ordered paracetamol g, orally. Stock on hand is 500 mg tablets. Calculate the number of tablets required. 2. Ordered: codeine 5 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 50 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 2.5 mg? 0. Digoxin 25 mcg is ordered. Tablets available are 0.25 mg. How many tablets should be given? Check that you have used the same unit of weight throughout a calculation. Are both weights in milligrams (mg)? Or are both weights in micrograms (mcg)? 45

Dosage Calculations Exercise 5 - Anwers All answers are in tablets. 2 5. 9. 2. 6. or 0.5 0. or 0.5 3. 7. 4. 2 8. or 0.5 46

Dosage Calculation Exercise 6 Calculate the volume of stock required. Give answers greater than ml correct to one decimal place; answers less than ml correct to two decimal places. Ordered Stock ampoule. Morphine 2 mg 5 mg/ml 2. Calciparine 7000 units 25 000 units in ml 3. Benzylpenicillin 500 mg.2 g in 0 ml 4. Heparin 3000 units 5000 units/ml 5. Phenobarbitone 70 mg 200 mg/ml 6. Pethidine 80 mg 00 mg/2 ml 7. Buscopan 0.24 mg 0.4 mg/2ml 8. Digoxin 200 mcg 500 mcg in 2 ml 9. Furosemide (frusemide) 50 mg 250 mg in 5 ml 0. Ondansetron 5 mg 4 mg in 2 ml. Capreomycin 800 mg g in 5 ml 2. Tramadol 20 mg 00 mg in 2 ml 3. Gentamicin 70 mg 80 mg in 2 ml 4. Vancomycin 800 mg g in 5 ml 5. Morphine 7.5 mg 0 mg in ml 6. Ceftriaxone 250 mg g/3 ml 7. Buscopan 25 mg 20 mg in ml 8. Dexamethasone 3 mg 4 mg/ml 9. Vancomycin.2 g 000 mg/5 ml 20. Naloxone 0.5 mg 0.4 mg/ml 47

Dosage Calculations Exercise 6 Answers All answers are in ml. 0.8 6..6. 4 6. 3.8 2. 0.28 7..2 2. 2.4 7..3 3. 2.5 8. 0.8 3..8 8. 0.75 4. 0.6 9. 3 4. 4 9. 6 5. 0.35 0. 2.5 5. 0.75 20..3 48

Dosage Calculations Exercise 7 Calculate the volume of stock to be drawn up for injection.. Pethidine 60 mg is ordered. Stock ampoules contain 00 mg in 2 ml. 2. An adult is ordered metoclopramide 5 mg, for nausea. On hand are ampoules containing 0 mg/ml. 3. A patient is prescribed erythromycin 250 mg, I.V. Stock on hand contains g in 0 ml, once diluted. 4. Tramadol hydrochloride 80 mg is required. Available stock contains 00 mg in 2 ml. 5. A patient is ordered benzylpenicillin 800 mg. On hand is benzylpenicillin.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 g in 3 ml. 7. Digoxin ampoules on hand contain 500 mcg in 2 ml. Digoxin 50 mcg is ordered. 8. Stock Calciparine contains 25 000 units in ml. 5 000 units of Calciparine are ordered. 9. Penicillin 450 mg is ordered. Stock ampoules contain 600 mg in 5 ml. Dosage Calculations Exercise 7 - Answers All answers are in ml..2 4..6 7. 0.6 2..5 5. 4 8. 0.6 3. 2.5 6..5 9. 3.75 (3.8 to dp) 49

Dosage Calculations Exercise 8. An injection of morphine 8 mg is required. Ampoules on hand contain 0 mg in 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 g in 0 ml, after dilution. Calculate the required volume. 5. Stock heparin has a strength of 5000 units per ml. What volume must be drawn up to give 6500 units? 6. Pethidine 85 mg is to be given I.M. Stock ampoules contain pethidine 00 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? Dosage Calculations Exercise 8 - Answers All answers are in ml. 0.8 4. 2.5 7..5 2..4 5..3 8. 3.0 3. 0.9 6..7 50

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 you ve found any errors, so that we can improve it for future use. Please contact the Maths Skills Adviser by email at skills@hull.ac.uk The information in this leaflet can be made available in an alternative format on request using the email above. 5