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gives an overview of the decimal system
revises the four basic methods of calculating
considers ways of expressing and calculating numbers less than one
describes how very large numbers can be expressed.
2.1 Introduction
The modern healthcare environment demands a good understanding of decimals and the ability to use them. This includes whole numbers like 18, 140 and 567 as well as parts of whole numbers. Some medications are prescribed in whole numbers, for example paracetamol 500 mg and some are prescribed in amounts that include less than one unit, for example bendroflumethiazide 2.5 mg.
2.2 The decimal system
A number is made up from individual digits and communicates a great deal of information. If 475 is used as an example, this isn’t simply a ‘4’, a ‘7’ and a ‘5’. The place of the digit within the number gives a value such as hundreds, tens and ones. Reading from left to right 475 has a value of 4 ‘hundreds’, 7 ‘tens’ and 5 ‘ones’. This is because we use a ‘base 10’ decimal system. This means that the value of each place in a number is 10 times greater than the number to the right of it. The place value of ‘7’ is tens and of ‘5’ is ones. Similarly, the value of each place is ten times smaller than the place to its left. Figure 2.1 presents this visually.
Figure 2.1. Whole numbers and decimal fractions.
Using the example of digoxin (a drug used to control an irregular heart rate), a common dosage is 125 micrograms. The place values in the number tell us the exact amount to give: 1 ‘hundred’, 2 ‘tens’ and 5 ‘ones’ or ‘one hundred and twenty-five’ micrograms. Figure 2.2 illustrates this.
Figure 2.2. Digoxin 125 micrograms.
The decimal point is used to signpost the end of the whole number and the beginning of amounts that are less than one. Returning to the digoxin example, this can also be prescribed as 62.5 micrograms. 62.5 tells us that there are six ‘tens’, two ‘ones’ and five ‘tenths’. Figure 2.3 illustrates the positions of whole numbers and parts of whole numbers or decimal fractions.
Figure 2.3. Digoxin 62.5 micrograms.
The role of zero
Within the decimal system, zeros play an important role when there are no values. If the number 702 is used as an example, ‘7’ indicates seven hundreds, ‘0’ indicates no tens and ‘2’ indicates two ones. The ‘0’ maintains the position of the other digits within the number.
When writing numbers that contain four digits or more, you will see various different formats:
four digits – one thousand may be seen written as 1,000 (using a comma), 1 000 (using a gap) or 1000 (closed up, no gap or comma); according to the metric (SI) system, all numbers up to 9999 should be written with no space and no comma
five or more digits – ten thousand five hundred may be seen written as 10,500, 10 500 or 10500; this book again uses the SI convention and presents all numbers above 10 000 with a space.
These standards should be used in healthcare and this approach has safety advantages because the comma cannot then be mistaken for a decimal point.
ERROR ALERT
Zeros can maintain the position of other digits, but they can also be the source of errors. When calculating and giving medications or fluids, it is critical to remove any trailing zeros – ones used after the decimal point that don’t maintain the position of other digits within a number. For example, five milligrams should be written as 5 mg and not 5.0 mg. If the decimal point is not clear, the result is a ten times overdose.
Self-assessment test 2.1: digit value
The recap questions below will help to consolidate your learning about the value of digits within a number. Answers can be found at the end of the book.
1 In the number 1.65, what value does the digit ‘5’ have?
2 In the number 6.079, what value does the digit ‘0’ have?
3 In the number 8.125, what value does the digit ‘5’ have?
4 In the number 4 012 000, what value does the digit ‘4’ have?
5 In the number 12.75, what value does the digit ‘2’ have?
6 In the number 2.09, what value does the digit ‘9’ have?
7 In the number 725.3, what value does the digit ‘3’ have?
8 In the number 7.005, what value does the digit ‘5’ have?
9 In the number 0.13, what value does the digit ‘3’ have?
10 In the number 9.125, what value does the digit ‘5’ have?
2.3 Addition
Most simple additions can be performed mentally, but where many individual numbers have to be added together, like when adding up the fluid intake of a patient over twenty-four hours, the potential for error increases. In these circumstances, it is sensible to perform the calculation on paper.
The numbers are written down in a column format as in the example below. This ensures that the numbers are lined up correctly for the calculation and maintains the place value of each digit within the number. When adding two or more numbers together, the calculation can be performed in any order: 74 + 26 gives the same answer as 26 + 74.
SENSE CHECK
As well as being able to perform calculations, you need to learn ways of checking to make sure you haven’t made a careless mistake. Remember that if you add numbers together, your answer must be greater than the numbers that you started with.
Use the following procedure to check your addition answers:
take one of the numbers that you added up away from the answer
for example, 6 + 9 = 15 and therefore 15 – 9 = 6.
| EXAMPLE 2.1 |
The columns are not usually labelled as hundreds (H), tens (T) or ones (O), but this helps to illustrate the calculation.
Method
The addition is calculated vertically from right to left, starting under the ‘ones’ column and involves three individual calculations, one for the ‘ones’ column, one for the ‘tens’ column and a final calculation for the ‘hundreds’ column.
Process
This gives the answer of 7 hundreds, 7 tens and 8 ones, or 778.
Checking
To check your answer: 778 – 62 = 716.
Not all additions are this straightforward as there will be times when the calculation results in ten or more in a column, such as in Example 2.2.
| EXAMPLE 2.2 |
Method
As before, the addition is calculated vertically from right to left, starting under the ‘ones’ column.
Process
The number ‘12’ is made up of one ‘ten’ and two ‘ones’. Because this column is only used to record the ‘ones’, the two ‘ones’ are recorded here and the ‘ten’ is carried over to the ‘tens’ column. The usual way of doing this is to write a small ‘1’ by the 6 under the ‘tens’ column.
This gives the answer of 7 hundreds, 8 tens and 2 ones, or 782.
Checking
To check your answer: 782 – 69 = 713.
| EXAMPLE 2.3 |
Fluid balance charts are used to monitor the fluid intake and output of patients. You will need to add up fairly large numbers, particularly when monitoring urine output.
If a patient passed 425 millilitres of urine after breakfast and has passed 485 millilitres just before lunch, how much urine have they passed during the morning?
Method
Perform the addition calculation vertically from right to left, starting under the ‘ones’ column.
Process
The number ‘10’ is made up of one ‘ten’ and no ‘ones’. Because this column is only used to record the ‘ones’, a zero is recorded here and the ‘ten’ is carried over to the ‘tens’ column. The usual way of doing this is to write a small ‘1’ by the 8 under the ‘tens’ column.
This part of the calculation is being performed under the tens column, therefore the number ‘11’ is made up of one ‘hundred’ and one ‘ten’ (eleven lots of ten). Because this column is only used to record ‘tens’, a one is recorded here and the hundred is carried over to the ‘hundreds’ column, identified as a small ‘1’ by the 4 under the ‘hundreds’ column.
This gives the answer of 9 hundreds, 1 ten and no ones or 910. So the total amount of urine passed during the morning is 910 millilitres.
Checking
To check your answer 910 – 485 = 425.
TOP TIP
It may seem tiresome adding up large amounts of numbers but persevere and don’t resort to using the calculator on your mobile phone. The NMC standards dictate that Registered Nurses must be able to perform calculations without the use of a calculator.
Self-assessment test 2.2: addition
The recap questions below will help to consolidate your learning about additions. Answers can be found at the end of the book.
1 23 + 77 =
2 156 + 239 =
3 17 + 3294 =
4 21 006 + 2005 =
5 179 + 642 =
6 130 + 150 + 190 + 250 + 80 + 225 =
7 125 + 145 + 155 + 68 + 95 + 300 =
8 500 + 200 + 150 + 45 + 60 + 120 + 397 =
9 220 + 140 + 50 + 65 + 72 + 168 =
10 85 + 33 + 120 + 235 + 128 + 50 =
11 During the course of a morning, a patient drinks the following amounts of fluid: tea 180 millilitres, orange 100 millilitres, water 120 millilitres, milk 125 millilitres and coffee 150 millilitres. What is the total amount of fluid that the patient receives?
12 A patient has a chest drain following surgery. Over the course of 24 hours, it drains: 125 millilitres, 70 millilitres and 40 millilitres. In addition, the patient is nauseous and vomits on four occasions, losing the following amounts of fluid: 250 millilitres, 150 millilitres, 120 millilitres and 90 millilitres. What is the total amount of fluid lost over the 24-hour period?
2.4 Subtraction
Subtractions involve taking one number away from another. In the clinical environment they are used in the calculation of fluid balance records to determine if the patient is in a negative or positive state of fluid balance and they are also used to calculate the stock levels of controlled drugs following administration.
The same basic rules apply as they did for additions. It is important that the digit positions are maintained to avoid errors and this is where the use of columns can help. Usually, but this is not always the case, a smaller number is taken away from a larger number, so place the larger number at the top and the smaller number below it. When calculating fluid balance records, for example, the fluid loss from the body can be greater than the input and so in this case you are calculating the difference between the input (this number at the top) and the output (this number at the bottom). Chapter 5 covers this in more detail.
SENSE CHECK
Don’t forget to check that you haven’t made a basic error with your calculation. If you subtract, the answer must be less than the number that you started with.
Use the following procedure to check your subtraction answers:
add your answer to the number that you took away
for example 34 – 22 = 12 so therefore 12 + 22 = 34.
| EXAMPLE 2.4 |
Method
The subtraction is calculated vertically from right to left, starting under the ‘ones’ column and involves three individual calculations, one for the ‘ones’ column, one for the ‘tens’ column and a final calculation for the ‘hundreds’ column.
Process
This gives the answer of 3 hundreds, 7 tens and 4 ones, or 374.
Checking
To check your answer: 374 + 311 = 685.
Not all subtractions are straightforward because there will be times when the calculation requires you to take a larger number from a smaller number; see Example 2.5 in which you need to take 5 from 3.
| EXAMPLE 2.5 |
Method
As before, the subtraction is calculated vertically from right to left, starting under the ‘ones’ column.
Process
Starting with the ‘ones’ column:
5 – 3 =
You cannot take 5 from 3 because 3 is less than 5. The way around this is to borrow 1 ‘ten’ from the 8 ‘tens’ under the ‘ten’ column. When this is transferred to the ‘ones’ column, this is added to the 3 to give 13. The usual way of doing this is to write a small ‘1’ by the 3 under the ‘ones’ column. The 8 under the ‘tens’ column needs to be reduced to 7 to account for 1 ten being borrowed by the ‘ones’ column. This involves crossing out the 8 and replacing it with a 7.
This gives the answer of 3 hundreds, 3 tens and 8 ones, or 338.
Checking
To check your answer: 338 + 445 = 783.
| EXAMPLE 2.6 |
When monitoring a patient’s state of hydration, fluid balance charts are invaluable. These involve calculating the amount of various types of fluid input and output as well as the total input and output from the body.
The overall fluid balance is calculated by subtracting the output from the input. If a patient’s total input over 24 hours was 2455 millilitres and their output was 2260 millilitres, you can calculate their fluid balance as follows.
Method
As before, the subtraction is calculated vertically from right to left, starting under the ‘ones’ column. Note that an additional column for ‘thousands’ has been included because the numbers are in the thousands.
Process
Moving left to the ‘tens’ column:
5 – 6 =
You cannot take 6 from 5, because it is a lower number. The way around this is to borrow 1 ‘hundred’ from the 4 ‘hundreds’ under the ‘hundreds’ column. When this is transferred to the ‘tens’ column, this is added to the 5 to give 15. This is recorded by writing a small ‘1’ by the 5 under the ‘tens’ column. The 4 under the ‘hundreds’ column needs to be reduced to 3 to account for 1 hundred being borrowed by the ‘tens’ column. This involves crossing out the 4 and replacing it with a 3.
This gives the answer of 0 thousands, 1 hundred, 9 tens and 5 ones, or 195. Therefore the patient’s input was 195 millilitres more than their output.
Checking
To check your answer: 195 + 2260 = 2455.
Self-assessment test 2.3: subtraction
The recap questions below will help to consolidate your learning about subtractions. Answers can be found at the end of the book.
1 155 – 42 =
2 1276 – 165 =
3 916 – 817 =
4 96 – 58 =
5 117 – 99 =
6 2139 – 126 =
7 6483 – 5261 =
8 2912 – 1915 =
9 792 – 689 =
10 542 – 454 =
11 If a patient’s total fluid input over 24 hours was 3205 millilitres and their output was 2410 millilitres, what is their overall fluid balance?
12 A patient takes morphine sulphate solution 10 mg in 5 millilitres orally, six times each day. The ward stock bottle contains 250 millilitres. How much will be left in the bottle after seven days?
2.5 Multiplication
Multiplication is the same as repeatedly adding the same number together, for example, 5 × 12 is the same as 12 + 12 + 12 + 12 + 12, but the process is less time consuming. Multiplications are used to calculate fluid balance charts where the same volume has been given repeatedly and when calculating drug doses that are prescribed per kilogram of body weight.
Many people remember their ‘times tables’ from school. If you didn’t learn these or have forgotten them, you might find that a copy of the ‘multiplication grid’ in Appendix 5 acts as a valuable resource. If you don’t know your times tables, it’s worth practising them until you are confident that you can do simple multiplications in your head.
When multiplying two or more numbers together, the calculation can be performed in any order, so 25 × 4 gives the same answer as 4 × 25.
To check your multiplications
Divide your answer by one of the numbers you multiplied.
For example 6 × 4 = 24 therefore 24 ÷ 4 = 6.
| EXAMPLE 2.7 |
Method
The multiplication is calculated vertically from right to left, starting under the ‘ones’ column and involves three individual calculations, one for the ‘ones’ column, one for the ‘tens’ column and a final calculation for the ‘hundreds’ column.
Process
This gives the answer of 4 hundreds, 6 tens and 8 ones, or 468.
Checking
To check your answer: 468 ÷ 2 = 234.
Not all multiplications are as straightforward as this. There will be times when the calculation results in ten or more in a column, such as in Example 2.8.
| EXAMPLE 2.8 |
Method
As before, the multiplication is calculated vertically from right to left, starting under the ‘ones’ column and involves three individual calculations, one for the ‘ones’ column, one for the ‘tens’ column and a final calculation for the ‘hundreds’ column.
Process
The number ‘15’ is made up of one ‘ten’ and five ‘ones’. Because this column is only used to record the ‘ones’, the five ‘ones’ are recorded here and the ‘ten’ carried over to the ‘tens’ column. The usual way of doing this is to write a small ‘1’ below the 2 under the ‘tens’ column.
This gives the answer of 3 hundreds, 7 tens and 5 ones, or 375.
Checking
To check your answer: 375 ÷ 3 = 125.
This process works well when the multiplication involves one number that is less than ten. When both of the numbers to be multiplied together are greater than ten, an extra stage of calculation is necessary.
| EXAMPLE 2.9 |
A fluid balance chart shows that a patient’s oral intake in one 24-hour period was 11 glasses of water. Each glass measures 85 ml. What is the total oral intake?
This calculation involves multiplying 85 and 11.
Method
As before, the multiplication is calculated vertically from right to left, making sure that individual digits are kept in position within the columns. The terms ‘ones’, ‘tens’ or ‘hundreds’ column used within the explanation only refer to the top number in the calculation.
Process
The multiplication has two stages, firstly multiplying the top number by the 1 ‘one’ belonging to the 11 and secondly multiplying the top number by the 1 ‘ten’ belonging to the 11.
Stage one
Stage two
Stage two starts by placing a ‘zero’ under the ‘ones’ column to maintain the place of the other digits. This is done because the number being used in the multiplication from the bottom line is a ten, not a ‘one’.
The results of the two separate multiplications are then added together:
So the total oral intake of fluid over the 24-hour period is 935 ml.
Checking
To check your answer: 935 ÷ 11 = 85.
TOP TIP
Remember that:
an even number multiplied by an even number always makes an even number
an even number multiplied by an odd number always makes an even number
an odd number multiplied by an odd number always makes an odd number.
Self-assessment test 2.4: multiplication
The recap questions below will help to consolidate your learning about multiplications. Answers can be found at the end of the book.
1 15 × 4 =
2 23 × 6 =
3 35 × 9 =
4 26 × 22 =
5 72 × 18 =
6 124 × 12 =
7 161 × 13 =
8 148 × 17 =
9 257 × 14 =
10 321 × 67 =
11 You need to order an inhaler to last a patient for the next 28 days. The patient takes four inhalations (doses) of the inhaler daily. How many inhalations does the patient take over 28 days and will an inhaler containing 200 doses be sufficient for that period?
12 A patient is due to go home for the weekend. They take 5 millilitres of an oral solution four times each day. Over the weekend they will need: one dose for Friday night, four doses for Saturday, four doses for Sunday and one dose for Monday morning. How many millilitres need to be in the bottle to last until the patient returns to hospital?
Multiplying decimals
Clinical calculations will sometimes involve multiplying decimals together. Treat this as you would any other multiplication and then use the technique below to get the decimal point in exactly the right place.
Imagine that you had to calculate the annual leave entitlement for a member of staff. Nurse Williams is a Practice nurse and is entitled to 2.33 days annual leave for each month that she works. She has worked at the surgery for 4.5 months. How much time off is she owed?
To calculate the amount of annual leave that she can take means multiplying 2.33 (days) by 4.5 (months):
Method
As before, the multiplication is calculated vertically from right to left, starting under the ‘hundredths’ column (h) and involves three individual calculations, one for the ‘hundredths’ column (h), one for the ‘tenths’ (t) column and a final calculation for the ‘ones’ column. The results of these three individual calculations are then added together.
Process
Starting with the bottom number at the ‘tenths’ column:
The digit 5 from the number 4.5 is used to multiply the digit 3 in the ‘hundredths’ column from the 2.33. It is then used to multiply the digit 3 in the ‘tenths’ column and then the digit 2 from the number 2.33.
5 × 3 = 15
Similar to the earlier multiplication examples, the 5 from the 15 is recorded below the line beneath the hundredths column and the 1 from the 15 is carried over to the ‘tenths’ column (written as a small ‘1’ in this column).
The digit 5 from the number 4.5 is then used to multiply the digit 3 in the ‘tenths’ column from the number 2.33
5 × 3 = 15
and the 1 carried over is added
15 + 1 = 16
The 6 ‘ones’ in this number are written below the line beneath the 5 of the number 4.5 under the ‘tenths’ column and the 1 ‘ten’ from this number is carried over to the ‘ones’ column
The digit 5 from the number 4.5 is then used to multiply the digit 2 in the ‘ones’ column from the number 2.33
5 × 2 = 10
and the 1 carried over is added
10 + 1 = 11
The 1 ‘one’ in this number is written below the line beneath the 4 of the number 4.5 under the ‘ones’ column and the 1 ‘ten’ from this number is written below the line in the ‘tens’ column
Moving left to the ‘ones’ column:
The first action is to write a zero in the ‘hundredths’ column below the 5 in the first part of the answer. The overall answer to this calculation is calculated by adding up three short answers and this zero acts as a placeholder within the short answer, maintaining the value of the digits.
The digit 4 from the number 4.5 is used to multiply the digit 3 in the ‘hundredths’ column from the 2.33. It is then used to multiply the digit 3 in the ‘tenths’ column and then the digit 2 from the number 2.33
4 × 3 = 12
Similar to the earlier multiplication examples, the 2 from the 12 is recorded below the line beneath the ‘tenths’ column and the 1 from the 12 is carried over to the ‘ones’ column (written as a small ‘1’ in this column).
The digit 4 from the number 4.5 is then used to multiply the digit 3 in the ‘tenths’ column from the number 2.33
4 × 3 = 12
and the 1 carried over is added
12 + 1 = 13
The 3 ‘ones’ in this number are written below the line beneath the 4 of the number 4.5 under the ‘ones’ column and the 1 ‘ten’ from this number is carried over to the ‘tens’ column (written as a small ‘1’ in this column).
The digit 4 from the number 4.5 is then used to multiply the digit 2 in the ‘ones’ column from the number 2.33
4 × 2 = 8
and the 1 carried over is added
8 + 1 = 9
This number is written below the line in the ‘tens’ column.
The next stage of the calculation involves adding up the individual answer numbers that are written below the original sum.
The final stage of the process is identifying the correct location for the decimal point. This is obtained by adding up the number of digits to the right of the decimal point in the numbers being multiplied together. The upper number 2.33 has two digits after the decimal point and the lower number 4.5 has one digit after the decimal point. Adding these together gives three digits, so there are three digits after the decimal point in the answer, making our final answer 10.485.
Checking
Checking the answer to your calculation can be done like other multiplications, by division: 10.485 ÷ 4.5 = 2.33 but this is a complex division. (The nurse checking this calculation could use a calculator but the nurse performing the calculation should not.)
An alternative is to estimate the answer. This doesn’t give an exact answer but guides you to the number you would expect to see the answer close to. A rough estimation of this answer involves rounding 4.5 up to 5 and 2.33 down to 2, which would give an estimate of 5 × 2 = 10. Remember that this estimation is to provide guidance for placing the decimal point. The alternative positions for the decimal point would result in the answer being 1.0485 or 104.85, and both of these are clearly far from the estimation.
If you’re feeling confident, a more accurate estimate would be to say that 2.33 is almost exactly 2; 2 × 4.45 is close to 9, and × 4.45 is 1.5 - adding 9 and 1.5 gives 10.5
Self-assessment test 2.5: multiplying decimals
The recap questions below will help to consolidate your learning about multiplying decimals. Answers can be found at the end of the book.
1 To help with the development of her academic work, a lecturer has agreed to provide Student Nurse Vipond with 0.75 hours of supervision each week for the next 8.5 weeks. How much supervision will she receive?
2 Student Nurse Jones has to complete an assignment over the next week. She has calculated that if she writes 2.5 pages per day for 5.5 days, she will achieve more than the minimum of 12 pages. Is she correct?
3 A patient with heart failure and fluid retention needs to lose 0.75 litres of fluid per day for the next 2.5 days. How much fluid will they lose in total?
4 A seriously ill patient is having their urine output measured hourly. They have excreted 0.06 litres of urine per hour over the last 3.5 hours. What is the total urine output over the 3.5 hours?
5 A patient is set the weight loss target of 1.25 kg per month for 2.5 months. How much weight in total should the patient lose over the 2.5 months?
6 A trauma patient has been receiving intravenous fluid at a rate of 0.25 litres of fluid per hour for the last 4.5 hours. How much intravenous fluid have they received in total?
7 A registered nurse earns £10.86 per hour. How much will she earn for 18.25 hours work?
2.6 Division
