Maths for Nurses: Fractions and Decimals

Transcription

1 Maths for Nurses: Fractions and Decimals This booklet will provide an overview of the basic numeracy skills for Nursing students. If you have any problems in answering the questions within the booklet please contact for personal help using the maths support drop-in sessions. Also check out these e-videos and quizzes: University of Leeds Maths for nurses Acknowledgement: The original document was designed and produced by Hull University Study Advice Service

2 Fractions Intro: A fraction is the ratio of two integers (whole numbers) eg, The number at the top is called the numerator, the number at the bottom is called the denominator. has a numerator and denominator. It is spoken two fifths. has a numerator and denominator. It is spoken three sevenths. a) 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. e.g. is equivalent to since is equivalent to since is equivalent to since Equivalent fractions are needed when we come to addition and subtraction of fractions. Usually the preferred form for a fraction is the simplest i.e. when numerator and denominator have no common factors so that we cannot divide any further. eg can be made simpler by dividing numerator and denominator by to get and we cannot make any simpler.this simplification is called cancelling. Cancelling fractions makes them easier to work with particularly when we multiply and divide fractions.

3 The four operations we need to look at are, +, and and in that order:- Fraction Multiplication We can get a simple rule for this operation as follows: The shaded area in the diagram Represents x = 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) The rule is:- multiply numerators, multiply denominators e.g. i) ii) iii) = (= after cancelling) Algebraically = c) Fraction Addition and Subtraction + - Addition type I If two (or more) fractions have the same denominator, then we just add the numerators e.g. i) It is easy to see this is the correct method by looking at the following diagram + / The shaded region is Similarly Addition type II

4 If the fractions have different denominators then it is not possible to use the above method. We do, however, have a technique for changing fractions to equivalent fractions and we use this to convert two fractions with different denominators into two fractions with the same denominator and then just use the Type I method. e.g. ii) If we multiply the numerator and denominator of by i.e. and if we multiply the numerator and denominator of by i.e. we have produced two equivalent fractions whose denominators are the same. We can simply add:- = e.g. iii) + Multiply numerator and denominator of by and multiply numerator and denominator of by to produce two equivalent fractions:-, and + = Choosing what we multiply by is not difficult. It is (usually) the denominator of the other fraction. e.g. iv) + Multiply numerator and denominator of by (the denominator of ) to get Multiply numerator and denominator of by (the denominator of to get = Then Do not forget to cancel your fractions, if possible, to produce the simplest form of your answer. e.g. v) = and So,, which equals, which equals (simplest form) Subtraction This is done in exactly the same way except instead of adding we subtract! e.g. i) (Same denominators, so Type I subtraction)

5 - (Different denominators so produce equivalent fractions) and so Algebraically, + = This looks a little awkward, but if we look at the answer we can, very quickly, produce it as follows + nd term A B + C D rd term st term + + = + = d) Division To find a method for this operation we proceed as follows: (Just rewriting the sum) = (Multiply numerator and denominator by the same number to produce an equivalent fraction)

6 x = x (Cancelling) x = = (simplifying) Before working this out (using the method from multiplying fractions) look at the number we have obtained. has become i.e. The first fraction has remained the same, the divide sign has become times x, and the second fraction has turned upside down. This happens in general and gives us a simple method for dividing fractions. e.g. i) ii) = iii) Algebraically Summarising i) = ii) iii) It is best NOT to remember these as formulas but as methods. e) Mixed Numbers 6

7 If we have a, so called, mixed number to deal with, i.e. a fraction and a whole number e.g.,, it is best to convert the whole number to a top heavy fraction or improper fraction, perform the operation using the above rules and then convert back to a mixed number if necessary. e.g. i) and So (cancelling) ii) and So = 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. Exercise. For each group of fractions, state which fractions are equivalent: a),,, 6,,, c) 9,, 0,. Cancel the following fractions down to their simplest form: 6 0 a) 0 c) 6. For each of the following pairs of fractions, state which one is the larger: a),, c) 6,. Convert the following mixed fractions into improper fractions:

8 a) 6 c) 6. Convert the following improper fractions into mixed fractions: 6 9 a) c) 6. Work out the following (simplify your answer if possible): a) 9 c) 6. Work out the following (simplify your answer if possible): a) c). Work out the following divisions (simplify your answer if possible): a) 9 c) 9. Work out the following divisions (simplify your answer if possible): a) c) 6

9 Decimals Intro:.,.,.,.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:.6. Here we have and a bit. The bit is 0.6. a) Place Value The value of a number is dependent upon its position. This is called place value. Thousands Tens Units Tenths Hundredths Thousandths The table above shows how place value works for decimals..0 has one unit and one hundredth.. has two units and tenths..9 has five tens, seven units and 9 tenths has one hundred, 6 tens, and thousandths Decimal Speak: It is usual to say the numbers after the decimal point as individual numbers. For example.9 would be said as four point nine three not four point ninety three Notice that where a number does not have a value for a column, a nought is used. This preserves the value of the following numbers. In this way 0. is different from 0.0 in the same way that 0 is different from. As with numbers in front of the decimal point, noughts not contained within a number are not usually written. i.e.. is really but we can just assume that the following noughts are there. Multiplying and dividing by 0 If you multiply a number by ten, its digits will remain the same, but they will move in relation to the decimal point. Example 0 = 0. As is the same as.0, all we have done is to move the decimal point one place to the right, so.0 becomes 0. 9

10 Alternatively you can think of this as the number moving one place to the left. Whichever you prefer, the end result is the same. This system works for numbers with decimal places too. Examples.6 0 = = = = = = 0. Notice that in the last example, the nought is treated in the same way as all other digits. When you are on a ward or in a clinic, you may be asked to measure doses of medication. For these calculations, a sound grasp of place value is essential, as 0. grams is 00 times the amount of a medicine that 0.00 grams would be. Multiplying by 00, 000 etc is performed in a similar way to multiplying by 0. Example We have seen that. 0 =. Multiplying by 0 again gives. 0 = As 00 = 0 0, multiplying by 00 is exactly the same as multiplying by 0, then multiplying the result by 0. Each time we multiply by 0 we move the decimal point one place to the right. Multiplying by 00 moves the decimal point one place to the right twice, so the overall effect is to move the decimal point two places to the right. So, looking at the example again:. 0 =.. 0 =. 00 = More examples.6 00 = =.09 00=0.9 The most common multiplication of this type you will have to do will be multiplication by 000. As 000 = 0 0 0, we can look at multiplying by 000 as multiplying by 0 three times in succession. Looking at our example:. 0 =.. 0 = 0 = = 0 The overall effect of multiplying by 000 is to move the decimal point three places to the right. 0

11 Division As division by 0 is the inverse process to multiplication by 0, we simply apply the same processes but in reverse. To multiply by 0, we move the decimal point one place to the right. To divide by 0 we move the decimal point one place to the left. Examples 0=. 0=.. 0= 0. In the same way we can divide by 00 and 000. Examples 00 =0. 00=.. 00= 0.0 Note- when dividing by 0, 00, 000 etc, it may be useful to write some noughts in front of your number so that you don t lose track. i.e = = = = 0.00, so. 000 = 0.00 c) Dividing by numbers smaller than Sometimes you may be asked to divide by numbers smaller than one. Example Evaluate 0.. Essentially this is asking us how many 0.s are in. The first thing that we do is note that 0. is one tenth. We know from our work on fractions that there are ten tenths in a unit. We have of these units. Hence our question can be changed to: Evaluate 0=0 When we divide a value by a number less than, our answer will be larger than the value we started with.

12 Exercise. Express the following in terms of hundreds, tens, units, tenths etc: a).9.0 c) Write these numbers in figures: a) One unit, six tenths and one thousandth Five tens and five tenths c) Three hundreds, six units, nine hundredths and one thousandth. Evaluate the following: a) 0. 0 c) d) e) f).9 00 g). 000 h) i) Evaluate the following: a) 0. 0 c) d) e) f).9 00 g). 000 h) i) Copy the procedure below to answer the following questions: The question asks for I am dividing by is one hundredth, so there are 00 of them in unit. I have 6. units, so I must have hundredths. The question is equivalent to = 60, so = 60 a) c)

13 Answers to exercises Exercise. a), 6, c), 6. a) 0 9 c). a) c). 6 a) 0 6 c). a) 6 c). a) 6 c). a) 9 c) 9. 6 a) 6 6 c). a) c) 6 Exercise. a) one hundred, two tens, five units and nine tenths eight tens, seven units, and three hundredths c) one hundred, two units, six hundredths and five thousandths.. a) c) a) 0 c) 0. d) 60 e) 09 f) 9 g) 00 h) 900 i) 00. a). 0. c) 0.00 d) 0.6 e) f) 0.09 g) 0.00 h) 0.09 i) a).9 0.=.9 x 0 = = x 000 = 000 c) = 0. x 00 =