BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Calculations 2 In this section we will look at - rounding numbers up or down - decimal places and significant figures - scientific notation - using calculators with fractions, rounding, significant figures and scientific notation - common mistakes with a calculator Helping you practice At the end of the sheet there are some questions for you to practice. Don t worry if you can t do these but do try to think about them. This practice should help you improve. I find I often make mistakes the first few times I practice, but after a while I understand better. Videos All the examples in this worksheet and all the answers to questions are available as answer sheets or videos. Good luck and enjoy! Videos and more worksheets are available in other formats from www.mathsupport.wordpress.com
1. Rounding numbers Sometimes numbers are given very precisely, for example, The population of Wales was estimated to be in 2010 and we would say this as 'two million, nine hundred and ninety nine thousand, three hundred and nineteen' but this is not particularly useful as a week later this number will be different. It is more useful to say the population of Cardiff in 2010 was about three million 3,000,000. Saying 'three million' instead of 'two million, nine hundred and ninety nine, three hundred and nineteen is called rounding and in this case we have rounded to the nearest million. Rounding up or rounding down To decide whether to round up or round down we need to see where the number is on the numberline, for example 120 121 122 123 124 125 126 127 128 129 130 Here we have drawn 123 pounds on the number line and we can see that it is nearer the number 120 than the number 130. So if we round to the nearest 10 it becomes 120 because it is nearest to 120 on the number line. 120 121 122 123 124 125 126 127 128 129 130 But if we draw the number 129 we can see that it is nearer to 130 than to 120 and so 129 rounded to the nearest 10 is 130 because it is the closest multiple of ten on the number line.
Rounding to the nearest ten: the less than 5 rule Here we will start by rounding to the nearest ten only. If we have 0, 1, 2, 3, or 4 pounds then to the nearest 10 we say that we have 0. But if we have 5, 6, 7, 8 or 9 pounds then rounded to the nearest ten we have 10 pounds. We can see this on the number line. 0 1 2 3 4 5 6 7 8 9 10 The number 4 is nearer 0 than 10 but the number 6 is nearer 10 than 0. Although 5 is in the middle, we round it to 10. Using the same ideas, if we have 22, its nearest multiple of 10 is 20 and if we have 27, then to the nearest 10 it is 30. 20 21 22 23 24 25 26 27 28 29 30 Remembering for example, if we have 155 then to the nearest 10 it is 160. 150 151 152 153 154 155 156 157 158 159 160 Easy way to round However if we draw numberlines each time we round a number it would take a lot of time, so instead we can round in the following way. Let's look at the number 27 and round it to the nearest 10. If we are rounding to ten we put a ring around the number of tens, which in this case is 2. 27 rounded to nearest 10 30 Now we look at the number to the right (which we underline). If this underlined number is 5 or higher we round the circled number up one to give 3. We then set numbers to the right of this circled number to zero.
Another example: We want to round 431 to the nearest ten. So we put a ring around the tens and underline the number to the right rounded to nearest 10 431 430 Is the underlined number five or more? No, so we don't round the 3 up, we just leave it as 3, but we change all the numbers to the right to zero to get 430. Rounding to the nearest 100 If we want to round a number to the nearest 100, then we ring the hundreds and underline the tens. If there are more than five tens we round up, and if there are less than five tens we round down. For example rounded to nearest 100 237 200 rounded to nearest 100 581 600 Rounding to any multiple of ten If we take a number such as the population of Dundee 141,937 we can round this to the nearest ten, hundred, thousand etc. rounded to nearest 10 141,937 141,940 rounded to nearest 100 141,937 141,900 rounded to nearest 1,000 141,937 142,000 rounded to nearest 10,000 141,937 140,000 rounded to nearest 100,000 141,937 100,000 2. Decimal places and significant figures Dealing with decimal places and significant figures is very similar, so we will start with decimal places and then move onto significant figures.
DECIMAL PLACES When dealing with decimals we use the words decimal point and decimal place to help us describe the number. Consider the following number. The decimal point is between the number 2 and 3. If we count the numbers after the decimal point, this tells us the number of decimal places. In this case we have 6 decimal places. decimal point 1 The number 3 is in the first decimal place The number 4 is in the second decimal place The number 0 is in the third decimal place The number 5 is in the fourth decimal place The number 8 is in the fifth decimal place The number 7 is in the sixth decimal place Try to work out how many decimal places the following numbers have: 1) 2.351 2) 9.01356 3) 14.2 4) -6.2 5) 130.01 6) 200.1002 The answers are: three, five, one, one, two and four. Rounding to decimal places Rounding to a number of decimal places works in exactly the same way as rounding to 10 or 100.
Consider the number, if we want to round this to one decimal place we are asking if the number is nearer 2.3 or 2.4. We could draw a number line to see which is nearer 2.3 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.4 Or we could look at the number directly, circle the decimal place we are rounding to and use the number to the right to tell us whether to round up or round down. We use d.p. to mean decimal place. So if we round to 3 decimal places we write 3 d.p. rounded to 5 d.p. rounded to 4 d.p. rounded to 3 d.p. rounded to 2 d.p. rounded to 1 d.p. 2.34059 2.3406 2.341 2.34 2.3 Notice that if we round to a d.p. the extra zeros to the right of the that decimal place can be left out. But it is important to remember we can t do this when rounding to tens, hundreds etc. Consider if we have 2,315 then rounded to the nearest thousand is definitely not 2 but is 2,000. SIGNIFICANT FIGURES
In the same way as decimal places counts the numbers after the decimal point, significant figures count the numbers from the first number in the expression. Consider the population of Wales again, second significant figure seventh significant figure first significant figure sixth significant figure It is usual to write Significant Figure is usually written as s.f. so the 3rd s.f = 9 and the 4th s.f. = 9 and the 5th s.f. = 3 To round this number to any significant figure we could use the numberline as before but this time we will just round by looking directly at the number. rounded to 6 s.f. rounded to 5 s.f. rounded to 4 s.f. rounded to 3 s.f. rounded to 2 s.f. rounded to 1 s.f. 2,999,320 2,999,300 2,999,000 3,000,000 3,000,000 3,000,000 Notice that we have a slightly special case here. Since there are three nines in a row when we round to 3 s.f. from 7 s.f. we have to round up from to 3,000,000. The same is true for rounding to 2 s.f. and 1 s.f. The questions at the end of this worksheet include plenty of practice at rounding significant figures and decimal places. 3. Scientific notation
Scientific notation uses the ideas we started to explore in indices to express large or small numbers in a convenient way that prevents scientists from making mistakes in calculations by missing out zeros when writing numbers. To write numbers in scientific notation - the number is always re-written as a number bigger than or equal to 1 and smaller than 10, so that there is only one number to the left of the decimal point - the magnitude of the number is always written as a power of ten For example Where the number in bold is in scientific notation. A quick trick An easy way to write any number in scientific notation is to count the number of places the decimal point has to move. For example if we write the number 338,100,000 we have to move the decimal point 8 places to the left to make sure that only one number is to the left of the decimal point, giving a power -8: 8 7 6 5 4 3 2 1 and if we now consider 0.00003193 we see that the decimal point has to move five places to the right, and so the power is -5. -1-2 -3-4 -5 Once we are able to put any number into scientific notation we can just do the opposite to get a number from scientific notation into decimal format.
For example 1) 2) 3) 4) Rounding with scientific notation Rounding when the number is in scientific notation is easy if we are happy rounding with numbers written in their usual format since we apply exactly the same rules but just to the decimal and not to the power, for example 1) 2) 3) 4) 5) And some trickier ones 1) 2) 3) Notice how we need to include the significant figures even if they are zero. For example if I say I have 200 pound to 3 s.f. then I have exactly 200, and not 201 and not 199. But if I have 200 to 2 s.f. then I could have 204 or 195. 4. Using calculators
Here we continue from the work in 02 Calculations 1 (so we assume that you are using a scientific algebraic calculator for example any of the Casio FX range). We first look at using rounding and significant figures and then how to type numbers in scientific notation in calculators Significant figures and rounding When using calculators for complicated calculations it is usual to get an answer that has as many significant figures as a calculator will allow. For example 1 inch = 2.54 cm, so 1 cm = 1 2.54 inches and if we enter this into a calculator we get an answer as 0.3937007874015... and the number will be longer if your calculator allows more numbers to be displayed. This reason is one of the reasons it is important to be able to correctly round numbers. As a rule of thumb 3 significant figures are good enough for most calculations (so in this case we would write 1 cm = 0.394 inches) but check or ask to see how many significant figures a particular calculation needs. Scientific notation You can test how your calculator displays numbers in scientific notation by entering 1,000,000,000 x 1,000,000,000, which should give an answer of To enter a number in scientific notation you should use the button [EXP] or [ ]. For example to enter the number type Common Mistakes: There are easy mistakes to make with calculators involving missing out brackets. Try the following calculations. If you get different answers making sure you are using brackets correctly. All answers are to 3 s.f. 1) 4) 2) 5) 3) 6) 5. Now your turn
Generally the more maths you practice the easier it gets. If you make mistakes don t worry. I generally find that if I make lots of mistakes I understand the subject better when I have finished. If you want to see videos explaining these ideas and showing the answers visit www.mathsupport.wordpress.com A) Rounding including decimal places and significant figures 1) round 1.38972 to 1 d.p. 2) round 14.90012 to 6 s.f. 3) round 0.056443 to 2 s.f. 4) round 0.01507 to 2 d.p. 5) round 500,419 to 3 s.f. 6) round 0.099501 to 3 d.p. 7) round 10101 to 4 s.f. 8) round 0.992 to 1 d.p. B) Write the following numbers in scientific notation 1) 230 2) 1,579 3) 0.00546 4) 11 5) 0.219 6) 0.00005007 7) 999.9 8) 0.000100012 C) Write the following numbers as one number in decimal format 1) 2) 3) 4) 5) 6) 7) 8) D) Carry out the following calculations using your calculator 1) 2)
3) 4) 5) 6) 7) = 8) E) General word problems involving calculations, indices and rounding 1) The cost of a second hand Skoda car is 5975. What is this amount to the nearest 10, to the nearest 100 and to the nearest 1,000? 2) The GDP (Gross Domestic Product) is the market value of all final goods and services from a country in one year. In millions of US Dollars the GDP of the UK is 2,258,565 and the GDP of Japan is 5,390,897. Using the exchange rate of 1 US Dollar = 0.6430 UK pounds, give both these numbers in pounds sterling in scientific notation to 3 significant figures. 3) In Berlin in 2009 the 100m world record was broken by Usain Bolt. His time was 9.58 seconds. This beat the previous world record of 9.69 seconds which was help by Tyson Gay. How much faster was Usain than Tyson and write this number in scientific notation. 4) Most bacteria species are spherical, and have a radius of about a micrometer. We can write 1 micrometer in scientific notation as. If the volume of a sphere can be calculate from the formula Volume =, where the symbol r=radius, and =3.14, calculate the volume of a single bacteria in and give your answer to 3 s.f in scientific notation 5) For certain drugs the dosage for children will depend on their weight and their temperature. For a fever less than the dose is 5mg per kilogram of weight of the child. Above the dose is 10mg per kilogram of weight of the child. The dose should be taken every 8 hours. If a child weighs 11kg and has a temperature of how many milligrams of the drug do they take in one day.