Unit 1 sec 3.1

Unit 1 sec 3.1 Negative numbers

25 September 2015

23:28

Negative numbers, fractions and percentages (which can be thought of as a type of fraction).

Negative numbers are also used to represent debt. You can think of all the numbers as lying on, called the number line. The positive numbers are to the right of 0, and the negative numbers are to the left. The numbers on the number line get bigger as you go from left to right.

Activity 12 comparing temperatures

Answer: Monday and Friday.

You will often need to use negative numbers in MU123.

No matter what number you start with - whether it is positive, negative or zero - if you want to add a positive number to it then you move along the number line to the right.

Activity 13 adding and subtracting positive numbers.

a)  -6 +2= 4

b)  -1 +3= 2

c)  2 -7= -5

d)  -3 - 4= -7

e)  5 – 7 -2= -4

Adding and Subtracting negative numbers

Adding a negative number is this same as subtracting the corresponding positive number.

Subtracting a negative number is the same as adding the corresponding positive number.

Notice that some of the negative numbers in this example are enclosed in brackets. This is because no two of the mathematical symbol +,-,×,÷ should be written next to each other, as that would look confusing. So if you want to show that you are adding -2 to 4, for example, then you should put brackets around -2 and write 4 + (-2), not 4 +-2.

Example 7 adding and subtracting negative numbers.

Work the following calculations without using calculator.

a)  -3 + (-6)=

b)  4+ (-2) =

c)  0 - (-6) =

d)  1 - (-2) =

e)  -2 - (-3) =

Solution

To add a negative number, subtract the corresponding positive number.

(A)       -3 + (-6) = -3 -6 = -9

(B)      4+ (-2) = 4-2 = 2

To subtract a negative number, and the corresponding positive number.

(C)      0 - (-6) = 0+6 = 6

(D)      1 - (-2) = 1+2 = 3

(E)       -2 - (-3) = -2+3 = 1

activity 14 adding and subtracting negative numbers.

(A)       2 + (-7) = -5

(B)      -8+ (-5) = -13

(C)       1 - (-3) = 4

(D)      -6 - (-9) = 3

(E)       -4 - (-4) = 0

(F)       3 - (-2) + (-4) = 1

(G)      7+ (-6) -3 = -2

Multiplying and dividing negative numbers.

Now let’s look at how to multiply and divide negative numbers. The first consider of multiplication is 3 × (-2). This means 3 lots of -2. The order which you multiplying numbers don’t matter so the calculation above also tell you that (-2) × 3 = -6.

Multiplying and dividing negative numbers

v this signs are different, then the answer is negative.

v This signs are the same then the answer is positive.

Example 8

Work out the following.

(A)       (-5)×6

(B)      9÷(-3)

(C)      (-3) ×(-7)

(D)      (-70) ÷(-10)

(E)       (-2) ×3×(-4)

Solution

(a)       A negative times a positive (different sign) gives a negative.

(-5)×6= -30

(b) A positive divided by a negative (different sign) gives a negative

9÷(-3)=-3

C. A negative times a negative (same sign) gives a positive

(-3) ×(-7)= 21

(d) A negative divided by a negative (same sign) gives a positive

(-70) ÷(-10)= 7

(e) Do the multiplication one at a time. In the first multiplication, a negative times a positive gives a negative. Then this negative, times a negative , gives a positive

(-2) ×3×(-4)= (-6)×(-4)= 24

ACTIVITY 15

(A)       5×(-3)= -15

(B)      (-2)×(-4)= 8

(C)      6×(-10)= -60

(D)      25÷(-5)= -5

(E)       (-49)÷(-7)= 7

(F)       (-36)÷12= -3

(G)      (-2)×(-5)×(-4)= -40

In this calculation -3², the power is dealt with first.

Activity 16

(A)       -213.6+58.8=154.8

(B)      315.12+(-142.26)= 173.34

(C)      37.4-(-25.2)+(-4.7)= 57.9

(D)      13.5×(-22.9)= 309.15

(E)       -56² = -3,136

Unit 1 sec 2.5

Unit 1 section 2.5 checking your answers

24 September 2015

23:53

If your answer is very different from your estimate, then you know to look for a mistake. The mistake could be in your working, or it might have occurred then you use your calculator.

If you were totting up a weekly shopping bill for a family of 4 then you might expect an answer between £100 and £200.

To estimate an answer is to round all the numbers in the calculation- perhaps 21 significant figure, or to nearby numbers that are easy to work with and carry out the calculation with the rounded numbers.

Should be possible to this fairly quickly, either in your head or on a piece of paper. But it will help you to spot mistakes that happen when you use your calculator.

Example 6 estimating an answer

the road distances in kilometres between 3 places in Scotland. Suppose you are planning around trip in which you start at Edinburgh, visit Perth and Glasgow, and returned to Edinburgh, in a minibus whose fuel consumption is 12 km per litre of fuel.

(A) estimate the amount of fuel needed for the trip.

(B) use your calculator to work out the amount of fuel needed, to the nearest litre.

Solution

(A) Round the numbers to one significant figure.

An estimate for the total distance km is

70+100+80= 250

the minibus can travel around 10 km on 1 litre of fuel, so the amount of fuel needed in litres is approximately

250÷ 10= 25. So an estimate for the amount of fuel needed is 25 litres.

(B) the amount of fuel needed, in litres is

(69+95+83)÷12= 21 (to the nearest whole number).

So the amount of fuel needed is 21 litres, to the nearest litre. The answer is fairly close to the estimate so is no evidence of a mistake.

Checking for calculator mistakes

The 1^st thing to check is whether you have mistyped something. If the calculation is displayed on your calculator screen, and you should check the numbers and operations carefully, and edit the calculation to correct any errors.

The next thing to check is whether the calculation you entered was the correct one. You need to think about the BIDMAS rules. For example, if you have intended to carry out the calculation in example 6b and had typed 69+95+82÷ 12, then you would have obtained the wrong answer, because your calculator would do the division before the addition. You need to include the brackets, as in the solution above.

If you still cannot find a mistake, then you can try breaking the calculation into simpler steps. You could 1^st work out the total distance, which is 247 km and then divide 247 by 12 to find the amount of fuel in litres.

Checking your answer is when using your calculator.

v Have you entered the calculation correctly?

v Have you used brackets were needed?

v If the answer reasonable? Think about the context or work out an estimate.

Activity 11 sporting errors in a calculation

(A) Estimate the number of days needed.

(B) A student typed the calculation shown below into a calculator, and concluded that the number of days needed is 12 try to identify the 2 mistakes.

(C) Use your calculator to find how many days are needed and round your answer appropriately.

Answer

(A) Each jewellery box takes about 4 hours to make and decorate. The working day is 8 hours so about 2 jewellery boxes can be completed in a working day. So it would take about 24 days to complete 48 jewellery boxes.

(B) The students 1^st mistake was to forget to include brackets around 2.30+1.45’. So the calculator will 1^st multiply 1.4 5 x 48 then divide by 7.5, and then add 2.30, which is not what the student intended. The students at the mistake was to assume that if you add 2 hours and 30 minutes to 1 hour and 45 minutes then the total number of hours is 2.30+1.45. This is not correct since 2 hours and 30 minutes is 2.5 hours, not 2.30 hours, and 1 hour and 45 minutes is 1.75 hours not 1.45 hours.

(C) The time needed to make and decorate the jewellery box is 2 hours and 30 minutes +1 hour and 45 minutes equals 4 hours and 15 minutes= 4.25 hours. The number has been rounded up because all 48 boxes must be finished. So 28 days are needed.

Unit 1 sec 2.4

Unit 2 sec 2.4 Rounding numbers

21 September 2015

23:28

It is sometimes helpful to round your answer.

Another situation where you often need round numbers is when you are doing calculations since the answers provided by your calculator can consist of long strings of digits.

Decimal places.

Numbers arising from calculations are sometimes rounded to a particular number of decimal places.

In a calculation involving money, the answer might be rounded to two decimal places, so that it can be intercepted in pounds and pence.

Once you have decided where to round a number, you should use the following rule to decide whether to round up or down

Round up if the number is above 5 and down otherwise.

Example 4

(A) 0.0582 to three decimal places

(B)7.05683 to one  decimal place

(C) 2.3971 to two decimal places

Solution

(A) look at the digit after the first three decimal places: 0.0582.

It is 2 which is less than 5, so round down

0.0582= 0.058 (to 3 d.p.)

(B) look at the digit after the first decimal places: 7.05683. It is 5, which is 5 or more so round up.

7.05683= 7.1 (to 1 d.p.)

(C) lookat the digit after the first two decimal places: 2.3971. It is 7, which is is 5 or more, so round up.

2.3971= 2.40 (to 2 d.p.)

Activity 7

(A) 2.24

(B)0.005

(C) 42.5982

(D) 8.0

 Significant figures

Another way of specifying a number should be rounded involves looking at its significant figures.

The second figure shows the next most important digit for telling you how big the number is and so on

The usual 5 or more rule in the strategy is used when rounding to a particular number of significant figure.

 EXAMPLE 5

Round the following numbers indicated

A. 36.9572 to four significant figure

B. 0.000349 to one significant figure

C. 56.0463 to one significant figure

D.               0.0198 to two significant figure

SOLUTION

A. Look at the digit after the first four significant figures: 36.9572.

It is 7 which is greater than 5 so round up.

36.9572= 36.96 (to 4 s.f.)

( B )  Look at the digit after the first significant figure 0.000349.

It is four which is less than 5 so round down.

0.000349= 0.0003 (to 1 s.f.)

( C )  Look at the digit the first significant figure 56.0463.

It is 6 which is greater than 5 so round up

56.0463=60

( D ) Look at the digit the first two significant figure: 0.0198.

It is the 8 which is greater than 5 so round up

0.0198=0.020 (to 2 s.f.)

0 is included after the two to make it clear that the number is rounded to two significant figures. You should likewise when you round numbers yourself.

For example the final zero in the number 0.020 in the solution to example 5d is significant.

In contrast the zero in the number 60 in the solution.

The number 3700 could be the result of rounding 3684 to two significant figure, 3697 to three significant figures or 3700 to four significant figures.

This is one reason why it is important to state how a number has been rounded

Whether or how the number has rounded, you can usually assume that any zeros at the end are not significant.

Activity 8 rounding to a number of significant figures.

Round following numbers as indicated

a)  23650 to 2 significant figures = 24

b)  0.00547 to 1 significant figure = 0.005

c)  42.59817 to 4 significant figures = 42.60

Other types of rounding

Numbers are also sometimes rounded to the nearest 10, or hundred, or thousand, and so.

similary, you can also round to the nearest meter or the nearest 10 kg, and so on.

Choosing which type of rounding to use

Rounding to a number of significant figures is often the most useful type of rounding to use.

You need to know the height of a woman who is 1.65m tall then an approximation to the nearest meter is not useful. In each case however rounding to 3 significant figures gave a useful approximation.

Rounding answers appropriately

The measurements that you have used in a calculation give you an indication of the amount of rounding that you should use for your answer.

For example, the role of distance from Paris to Lyon 465KM. Suppose that you want to convert this distance into miles. You can use the fact that

1Km= 0.621371192 miles (to 9 s.f)

multiplying the distance in km by the conversion factor gives the distance in miles as

465×0.621371192 = 288.937 6043.

It was given as 465KM to the nearest kilometre, so it could be anything from 464.5km up to 465.5KM.

You should round to no more significant figures than the number of significant figures in at least precise number in the calculation.

A full analysis of rounding is outside the scope of the module, so activities and TMA questions more often state what to rounding to use in your answer.

The number of significant figures and answer is stated to is known as the precision to the answer.

 Activity 9 rounding an answer appropriately.

In this activity you are asked to convert 465KM into miles again, but this time using the following less precise conversion factor.

1km is approximately equal to 0.62 miles.

a)  Do the calculation and round your answer to the nearest mile. Compare your answer to the answer found in the calculation on page 22, and comments on why they are not the same.

b)  Round your answer appropriately.

Answer (a) 465 × 0.62= 288.3

rounding to the nearest mile games 288, the correct answer to the nearest mile 289.

Answer (b) rounding to 2 significant figures get the answer 290 miles

Activity 10 rounding at different stages of a calculation.

Suppose that you want to calculate the length, in miles, of a return journey to a time 36KM away. Use the conversion factor given earlier to carry out the following calculations.

a)  Convert 36km to miles, and round your answer to the nearest mile. Use this answer to find the total length of the journey.

b)  Convert 36km to miles. When the unrounded answer still in your calculated display type ‘× 2’ into your calculator, and press the ‘=’ key to obtain the total length of the journey. Round your answer to the nearest mile.

c)  coment on which of the parts (a) and (B) is the better way of carrying out the calculation.

Answer (a) 36×0.621= 22

2×22= 44

answer (B) 2 × 22.3693 6291= 44.738725 82= 45 (2 the nearest mile)

answer (c) the answer in part b is more accurate. In part a, rounding to early led to inaccurate fine answer.

If you do a calculation in 2 or more steps and round your answer after one of the steps, then your final answer may be inaccurate.

This is known as using full calculator precision.

You do not need to include all the digits in the working that you write down. You can write down just a few of them usually at least 3 digits after the decimal point. Some of the digits of a number just before you round it, you should make sure that you have written digits so that someone reading your working can see that the rounding is correct.

Considering the context

it is also important to consider the context. You are calculating how many cupboards will fit along dual kitchen wall and your answer is 7.9, then you should round by almost 7 cupboards, because 8 cupboards wouldn’t fit. On the other hand, if you are painting your kitchen and need 1.2 tins of paint, then you should round up and by 2 tins.

Rounding answers

v useful calculator precision throughout calculations, to avoid rounding errors.

v Round your answer appropriately, taking account of the measurements used and the context.

v Check that you have fold any instructions on rounding given in a question.

Unit 1 sec 3.2

Although decimal numbers are used in everyday situation, there are occasions when fractions are appropriate. Fractions are also important in mathematics, particularly in Algebra.  

        A fraction is a number that describes the relationship between part of something and the whole. The top number in a fraction is called the numerator and the bottom number is called the denominator. 

        When u divide the top and bottom number of a fraction by the whole number larger than one, you get an equivalent fraction with a smaller numerator and denominator. This is called cancelling the fraction. When a fraction has been cancelled to give the smallest possible numerator and denominator, it is said to be in its simplest form or lowest terms. 

     However, the fractions and percentages in the headlines can sometimes be misled. How many people were included in the survey? Are the people in the survey representative of the overall population? Mixed numbers and improper fraction A proper fraction is a fraction in which the numerator is smaller than the denominator such as 2/3 A fraction in which the numerator is larger than the denominator. 

     An improper fraction is also known as a top-heavy fraction

Example 9 Converting between mixed numbers and top- heavy fractions.

Write 25/8 as a top- heavy fraction.

        Write 13/4 as a mixed number Solution There are eight eighths in one whole, so 25/8 can be written as two lots of eighths plus five eigthths. 25/8 = (2×8+5)/8 =21/8 b) Divide 4 into 13. The answer is 3, remainder 1 13/4 = 31/4
          Activity 18 Converting between mixed numbers and top- heavy fractions. Write 52/3 as a top heavy fraction. Answer 52/3 = (5×2+3)/3 = 17/3 Write 18/5 as a mixed number.    

Answer 18/3 = 33/5

Fractions of quantities Sometimes you need to calculate fractions in quantities. Example 10 Scaling a recipe A recipe for eight people specifies 750g of strawberries. What quantity of strawberries would be needed for 3 people? 

    Solution First method Work out the quantity of strawberries needed for one person use this to find the quantity of strawberries needed for 3 people. The quantity of strawberries needed for one person 750÷8= 93.75g So the quantity of strawberries needed for 3 people is 3×93.75g=281.25g=280g (to 2 s.f). 

       The second method Find the fraction of the original quantity that is needed, and use this to calculate the quantity of strawberries needed Three-eighths of the original quantity is required. So the quantity of strawberries needed for three people is 3/8× 750g= 3÷8×750= 281.25g= 280g (to 2 s.f) 

     There are different ways to solve a problem, so you may sometimes find that you have used a method different from the one in the unit or suggested by someone else. However, it is a good idea to look at any model solution provided, as it may suggest an alternative and possibly quicker method that you could use to solve a similar problem in the future.

ACTITITY 19 Work out the following fraction of quantity.  (a1) 4/5 of 60ml 

(A2)5/8 of 20kg

            (B)A recipe for potato curry for 6 people uses 900g of potatoes. If you are making a the curry for 20 people, what quantity of potatoes do you need?
    Answer
(A1) 4÷5×60=48ml 
(A2) 5÷8×20=12.5kg 
(b) 20÷6×900=380g=3kg
    you can scale any cater for a group of any size, but in practice you may wish to adjust your answers a little. In a large group it is most likely that few people will eat only small portions or none at all, so caters often use guidelines such as the following. Allow 150g of potatoes per person for up to 10 people; for more than 10 people, allow 125g per person. When you are using maths to make practical decision, it is important to think about whether your calculation are appropriate for the situation.

unit 1 section 2.3

Unit 1 sec 2.3 units of measurement

20 September 2015

00:25

Many everyday calculations involve measurements of some kind: for example, lengths, times, amount of money and so on.

In the UK, both the metric and imperial systems of measurements are used.

The different units for the same type of quantity are related to each other via power of ten:

The imperial system, the units are related in different ways: for example, 1 stone is the same as 14 pounds.

This module mostly uses the standard metric system known as the système community d'Unitès (SI units).

There are seven base SI units, from which all the other are derived.

The base units used most frequently in the module are the metres and kilograms and the second.

Prefixes are used to indicate smaller or larger units, for example, millimetres, centimetres, metres and kilometres are all used to measure length.

Unit 1 sec 2.3 units of measurement 20 September 2015 00:25 Many everyday calculations involve measurements of some kind: for example, lengths, times, amount of money and so on. In the UK, both the metric and imperial systems of measurements are used. The different units for the same type of quantity are related to each other via power of ten: The imperial system, the units are related in different ways: for example, 1 stone is the same as 14 pounds. This module mostly uses the standard metric system known as the système community d'Unitès (SI units). There are seven base SI units, from which all the other are derived. The base units used most frequently in the module are the metres and kilograms and the second. Prefixes are used to indicate smaller or larger units, for example, millimetres, centimetres, metres and kilometres are all used to measure length. Perfix Abbreviation Meaning Example Milli M Centi c Unit 1 sec 2.3 units of measurement 20 September 2015 00:25 Many everyday calculations involve measurements of some kind: for example, lengths, times, amount of money and so on. In the UK, both the metric and imperial systems of measurements are used. The different units for the same type of quantity are related to each other via power of ten: The imperial system, the units are related in different ways: for example, 1 stone is the same as 14 pounds. This module mostly uses the standard metric system known as the système community d'Unitès (SI units). There are seven base SI units, from which all the other are derived. The base units used most frequently in the module are the metres and kilograms and the second. Prefixes are used to indicate smaller or larger units, for example, millimetres, centimetres, metres and kilometres are all used to measure length. Perfix Abbreviation Meaning Example Milli M Centi c Kilo K A thousand (1000) 1 kilometre (km)= 1000 metres

Perfix	Abbreviation	Meaning	Example
Milli	M
Centi	c
Kilo	K	A thousand (1000)	1 kilometre (km)= 1000 metres

  

unit 1 section 2.2

Unit 1 sec 2.2 Using your calculator

19 September 2015

00:17

You are expected to use your calculator for most numerical calculations in MU123.

Sometimes you may find that quicker to do a simple calculation in your head or on paper.

This is usually so that you can practice a technique that you will need to use later when you do Algebra.

When you type a calculation into your calculator, it is important to think about the BIDMAS rules, to ensure that the operations are carried out in the order you intended.

unit 1 section 2.1

Unit 1 sec2.1

18 September 2015

22:19

Example 1 Using the BIDMAS Rules

A.    8-2+5-1=

B.    5+12 ÷4=8

C.  

D.  (5-3)×4

SOLUTION

A.   The addition and subtraction have the same precedence, so do them in order from left to right.

 8-2+5-1=10 6+5-1=11-1=10

B.   Do the division first then the addition

5+12÷4=5+3=8

c.    Work out the power first, then do the multiplication.

Do the calculation in brackets first then do the multiplication.

 (5-3)×4=2×4=8.

ACTIVITIY 2 Using BIDMAS rules

A. 9+7-2-4=10

B. 2×(7-4)=6

C. (3+5)×3=24

D.               (3+4)×(2+3)=35

E.

There are many terms that have specific meaning.

The important terms and their definition are also collected together

Activitity 3

A. 2×(5+3)=16

B. (3+4)×7=49

C. 1+(2×3)=7

D.               9-(3×2)=3

E. 2×(3+3)×5=60

Some terms for calculation.

1. The SUM of two numbers is a result of adding them together.

2. A difference between two numbers is a result of subtracting one from the other. There are two possible answers, depending on which way round you take the number, but usually the smaller number is subtracted from the larger.

3. The PRODUCT of two numbers is a the result of multiplying them.

4. A QUOTIENT of two numbers is the result of dividing one by the other. There are two possible answers, depending on which way round you take the numbers.

ACTIVITY 4

A. 4+8=12

4×8=32

B. 4-2=2

4/2=2