improve your numeracy november 2010

8/2/2019 Improve Your Numeracy November 2010

1/34

University of Bristol


2/34

2

Many students worry about anything to do with numbers, having done little since theirGCSEs. This booklet has been designed to offer practice and explanation in basicprocesses, particularly to anyone facing employers selection tests. It uses materialdeveloped by Clare Wright, Tom Frank, and Eric Williams of Birmingham University CareersCentre, where it has been successfully used for some time.

Regaining numerical confidence can take a little while but, if you once had the basic skills,practice should bring them back. If, however, you encounter real problems perhaps youshould question your motives. Employers dont test you just to make life difficult. They do

it because their jobs demand a particular level of proficiency. If youre struggling to meet

that level, or not enjoying it, is their job right for you?

1. Decimals . page 3

2. Fractions 6

3. Approximations 11

4. Averages . 13

5. Percentages . 15

6. Ratios . 18

7. Progressions 20

8. Tables 24

9. Graphs .. 26

10. Answers 28

University of Bristol Careers Service: www.bristol.ac.uk/careers

CONTENTS

INTRODUCTION


3/34

3

SECTION 1 - DECIMALS

The most common use of decimals is probably in the cost of items. If youve worked in ashop or pub, youre probably already familiar with working with decimals.

Addition and Subtraction

The key point with addition and subtraction is to line up the decimal points!Example 1

2.67 + 11.2 = 2.67+11.20 in this case, it helps to write 11.2 as 11.2013.87

Example 2

14.73 12.157 = 14.730 again adding this 0 helps- 12.157

2.573

Example 3

127.5 + 0.127 = 127.500+ 0.127

127.627

Multiplication

When multiplying decimals, do the sum as if the decimal points were not there, and thencalculate how many numbers were to the right of the decimal point in both the original

numbers - next, place the decimal point in your answer so that there are this number ofdigits to the right of your decimal point.

Example 4

2.1 x 1.2.Calculate 21 x 12 = 252. There is one number to the right of the decimal in each of theoriginal numbers, making a total of two. We therefore place our decimal so that there aretwo digits to the right of the decimal point in our answer.Hence 2.1 x 1.2 =2.52.Always look at your answer to see if it is sensible. 2 x 1 = 2, so our answer should be closeto 2 rather than 20 or 0.2 which could be the answers obtained by putting the decimal inthe wrong place.

Example 5

1.4 x 6Calculate 14 x 6 = 84. There is one digit to the right of the decimal in our original numbersso our answer is 8.4Check 1 x 6 = 6 so our answer should be closer to 6 than 60 or 0.6


4/34

4

Division

When dividing decimals, the first step is to write your numbers as a fraction. Note thatthe symbol / is used to denote division in these notes.Hence 2.14 / 1.2 = 2.14

1.2Next, move the decimal point to the right until both numbers are no longer decimals. Dothis the same number of places on the top and bottom, putting in zeros as required.Hence 2.14 becomes 214

1.2 120This can then be calculated as a normal division.

Always check your answer from the original to make sure that things havent gone wrongalong the way. You would expect 2.14/1.2 to be somewhere between 1 and 2. In fact, theanswer is 1.78.

If this method seems strange, try using a calculator to calculate 2.14/1.2, 21.4/12,214/120 and 2140 / 1200. The answer should always be the same.

Example 6

4.36 / 0.14 = 4.36 = 436 = 31.140.14 14

Example 7

27.93 / 1.2 = 27.93 = 2793 = 23.281.2 120

Rounding Up

Some decimal numbers go on for ever! To simplify their use, we decide on a cut off pointand round them up or down.If we want to round 2.734216 to two decimal places, we look at the number in the thirdplace after the decimal, in this case, 4. If the number is 0, 1, 2, 3 or 4, we leave the lastfigure before the cut off as it is. If the number is 5, 6, 7, 8 or 9 we round up the lastfigure before the cut off by one. 2.734216 therefore becomes 2.73 when rounded to 2decimal places.

If we are rounding to 2 decimal places, we leave 2 numbers to the right of the decimal.If we are rounding to 2 significant figures, we leave two numbers, whether they aredecimals or not.


5/34

5

Example 8

243.7684 = 243.77 (2 decimal places)= 240 (2 significant figures)

1973.285 = 1973.29 (2 decimal places)= 2000 (2 significant figures)

2.4689= 2.47 (2 decimal places)= 2.5 (2 significant figures)

0.99879 = 1.00 (2 decimal places)= 1.0 (2 significant figures)

Try these examples. Give all your answers to 2 decimal places and 2 significant figures

1. 2.45 + 7.68 2. 3.17 + 12.15 3. 2.421 + 13.1

4. 162.5 + 2.173 5. 12.5 3.7 6. 9.6 7.8

7. 163.5 2.173 8. 2.416 1.4 9. 26.95 1.273

10. 1.5 x 7.2 11. 2.73 x 8.14 12. 6.25 x 17 x 3

13. 2.96 x 17.3 14. 4.2 / 1.7 15. 53.9 / 2.76

16. 14.2 / 6.1 17. 2.5 / 0.03 18. 250/2.35

Answers on page 28


6/34

6

SECTION 2 - FRACTIONS

Cancelling Down

When we use a fraction, we usually give it in its simplest form. To do this we look at thetop (the numerator) and the bottom (denominator) and see if there is a number by whichboth can be divided an exact number of times.

Hence 2 = 1 x 2 = 1 since the twos cancel out8 4 x 2 4

E.G. 6 = 3 x 2 = 38 4 x 2 4

15 = 3 x 5 = 3

35 7 x 5 7

16 = 8 x 2 = 2 OR 16 = 4 x 4 = 4 = 2 x 2 = 224 8 x 3 3 24 6 x 4 6 3 x 2 3

Use as many steps as you need to reach the answer.

Adding Fractions

When the denominators (the bottom lines) are all the same, you simply add the top line

(numerators)

Eg: 2 + 3 = 2 + 3 = 5 5 + 3 = 5 + 3 = 86 6 6 6 9 9 9 9

Remember to cancel down if necessary.

When the denominators are different, we need to change the fractions so that thedenominators are the same then we can add the top line as above.

Suppose we wish to calculate 1 + 12 4

From the cancelling down process, we know that 1 = 1 x 2 = 22 2 x 2 4

The denominators of both fractions are now the same so we can calculate

1 + 1 = 2 + 1 = 2 + 1 = 3

2 4 4 4 4 4


7/34

7

Sometimes the denominators are not multiples of each other

Eg: 1 + 24 3

In this case we can make 12 the common denominator using

1 = 1 x 3 = 3 2 = 2 x 4 = 84 4 x 3 12 3 3 x 4 12

We can then add these two fractions directly:

1 + 2 = 3 + 8 = 3 + 8 = 11

4 3 12 12 12 12

Eg: 2 + 1 = ?5 62 = 2 x 6 = 12 1 = 1 x 5 = 55 5 x 6 30 6 6 x 5 30

2 + 1 = 12 + 5 = 12 + 5 = 175 6 30 30 30 30

Subtracting Fractions

This works in the same way as addition. If the denominators are the same, simply subtractalong the top line:

5 - 3 = 5 3 = 2 = 1 x 2 = 16 6 6 6 3 x 2 3

12 - 1 = 12 - 3 x 1 = 12 - 3 = 12 3 = 915 5 15 3 x 5 15 15 15 15

Cancelling down gives 9 = 3 x 3 = 315 3 x 5 5

NB: an alternative method would have been to cancel down 12/15 to 4/5 initially leaving an

easier sum of 4/5 1/5 = 3/5


8/34

8

Multiplication of Fractions

It may help to understand multiplication if you interpret the x sign as of.

Hence:1 x 2 means 1 of 2 = 1

2 5 2 5 5

The calculation involves multiplying both numerators and both denominators then cancellingdown:

Eg: 1 x 2 = 1 x 2 = 21 = 12 5 2 x 5 105 5

4 x 2 = 4 x 2 = 8

5 3 5 x 3 15

7 x 3 = 7 x 3 = 2111 8 11 x 8 88

Note that when multiplying, you can cancel down during the sum as well as at the final stage it often makes the calculation easier:

Eg: 31 x 2 = 213 31 13

41 x 3 = 1 x 31 = 1 x 1 = 19 82 93 2 3 2 6

5 x 61 = 5 x 1 = 5244 7 4 x 7 28

Dividing Fractions

The trick with division of fractions is to turn the second fraction upside down and then to

multiply:

Eg. 2 1 = 2 x 5 = 103 5 3 1 3

Note that this answer can also be written as 31/32 4 = 21 x 5 = 57 5 7 42 14

2 7 = 2 x 8 = 163 8 3 7 21


9/34

9

Improper Fractions

An improper fraction is one where the numerator is larger than the denominator, eg. 10/3To convert this to a mixed number (one combining a whole number and a fraction), think of10/3 in the following way:

3 x 1/3 = 1 so out of the ten 1/3s, nine can be grouped into three whole units, leaving only1/3 left over. Hence, 10/3 = 31/3

Similarly, 12/5 = 22/516/11 = 15/1121/15 = 16/15 = 1 2/5 (by cancelling)

We can also go from mixed numbers to improper fractions:

21/8 = (2 x 8) + 1 = 16 + 1 = 17

8 8 8 8 8

31/4 = (3 x 4) + 1 = 12 + 1 = 134 4 4 4 4

Addition of mixed numbers

Eg. 2 + 31/5 = 2 + 3 + + 1/5

= 5 + 5/10 + 2/10

= 5 + 7/10

= 57/10

17/8 + 52/3 = 1 + 5 + 7/8 + 2/3

= 6 + 21/24 + 16/24

= 6 + 37/24

= 6 + 113/24

= 713/24

Multiplication and Division of mixed numbers

Here, it is usually easiest to convert to improper fractions and multiply or divide as normal:

Eg: 25/6 x 31/8 = 17/6 x 25/8 = 425/48 = 8 41/48

12/3 2 = 5/3 11/4 = 5/3 x 4/11 = 20/33

2 x 41/3 1 = 5/2 x 13/3 5/4 = 5/2 x 13/3 x 4/5 = 260/30 = 26/3 = 8 2/3

13

13

13

13

13

13

13

13

13

13


10/34

10

Now try these examples:

Note that in these questions 2/5 is the same as 2

5

1. 2/5 + 1/3 2. 7/8 1/5 3. 5/8 + 2/3 4. 2/9+4/11

5. 6/7 4/9 6. 7/8 1/3 7. 2 - 1 5/8 8. 7 - 5 2/3

9. 2 4/5 + 6 7/8 10. 2/3 x 4/5 11. 8/9 x 1/3 12. 7/12 x 4/5

13. 8/9 2/3 14. 2/11 4/5 15. 6/7 9/8 16. 2 x 3

17. 5 1/7 x 2 1/5 18. 3 2/3 x 4 4/9 19. 7 2/3 3 4/5 20. 2 6 1/5

21. 3 1/3 4 1/3 22. 2 1/3 23. 5 2/7 x 11/3 24. 6 x 2 1/3

Answers on page 28


11/34

11

SECTION 3 - APPROXIMATING & ESTIMATING

Estimates and approximations bring 2 major benefits:

1. They enable you to check that the final answer is near to the figure that yourestimate led you to expect. This is particularly valuable if you are unsure in the useof the keys on the calculator.

2. They can provide you with a short cut, by showing plainly that you do not need to doall the calculations (eg. when comparing figures in tables).

Example 1

Calculate 51/73 of 300steps

i. Approximate 51/73 to 50/75 (move to figures that make easily handledfractions)

ii. simplify 50/ 75 to 2/3 (by dividing top and bottom by 25)

iii. Approximate answer is 2/3 300 i.e 200

iv. Calculator answer 209.59 (in fairly close range to 200, your approx.answer)

Example 2

The following table shows the number of items sold in one year by the Beans Co. ineach of its operating districts. Column Y shows the number of salespeople employedin eachdistrict.

Column X Column Y Column Z

District Number of Number ofsalespeople items sold

SW 57 53,971

NW 38 64,562

Mid 31 21,123

SE 27 47,051

NE 23 42,510


12/34

12

Task 1. In which district was the highest number of items sold per salesperson?

Task 2. In which district was the lowest number of items sold per salesperson?

Steps:

i. To make handling of figures more manageable, make approximations of the items

sold figures.Approximation table

District Salespersons Items sold (in thousands)

SW 57 54

NW 38 64.5

Mid 31 21

SE 27 47

NE 23 42.5

Approximation and comparison can eliminate certain districts from the need tocalculate accurately.

Task 1. The highest average sales

i. Mid and SE have similar numbers of salespeople, but Mid sold far feweritems. Hence Mid can be eliminated.

ii. SW sold fewer items than NW, but had far more salespeople. Hence SW canbe eliminated.

iii. The districts left in for consideration are SE, NW, NE

iv. Use a calculator to do the division, USING THE ACCURATE FIGURES,to find the averages of the 3 districts.

SE 47,051/27 = 1742.6

NW 64,562/38 = 1699

NE 42,510/23 = 1848

Hence the highest average sales figures per salesperson was in the NE district.

Task 2. The lowest average sales per person per district

From your approximation table it is evident that Mid sold far fewer items thaneach of NW,SE,NE, despite having similar numbers of salespeople.

It is not so clear as to how Mid compares with SW, so two calculations are desirable.

SW 53,971/57 = 946.86Mid 21,123/31 = 681

Hence Mid district had the lowest average number of items sold per salesperson


13/34

13

SECTION 4 AVERAGES

There are several ways of expressing an average. The most common way is called a mean.This is one value which is representative of all the numbers in a group. The mean iscalculated by adding up all the numbers in a group, then dividing by how many numbersthere are.

Example 1 Find the average, or mean, of the following numbers:

Mean = 2+6+4+8+5+5 = 30 = 56 6

Check your answer is an answer of 5 representative of the numbers?

Example 2 Find the average, or mean, of the following numbers:

1 3 4 2 1 20

Mean = 1+3+4+2+1+20 = 31 = 5.176 6

Check your answer in this case, most of the numbers are less than 5 but the very highvalue of 20 pulls the mean upwards, so the answer is sensible.

Example 3 In a test of numerical ability, students are tested in two groups. Which group

has the higher average score?

Group A Group B3 715 68 109 1112 1913 20

3+15+8+9+12+13 7+6+10+11+19+206 6

= 60 = 736 6

= 10 = 12.17

Group B has the higher average score


14/34

14

TRY THESE QUESTIONS:

Find the mean of each of these sets of data:1. 2 4 6 7 4 3 5

2. 49 50 63 61 67 59 73 68 42

3. 6 7 8 7 8 6

4. 15 16 12 19 14 20 18 17 18 16

5. 107 109 117 119 112

6. Which group of students scored best in this numerical reasoning test?Group A: 11 12 13 6 18 17 4 12 10 15Group B: 10 11 9 12 14 13 12 10 11 14

7. A lecturer decided to find out how punctual students really are. He recorded howlate his seminar group arrived on three occasions. The table shows how manyminutes late each of the ten students were on these days. What was the overallaverage lateness? Which day was worst? On which day were most students morethan 5 minutes late?

Day 1 Day 2 Day 32 5 40 10 14 8 315 6 010 0 04 0 55 10 40 9 10 0 0

1 0 15

Answers on page 29


15/34

15

SECTION 5 - PERCENTAGES

Percentage is one of three ways of expressing a value

e.g. 17% has the same value as 17/100 or 0.17

It is recommended that, before using the calculator, you first understand, throughmental calculation, the principles of layout of expressions and the functions ofaddition, subtraction, multiplication and division in fractions and decimals.

Percentage and decimal - the relationship

Examine the meaning of the figure 425. 631. The position of each digit gives it itsvalue. Regard each digit as being a member of a column

i.e. a b c d e f4 2 5 6 3 1

Column a expresses whole HUNDREDS. b expresses whole TENS..c expresses UNITS (ONES). d. expresses 1/10ths e. expresses 1/100thsF. expresses 1/1000ths.

1% is expressed as a fraction as 1/100. Hence, from the column rule, 1% is 1 in

column e i.e 0.01

Try to memorise the following table. It shows how identical values are expressed indifferent forms percentages, fractions and decimals

100% 100/100 1.00

10% 10/100 1/10 0.1

1% 1/100 0.01

0.1% 0.1/100 1/1000 0.001


16/34

16

Example

Calculate 17% of 3/4 Give the answer as a percentage.

i. Express 17% as 17/100

ii. 17% as 17/100 3/4 (note that of becomes )

iii. The rule of multiplication of fractions applies. Multiply the top values(numerators), and multiply the bottom values (denominators)i.e (17 3 )/(100 4 ) = 51/400

iv. Approximate 51/400 to 50/400

v. Simplify 50/400 to 1/8

vi. Convert 1/8 to a percentage. The principle is that the 8 (denominator) has tobe converted to 100 ( in this case by multiplying by 12.5). You treat the topfigure in the same way (i.e 1 multiplied by 12.5 )

vii. You thus produce a fraction 12.5/100 (NB. you would not normally leave afraction in this format). This indicates 12.5% as an APPROXIMATEANSWER to the question.

viii. Use the calculator to find the ACCURATE answer

Answer 12.75%


17/34


18/34

18

SECTION 6 - RATIO

Ratio is crudely explained as the proportion in which something or some things are shared.i.e. an expression of how the shares of the total compare with each other.

Example 1

Ann, Betty and Chris win a total of 1200 in a lottery. The money is shared according tohow much each staked in the lottery.Ann is to receive 2 times as much as Chris. Betty is to receive 3 times as much as Chris.

How much should each receive?

You need to begin by establishing the total number of shares.Begin by allocating 1 share to Chris

Ann will then have 2 shares (2 times Chriss share)Betty will receive 3 shares (3 times Chriss share)

Thus the total prize is to be shared 6 ways (1+2+3)

So one share will be 1200 divided by 6 i.e 200Chris will receive 200, Ann 400 (2 times Chriss share), and Betty 600 (3 times

Chriss share)

Check that the shares (200 +400 + 600) total 1200

Different phraseology of this question could be.....

1200 is to be shared out between Chris, Ann and Betty, in the RATIO of 1:2:3respectively.How much will each receive?

Example 2

A school collected money for a new laboratory, using three methods, raffle, sponsored walkand fete. The total collected was 5200 in the ratio of 5:2:3 for each method in the order- raffle, walk, fete.

How much did each method raise?

i. 5:2:3 expresses the proportions (shares)

ii. The total number of shares is 5+2+3 i.e. 10

iii. The total collected was 5200

iv. Each share was 5200/10 i.e. 520


19/34

19

v. The raffle raised 5 shares of 520 i.e. 2600

The walk raised 2 shares of 520 i.e. 1040

The fete raised 3 shares of 520 i.e. 1560

vi. CHECK the total by addition i.e. 5200

Try these questions:

i. 5 brothers inherit a total of 70000 in the proportions 1:1:2:3:3How much did each receive?

ii. In a darts game Ben scored twice as many points as Alan. Chris scored threetimes as many as Ben. Doug scored half as many as Chris.The total number of points of the 4 players added together was 480.

How many points did each player score?

Answers on page 29


20/34

20

SECTION 7 - NUMERICAL PROGRESSIONS

Youll often find progressions (or series as they are sometimes called) in aptitude tests.

Typically you are given a series of numbers and asked to fill in the missing ones.Progressions require creative thinking in finding relationships between numbers.

The following general tips may help. They are followed by worked examples, designed toillustrate some of the processes involved:

i. Begin by writing down the differences between the numbers in the series.

ii. Try all the combinations of multiplication, subtraction, addition, division evensquare roots etc. that could account for the differences.

iii. Remember that any series beginning with 0 is unlikely to develop throughmultiplication or division alone (multiply or divide zero by any number - itll stay zeroforever).

iv. If differences do not provide a clue, try developing the number itself, by adding,subtracting, multiplying or dividing.

The following examples illustrate some of the processes involved:

Example 1 Which is the next number in the series?

1 2 6 15 31 56

differences 1 4 9 16 25

i.e. 11 22 33 44 55

the next difference is therefore 66 (=36)the next number in the series is therefore 56 (66) = 92

Example 2 Which number is missing in the series?

1 1 2 6 24 720

differences 0 1 4 18

the differences do not provide a clue.try on the numbers themselves

1 1 2 6 24 720

1 2 3 4

hence it can be seen that each number is multiplied by numbers that increase by 1.

Applying this rule, the next function would be 5 . The missing number is therefore24 5 i.e. 120. Check the final number. i.e 120 6 i.e. 720.


21/34

21

Example 3

If none of the functions ( ) provides the answer, look at other patterns.

Which number is missing in the series ?

9 15 23 47 47 23 ..... 9

there is no clue from differences. The mirror pattern is evident either side of

Example 4

Look for alternate pattern progression.

Which number is missing from the series ?

2 17 4 15 6 13 ..... 11 10

The numbers printed bold are the even numbers increasing.

The numbers printed in normal type are odd numbers decreasing.

Hence the missing number in the series is 8. The inclusion of the odd numbers is merely adistraction.

Example 5

Look for functions used alternately in the series.

Which number is missing in the series?

6 12 13 26 27 ..... 55

The bold numbers are formed by multiplying the previous number by 2

The numbers printed in normal type are formed by adding 1 to the previous number in theseries.Hence the missing number is 54 (i.e 27 2). Check the answer with the final numberin the series ( i.e 54 1 = 55 ).


22/34

22

Example 6

Examine large numbers. Do not assume that they express the whole value.

e.g. 124 could express......... One hundred and twenty four

But in series puzzles could also express1 - 2 - 4 ( i.e without place value ) or

12 - 4 (i.e place value only in part ) or

1- 24 ( i.e place value only in part )

Which number is next in the series?

124 7 239 14 341 8 621

1 2 4 =7 2 3 9 = 14 3 4 1 = 8

hence the next number in the series is formed as 6 2 1 = 9

Example 7

Watch the spacing.

e.g. i. 2 9 11 7 2 9 6 6 12 5 ... 9

Each triplet applies the rule independently. The spacing provides the clue

i.e. 2 9 =11 7 2 = 9 6 6 = 12 5 4 = 9

e.g. ii. Which number is next in the series ?

2 3 5 16 3 1 4 15 623 30 524 .....

2 (3 5) = 16 3 (1 4) = 15 6 (2 3) = 30 5 (2 4) 30


23/34

23

Example 8

Numbers in a series can sometimes appear at first sight to be fractions.

i.e 1/4 1/2 = 3/4

But in series puzzles they may carry no value as fractions, but show instead two series ofnumbers, one along an imaginary top line and one along the bottom line

Which is the missing item in the series?

2/3 4/5 6/7 .... 10/11 12/13

2 4 6 ... 10 12 are ascending even numbers along the top line

3 5 7 ... 11 13 are ascending odd numbers along the bottom line

hence the missing item is 8/9

TRY THESE EXAMPLES

Fill in the missing items.

i. 0 3 7 12 18 .. 33 42

ii. 1 1 2 7 34 .. 1420

iii. 100 81 ... 49 36 25 16 4 1

iv. 2 4 10 22 42 .. 114

v. 2 3 4 6 8 12 16 24 .. 48 64

vi. 143 6 257 10 492 .. 761 0 837 2

vii. 721 208 361 5..2 901 622

viii. 1/2 3/8 9/32 .. 81/512

ix. 0 1 3 7 15 .. 63 127

x. 10 7 4 1 .. 5 8

Answers on page 29


24/34

24

SECTION 8 - TABLES

Tables are a way of presenting data in an ordinary and precise form, making interpretationeasier.

Here is the end-of-season league table for the six-team Motor Manufacturers

League:

Each team has played each opponent twice: once at home and once away.3 points are awarded for a win.1 point is awarded for a draw.

HOME AWAYMatchesPlayed

Won Drawn LostGoalsFor

GoalsAgainst


GoalsAgainst

POINTS

Rover Rangers 10 3 2 0 11 4 4 0 1 12 2 23

Citroen City 3 0 2 7 6

Austin Albion 10 2 1 2 7 11 1 4 0 7 6 11

Toyota Town 10 2 2 1 8 5 0 2 3 4 11 10

Opel Orient 10 1 1 3 3 8 1 2 2 6 7 9

WolseleyWanderers

10 1 2 2 6 9 0 1 4 6 16 6

Substitute the correct figures for the blanks above (it will help you if you write youranswers into the table as you complete each question):1. Citroen City - Matches played:

A B C D E

6 8 10 12 14

2. Citroen City Home wins:A B C D E

1 2 3 4 5

3. Citroen City Home draws:A B C D E

0 1 2 3 54. Citroen City Home losses:

A B C D E

0 1 2 3 4

5. Citroen City Home goals for:A B C D E

9 11 13 15 17

6. Citroen City Home goals against:A B C D E

1 3 5 7 9

7. Citroen City Total points:A B C D E

13 15 17 19 21


25/34

25

HOME AWAYMatchesPlayed


GoalsAgainst


GoalsAgainst

POINTS

Rover Rangers 10 3 2 0 11 4 4 0 1 12 2 23

Citroen City 10 3 1 1 13 5 3 0 2 7 6 19

Austin Albion 10 2 1 2 7 11 1 4 0 7 6 11Toyota Town 10 2 2 1 8 5 0 2 3 4 11 10

Opel Orient 10 1 1 3 3 8 1 2 2 6 7 9

WolseleyWanderers

10 1 2 2 6 9 0 1 4 6 16 6

Here are some further questions now the blanks in the table have been filled in.8. Which team has scored fewest goals?

A B C D EWolseley W Opel O Toyota T Austin A Citroen C

9. Which team has gained the highest proportion of its points from drawn matches?A B C D E

Wolseley W Opel O Toyota T Austin A Citroen C

10. Which team has lost fewer matches than the one immediately above it in the table?A B C D E

Wolseley W Opel O Toyota T Austin A Citroen C

Answers on page 30


26/34

26

SECTION 9 - GRAPHS

To those unfamiliar with and therefore perhaps fearful of graphs, it is important tobear in mind that they exist to clarify rather than to confuse numbers. Simple graphs ofthe sort found in numeracy tests should be seized upon as being probably among the easierquestions in any paper. They normally require only a careful reading of the question and a

confidence in handling the graphically presented data.

The following graph shows the number of bikes sold in each month between July andDecember:

SELLING PRICES:

Mountain bikes 120Racing bikes 240Folding bikes 180

Childrens bikes 60(The term sales

revenue used in the

questions below means

the sum of the selling

prices of the bikes sold

during the period in

question.)

TRY THESE QUESTIONS:

1. Which month produced the highest total sales revenue?A B C D EJul Dec Aug Oct Nov

2. Which model generated the highest sales revenue in the month of December?

A B C D EChildrens Mountain Folding Racing Insufficient data available

3. Owing to a rationalisation policy, only 3 of the current 4 models will be sold duringthe same period next year. Predicting on the basis of the current years figures shown

above, which model should be dropped in order to have the minimum effect on total profitover the six month period?A B C D EChildrens Mountain Folding Racing Insufficient data available

Acme Cycle Shop - Monthly Sales

0

5

10

15

20

25

30

35

Jul Aug Sep Oct Nov Dec

Mountain bikes

Racing bikes

Folding bikes

Children's bikes


27/34

27

4. In the month of August, approximately what percentage did childrens bikesrepresent of the total number of bikes sold?A B C D E2% 5% 7.5% 10% 13%

5. In the month of December, approximately what percentage of sales revenues didthe combined sales of folding bikes and childrens bikes represent?A B C D E11 22 33 39 48

6. The manager considered (and eventually decided against) a discount of 10 on eachbike sold during the month of December. On the assumption that this would have resultedin a 25% increase (to the nearest unit) in the number sold of each model, how would theDecember sales revenue have been affected (approximate percentages):

A B C D EDown by 8% Down by 2% No change Up by 8% Up by 15%

Answers on page 31


28/34

28

ANSWERS

SECTION 1 DECIMALS

Question 2 decimal places 2 significant figures

1 10.13 102 15.32 15

3 15.52 16

4 164.67 160

5 8.80 8.8

6 1.80 1.8

7 161.33 160

8 1.02 1.0

9 25.68 26

10 10.80 11

11 22.22 2212 318.75 320

13 51.21 51

14 2.47 2.5

15 19.53 20

16 2.33 2.3

17 83.33 83

18 106.38 110

SECTION 2 FRACTIONS

1. 11/15 2. 27/40 3. 1 7/24 = 31/24

4. 58/99 5. 26/63 6. 13/24

7. 7/8 8. 1 7/12 = 19/12 9. 9 27/40 = 387/40

10. 8/15 11. 8/27 12 7/15

13. 4/3 14. 5/22 15. 16/2116. 7 7/8 = 63/8 17. 11 11/35 = 396/35 18. 16 8/27 = 440/27

19. 2 1/57 = 115/57 20. 25/62 21. 10/13

22. 6 = 27/4 23. 19 8/21 = 407/21 24. 14


29/34

29

SECTION 4 AVERAGES

1. 4.43 2. 59.11 3. 7 4. 16.5 5. 112.86. Group A = 11.9, Group B = 11.6 so the answer is Group A (just!)7. Overall average = 122/30 = 4.067. Day 1 average = 41/10 = 4.1. Day 2 average = 48/10 = 4.8Day 3 average = 33/10 = 3.3, so Day 2 has the worst overall average.On Day1, 2 students were more that 5 minutes late, Day2 = 5 students and Day 3 = 1

student so Day 2 is also the day on which most students were more that 5 minutes late.

SECTION 5 PERCENTAGES

i. 16 v. 3.15 ix. 58%

ii. 2.5% vi. 243 x.a 20.781

iii. 5/16 vii. 0.6% x.b 3958

iv. 5 viii. 0.0025 x.c 3%

SECTION 6 RATIOS

i. 7,000, 7,000, 14,000, 21,000, 21,000

ii. Alan 40, Ben 80, Chris 240 and Doug 120

SECTION 7 PROGRESSION

i. 25 Differences are +3, +4, +5, +6, +7, +8, +9

ii. 203 Differences are ( 2 1), ( 3 1), ( 4 1) etc.

iii. 64 Squares in descending order

iv. 72 Differences are (11) 1, (22) 2, (3 3) 3 etc

v. 32 Alternate progressions 3 2, 6 2, 12 2 etc

2 2, 4 2, 8 2 etc

vi. 7 Groups of three 3 4 1, 7 5 2, 2 9 4 etc

vii. 3 The sum of the figures in each group is 10

viii. 27/128 Top numbers 3, bottom numbers 4

ix. 31 Differences are 2 1

x. 2 Differences are 3


30/34

30

SECTION 8 TABLES

1. Since each team has played each other team twice (once home and once away)Citroen City has played a total of 10 matches. The correct answer is C.

2. If you add up the column of home wins the total must be equal to the total of awaylosses (for each home winner there must be an away loser!) The total of away losses

is 12 (1+2+0+3+2+4=12). So the total of home wins must also be 12. Adding up thecolumn we get 3+?+2+2+1+1=12. The correct answer is C.

3. For each team gaining a home draw there must be a team gaining an away draw. Sothe total of home draws must equal the total of away draws. The correct answer isB.

4. By analogy with (2) above, total of home losses must be equal to the total of awaywins: so you can add up the columns and deduce the missing value. More simply atthis stage, however, having worked out that the team has played 5 home games andthat they won 3 and drew 1 (see questions (2) and (3) above) it is evident that theyhave lost 1. The correct answer is B.(It is worth checking now that home wins+homelosses+home draws=5. Otherwise you have got something wrong)

5. Using similar logic to that in (2) above, it is evident that for each home goal forthere must be an away goal against. Adding up the away goals against column we get

2+6+6+11+7+16=48. Thus the home goals for column is 11+?+7+8+3+6=48. Thecorrect answer is C.

6. By analogy with (5) above, total home goals for must equal total away goals against.The correct answer is C.

7. 3 home wins+3 away wins=6x3 points=18 points from wins. 1 home draw=1x1point=1point from draws. Total=19 points. The correct answer is D.

8. In each row, add up the number of home goals for and away goals for. Opel Orient,with a total of 6+3=9, scored fewest. The correct answer is B.

9. A quick scan of the Home Drawn and Away Drawn columns show that the top 2

teams only had 2 and 1 draws respectively, making the proportion of points fromtheir drawn matches relatively low. They can be ignored. The rest of the tableshows Austin Albion with a total of 5 draws (5 points out of a total 11), ToyotaTown with a total of 4 draws (4 points out of a total 10), Opel Orient with a total of3 draws (3 points out of a total 9) and Wolseley Wanderers with a total of 3 draws(3 points out of a total 6). DONT PICK UP YOUR CALCULATOR AT THIS POINT!

Wolseley Wanderers have gained 3 out of their 6 points with draws i.e. one half,or 50%. The others (5 out of 11, 4 out of 9 and 3 out of 9) are all proportions lessthan 50%. The correct answer is A.

10. Add up the Home Lost and Away Lost figures for each team. Austin Albion lost atotal of 2 while Citroen City lost a total of 3. The correct answer is D.


31/34

31

SECTION 9 GRAPHS

1. This one is easy. No calculations at all needed. The sales graphs for the individualmodels show December as the best sales month for each. Therefore, irrespectiveof the selling price of each model, the total revenue for the month of Decemberhas to be the highest.

DO NOT FALL INTO THE TRAP OF CALCULATING WHEN YOU DONT HAVE TO!The correct answer is B

2. Again, dont calculate more than you have to. Notice that the selling prices happento be multiples of 60 (Childrens 1x60,Mountain 2x60, Folding 3x60, Racing

4x60). Since we are asked here simply to find the highest figure, we dont have to

calculate absolute values but simply to compare relative values. Thus we can work inunits of 60 rather than in units of 1. This means that we can multiply thenumber of bikes sold by 1,2,3 or 4 rather than by 60, 120, 180 or 240 to calculaterevenues a much simpler task!

Notice also that you can take short cuts by eliminating some of the possibleanswers almost at a glance.For instance, childrens bikes (60) sold 18, generating 18 units of 60. Folding

bikes(180) sold 15, generating 15x3=45 units of 60. So you can immediatelyeliminate childrens bikes. You didnt REALLY have to calculate that: you could seeit at a glance.Racing bikes (240) sold 9, each generating 4 units of 60 making around 36units. This is well below the 45 generated by folding bikes. So folding bikes are stillin the lead.Finally, mountain bikes (120) sold 32, each generating 2 units of 60 makingaround 64 units the clear winner.DONT WASTE TIME BY USING YOUR CALCULATOR UNLESS YOU REALLY

NEED IT!The correct answer is B.

3. If you made any calculations AT ALL, you wasted precious time! Nowhere in thedata is there any mention of profit - only of selling price.The correct answer is E.

4. First, add up the sales bearing in mind that you cannot be more accurate than thescale of the graph allows. Do dont worry about having to be approximate:Childrens: 3Racing: 5Folding: 11Mountain: 21TOTAL: 40Childrens bikes made up 3 out of 40 which is 3/40x100=7.5% (But you probablyknow that 3/4=75%, so 3/40 must be 7.5%! Again, a calculator should beunnecessary.)

The correct answer is C.


32/34

32

5. As in question 2, you can work in 60 units rather than in actual prices.Approx number sold Revenue in 60 units

Childrens: 18 18x1= 18Racing: 9 9x4= 36Folding: 15 15x3= 45Mountain: 32 32x2= 64

TOTAL: 163

Combined revenue from folding and childrens bikes=45+18=63 units of 60Percentage of total=63/163x100=38.7%The correct answer is D

6. WITHOUT DISCOUNTModel Number sold Price Revenue

Mountain 32 120 120x32=3840Racing 9 240 240x9=2160

Folding 15 180 180x15=2700Childrens 18 60 60x18=1080Total 9780WITH DISCOUNTModel Number sold Price Revenue

Mountain 32+8=40 110 110x40=4400Racing 9+2=11 230 230x11=2530Folding 15+4=19 170 170x19=3230Childens 18+4=22 50 50x22=1100Total 11260Increase in revenue: 11260-9780=1480% Increase in revenue=1480/9780x100=15.1%The correct answer is E


33/34

33

Countdown to Mathematics and Mathematicsthe Basic Skills. Both these booksprovide clear explanations and plenty of practice examples.

GCSE Revision books such as Collins Study and Revision Guide GCSE Mathematics orthe BBCs GCSE Maths Revision Guide

Selection of books available from the Careers Service Information Centre: The Numeracy Test Workbook: Intermediate Level Mike Byron How to pass Numeracy Tests: Intermediate Level Harry Tolley and Ken Thomas The Advanced Numeracy Test Workbook: Review key quantative operations and

practise for accounting and business tests Mike Byron Test Your Numerical Aptitude: How to assess your Numeracy skills and plan your

careers Jim Barrett How to pass Graduate Psychometric Tests: Essential preparation for numerical and

verbal ability tests plus personality questionnaires Mike Byron Psychometric Tests for Graduates: Gain the confidence you need to excel at graduate-

level psychometric and management tests Andres Shavick

Occupation specific resources:

How to Pass the Civil Service qualifying tests: The essential guide for Clerical andFast Stream applicants Mike Byron

How to Pass the New Police Selection system Harry Tolley, Billy Hodge andCatherine Tolley

How to pass the Fire-fighter Selection process Mike Byron Passing the UK Clinical Aptitude Test and BMAT Felicity Walker-Buckton,

Rosalie Hutton and Glenn Hutton

Note: New Information Centre resources are acquired and old Information Centreresources are removed on a regular basis so actual stock items may differ. Search the

Careers Catalogue for current resources:www.bris.ac.uk/careers/essentials/heritage/index.asp

Online psychometric testing: Current University of Bristol students and graduates canrequest access to try an online psychometric test. Details of this and other onlinepsychometric test resources are available at:www.bristol.ac.uk/careers/advice/numeracy.asp

If you want to continue developing your numeracy you could try:

If you want more practice with numeracy tests:
http://www.bristol.ac.uk/careers/advice/numeracy.asphttp://www.bristol.ac.uk/careers/advice/numeracy.asphttp://www.bristol.ac.uk/careers/advice/numeracy.asp


34/34

2000University of BirminghamUniversity of Bristol

improve your numeracy november 2010

Documents