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Contents : Calculator questionsBest buy questionsLong multiplicationLong divisionSimple fractions questionsNegative numbersRoundingEstimatingPercentagesTypes of numberProducts of primesHCF and LCMIndicesSimplifying rootsStandard FormRatioFractions with the four rulesRecurring decimals as fractionsUnitsDistance, Speed, Time questionsDensity, Mass, Volume questions

Calculator questions Which buttons would you press

to do these on a calculator ?

2.5 + 4.1 3.5

1.7 + 2.82.3 – 0.2

1.56 2.5

3

10004 1.72 +

5.22

6.31

8.5 x 103

3.4 x 10-1

9.2 6.17.5 8

–

1 1 3.6 2.3

–

0.95L2.1L

78p32pOR

OR

34p

87pAlways divide by the price to see how much 1 pence will buy you

Beans Large 40087 = 4.598g/pSmall 15034 = 4.412g/pLarge is better value(more grams for every penny spent)

Milk Large 2.178 = 0.0269L/pSmall 0.9532 = 0.0297L/pSmall is better valueOR (looking at it differently){Large 782.1 = 37.14p/L Small 320.95 = 33.68p/L}

Best buy questions

Long multiplication Use the method that gives you the correct answer !!

Question : 78 x 59

9

50

70

8

Total = 3500 + 400 + 630 + 72 Answer : 4602

3500 400

630 72

Now try 84 x 46 and 137 x 23 and check on your calculator !!

Long division Again use the method that gives you the correct

answer !!Question : 2987 23

23 46 69 92 115 138161 184207230

23 times table

2 9 8 72329

16

68

222

227

920

Answer : 129 r 20

Now try 1254 17 and check on your calculator – Why is the remainder different?

Simple fractions questions Equivalent fractions

3?

= 1220

1. 56

= ?24

2.

9 .12

= ??

3. ??

= 5 .25

4.

Fractions into decimals

Divide top by bottom

9 .12

= 9 12 = 0.75

45

1. 67

2. 59

3.

Put the following fractions in order of size, smallest to largest:

4.

47

23

58

34

Fractions of amounts

12

of £301.

16

of £302.

56

of £303.

59

of $724.

Divide by the bottomthen times by the top

Negative numbers Put these in order - smallest first

- 4 , - 3.6 , - 0.4 , - 0.36 , - 1 , 0 , 2 , - 1.4Up and down the scale

1. - 3 + 5 =

2. - 7 + 2 =

3. - 2 - 4 =

4. 5 - 12 =

5. - 6 + 4 =

Two signs next to each other

1. - 3 + - 1 = - 3 - 1 =

2. - 4 - - 2 = - 4 + 2 =

3. - 9 - - 11 =

4. 5 + - 7 =

Multiplication and Division

1. - 4 3 = 2. - 5 - 2 = 3. - 8 - 4 =

4. 20 - 5 = 5. - 4 4 = 6. (- 7)2 =

Signs same +ve answerSigns different -ve answer

Rounding 4715.692803 cmRound this number off to :

(a) 1 decimal place(b) 1 significant figure(c) 2 decimal places (d) 2 significant figures(e) the nearest centimetre (f) the nearest metre(g) 3 decimal places (h) 3 significant figures

(a) 4715.7 cm(b) 5000 cm(c) 4715.69 cm (d) 4700 cm(e) 4716 cm (f) 47 m(g) 4715.692 cm (h) 4720 cm

But there is always a trickier one14.9999

Round this number off to :(a) the nearest whole number(b) 3 significant figures(c) 2 decimal places

(a) 15

(b) 15.0(c) 15.00

Estimating If you are asked to estimate an answer to a calculation – Round all the numbers off to 1

s.f. and do the calculation in your head. DO NOT USE A CALCULATOR !!

e.g. Estimate the answer to 4.12 x 5.98 4 x 6 = 24

Always remember to write down the numbers you have rounded off

Estimate the answer to these calculations

1. 58 x 21

2. 399 x 31

3. 47 x 22

4. 4899 46

5. 7.12 x 39.2 0.87

6. 377 19

7. 1906 44

8. 4.89 x 6.01 1.92

9. 360 x 87

10. 58 x 21

Percentages Percentage increase and decrease

A woman’s wage increases by 13.7% from £240 a week. What does she now earn ?

13.17% of £240Increase:

13.17 100

240 =x

New amount:

31.608Her new wage is £271.61 a week

240 + 31.608 =

271.608

Percentages of amounts

25% =

20% = 50% = 2% =

45% =

1% =

75% =

30% = 10% =

85% =

5% = £600

(Do these without a calculator)

Percentages Fractions, decimals and percentages

83%

9500.04

56%

28%

19200.92

425

0.17

%

FracDec

50%

0.512

Copy and complete:

Reverse %

e.g. A woman’s wage increases by 5% to £660 a week. What was her original wage to the nearest penny?

Original amount = 660 ÷ 1.05 = £628.57

Originalamount x 1.05 £660

Originalamount

£660÷ 1.05

e.g. A hippo loses 17% of its weight during a diet. She now weighs 6 tonnes. What was her former weight to 3 sig. figs. ?

Original weight = 6 ÷ 0.83 = 7.23 tonnes

Originalweight x 0.83 6 ton.

Originalweight

6 ton.÷ 0.83

Repeated %

e.g. A building society gives 6.5% interest p.a. on all money invested there. If John pays in £12000, how much will he have in his account at the end of 5 years.

He will have = 12000 x (1.065)5 = £16441.04

e.g. A car loses value at a rate of approximately 23% each year. Estimate how much a $40000 car be worth in four years ?

The car’s new value = 40000 x (0.77)4 = $14061 (nearest $)

£12000 x 1.065 ?x 1.065 x 1.065 x 1.065 x 1.065

This is not the correct method: 12000 x 0.065 = 780

780 x 5 = 390012000 + 3900 = £15900

£40000 x 0.77 ?x 0.77 x 0.77 x 0.77

This is not the correct method: 40000 x 0.23 = 9200

9200 x 4 = 3680040000 – 36800 = $3200

Types of number

From this set of numbers list the:• Odd numbers• Even numbers• Multiples of 8• Factors of 12• Prime numbers• Square numbers• Cube numbers

9 1261 20

100

7

25311

13 162 27

Some useful words to know the meaning of: Sum = add together Product = multiply together Difference = subtract one number from another Reciprocal of a number = 1 divided by the number (e.g. Reciprocal of 4 = ¼ or 0.25)

Products of primes

Express 40 as a product of primes

40

2 20

2 10

2 540 = 2 x 2 x 2 x 5 (or 23 x 5)

Express 630 as a product of primes 630

2 315

3 105

3 35

5 7630 = 2 x 3 x 3 x 5 x 7 (or 2 x 32 x 5 x 7)

Now do the same for 100 , 30 , 29 , 144

Finding the HCF and LCM of a pair of numbersHCF stands for the Highest Common Factor (the biggest number that will go into both numbers)LCM stands for Lowest Common Multiple (the first number to appear in both numbers times table)

e.g. Find the HCF and LCM of the two numbers 140 and 112

140 = 2 x 2 x 5 x 7 and 112 = 2 x 2 x 2 x 2 x 7

Write both numbers as a product of primes

For the HCF write out all the primes that appear in both answers

HCF = 2 x 2 x 7 = 28

For the LCM write out the largest number of each prime that exists in either number

LCM = 2 x 2 x 2 x 2 x 5 x 7 = 560

Indices

104

190

75 73102

23

91

5232

53

25

26

9

43171

Simplifying roots Tip: Always look for square numbered factors (4, 9, 16, 25, 36 etc)

12 4 x 3 2 3

8 4 x 2 2 2

45 9 x 5 3 5

72 36 x 2 6 2

700 100 x 7 10 7

e.g. Simplify the following into the form a b

Standard form

Write in Standard Form

3 600 0.041

46.70.003

0.0001

8 900 000 000

23 600

9.6

0.2

Write as an ordinary number

2 x 100

8.6 x 10-1

1 x 102

6 x 106

7 x 10-2

5.1 x 104

9.2 x 103 3.5 x 10-3

4.7 x 109 8 x 10-3

Do 3 x 104 x 7 x 105 with and without a calculator

Ratio

Equivalent Ratios

1 : ?

0.5 : ?

? : 1 2100 : ?

? : 12

14 : ?

? : 12? : 6

? : 10

21 : ?

49 : ?7:2

Splitting in a given ratio

£600 is split between Anne, Bill and Claire inthe ratio 2:7:3. How much does each

receive?

Total parts = 12

Anne gets 2 of 600 = £100 12

Claire gets 3 of 600 = £150 12

Basil gets 7 of 600 = £350 12

Fractions with the four rules + – × ÷

• Always convert mixed fractions into top heavy fractions before you start

• When adding or subtracting the “bottoms” need to be made the same

• When multiplying two fractions, multiply the “tops” together and the “bottoms” together to get your final fraction

• When dividing one fraction by another, turn the second fraction on its head and then treat it as a multiplication

Learn these steps to complete all fractions questions:

Fractions with the four rules

4⅔ + 1½

14 3

32

+=

37 6

=

96

28 6

+=

= 6 16

4⅔ 1½

14 3

32

=

14 3

23

=

28 9

=

= 3 19

Recurring decimals as fractions

Express 0.77777777….. as a fraction.

Let n = 0.77777777….. so 10n = 7.77777777…..so 9n = 7 so n = 7/9

Express 2.34343434….. as a fraction.

Let n = 2.34343434….. so 100n = 234.34343434…..so 99n = 232 so n = 232/99

Express 0.13213213….. as a fraction.

Let n = 0.132132132….. so 1000n = 132.132132132…..so 999n = 132

so n = 132/999 n = 44/333

Learn this technique which changes recurring decimals into fractions:

Units

Metric length conversions

km m cm mm

x 1000 x 10x 100

÷ 1000

÷ 10÷ 100

Metric weight conversions

kg g cg mg

x 1000 x 10x 100

÷ 1000

÷ 10÷ 100

Metric capacity conversions

kl l cl ml

x 1000 x 10x 100

÷ 1000

÷ 10÷ 100

Learn this pattern for converting between the various metric units

Learn these rough conversions between imperial and metric units

1 inch 2.5 cm1 yard 0.9 m5 miles 8 km2.2 lbs 1 kg1 gallon 4.5 litres

Speed, Distance, Time questions

Speed, Distance and Time are linked by this formula

To complete questions check that all units are compatible, substitute your values in and rearrange if necessary.

S = DT

1. Speed = 45 m/sTime = 2 minutesDistance = ?

2. Distance = 17 milesTime = 25 minutesSpeed = ?

3. Speed = 65 km/hDistance = 600kmTime = ?

S = D T

45 m/s and 120 secs

45 = D . 120

45 x 120 =

D D = 5400

m

S = D T

65 = 600 . T

T = 9.23

hours

S = D T

17 miles and 0.417 hours

S = 17 . 0.417 S = 40.8

mph

T = 600 . 65

Density, Mass, Volume questions

Density, Mass and Volume are linked by this formula

To complete questions check that all units are compatible, substitute your values in and rearrange if necessary.

D = MV

1. Density = 8 g/cm3

Volume = 6 litresMass = ?

2. Mass = 5 tonnesVolume = 800 m3

Density = ?

3. Density = 12 kg/m3

Mass = 564 kgVolume = ?

D = M V

8 g/cm3 and 6000 cm3

8 = M . 6000

8 x 6000 =

M M = 48000 g

D = M V

12 = 564 . V

V = 47 m3

D = M V

800 m3 and 5000 kg

D = 5000 . 800 D = 6.25

kg/m3

V = 564 . 12

( or M = 48 kg)