Higher Tier - Number revision

Contents : Calculator questionsLong multiplication & divisionBest buy questionsEstimationUnitsSpeed, Distance and TimeDensity, Mass and VolumePercentagesProducts of primesHCF and LCMIndicesStandard FormRatioFractions with the four rulesUpper and lower boundsPercentage errorSurds Rational and irrational numbersRecurring decimals as fractionsDirect proportionInverse proportionGraphical solutions to equations

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

–

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?

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

Estimation 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

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)

Simple % 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)

Simple % 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

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

HCFExpressing 2 numbers as a product of primes can help you calculate their Highest common factor

LCM Expressing 2 numbers as a product of primes can also help you calculate their Lowest common multiple

e.g. Find the highest common factor of 84 and 120.

84 = 2 x 2 x 3 x 7

120 = 2 x 2 x 2 x 3 x 5

Highest common factor = 2 x 2 x 3 = 12

Pick out all the bits that are common to both.

e.g. Find the lowest common multiple of 300 and 504.

300 = 22 x 3 x 52

504 = 23 x 32 x 7

Lowest common multiple = 23 x 32 x 52 x 7 = 12600

Pick out the highest valued index for each prime factor .

LCMConsider the numbers 16 and 20.

Their multiples are: 16, 32, 48, 64, 80, 96 and 20, 40, 60, 80, 100

Their lowest common multiple is 80

HCFConsider the numbers 20 and 30.

Their factors are: 1, 2, 4, 5, 10, 20 and 1, 2, 3, 5, 6, 10, 15, 30

Their highest common factor is 10

Indices

10-4

190

75 73

163/2

2-3

91/2115

3-2

36-1/2

2-143

811/4

Evaluate:

25 x 21

Standard formWrite in Standard Form

3 600 0.041

0.0001

8 900 000 000

56 x 103

9.6

0.2

Write as an ordinary number

8.6 x 10-1

1 x 102

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

Calculate 4.6 x 104 ÷ 2.5 x 108 with a calculator

Calculate 3 x 104 x 7 x 10 -1 without a calculator

Calculate 1.5 x 106 ÷ 3 x 10 -2 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

A journey of 37 km measured to the

nearest km could actually be as long as 37.49999999…. km or as short as 36.5 km. It could not be 37.5 as this would round up to 38 but the lower and upper bounds for this measurement are 36.5 and 37.5 defined by: 36.5 < Actual distance < 37.5 e.g. Write down the Upper and lower bounds of each of these values given to the accuracy stated:

Upper and lower bounds

9m (1s.f.) 85g (2s.f.)

180 weeks (2s.f.)

2.40m (2d.p.)4000L (2s.f.)60g (nearest g)

8.5 to 9.5 84.5 to 85.5

175 to 185

2.395 to 2.4053950 to 4050

59.5 to 60.5

e.g. A sector of a circle of radius 7cm makes an angle of 320 at the centre. Find its minimum possible area if all measurements are given to the nearest unit. ( = 3.14)

Area = (/360) x x r x rMinimum area = (31.5/360) x 3.14 x 6.5 x 6.5Minimum area = 11.61cm2

320

7cm

% errorIf a measurement has been rounded off then it is not accurate. There is a an error between the measurement stated and the actual measurement.

The exam question that occurs most often is: “Calculate the maximum percentage error between the rounded off measurement and the actual measurement”.

e.g. This line has been measured as 9.6cm (to 1d.p.). Calculate the maximum potential error for this measurement.

Upper and lower bounds of 9.6 cm (1d.p.) Maximum potential difference (MPD) between actual and rounded off measurements Max. pot. % error = (MPD/lower bound) x 100

9.55 to 9.65

= (0.05/9.55) x 100= 0.52%

0.05

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

20 4 x 5 2 5

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

Surds A surd is the name given to a number which has been left in the form of a root. So 5 has been left in surd form.

A surd or a combination of surds can be simplified using the rules:M x N = MN and visa versaM ÷ N = M/N and visa versa

Tips: Deal with a surd as you would an algebraic term and always look for square numbers

SIMPLIFYING EXPRESSIONS WITH SURDS IN

(3 – 1)2

5(5 + 20)

135 ÷ 3

12 4 x 3

2 3

9 x 5

3 5

5 + 100

15

(3 – 1) (3 – 1)

3 – 3 – 3 + 1

4 – 23

45

LEAVING ANSWERS IN SURD FORM Calculate the length of side x in surd form (non-calculator paper):

x

6

14Pythagoras (14)2 = (6)2 + x2

14 = 6 + x2

x = 8Answer: x = 22

Rational and irrational numbers

Rational numbers can be expressed in the form a/b. Terminating decimals (3.17 or 0.022) and recurring decimals (0.3333..or 4.7676..) are rational.

Irrational numbers cannot be made into fractions. Non-terminating and non-recurring decimals (3.4526473… or or 2) are irrational.

16 1/53 20 2.3/5.752 2.7

State whether the following are rational or irrational numbers:

What do you need to do to make the following irrational numbers into rational numbers:

3 20 253 63

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 0.413213213….. as a fraction.

Let n = 0.4132132132….. so 10000n = 4132.132132132…..and 10n = 4.132132132……so 9990n = 4128

so n = 4128/9990 n = 688/1665

Learn this technique which changes recurring decimals into fractions:

Direct proportion

If one variable is in direct proportion to another (sometimes called direct variation) their relationship is described by:

p t

p = kt

Where the “Alpha” can be replaced by an “Equals” and a constant “k” to give :e.g. y is directly proportional to the square of r. If r is 4 when y is 80,

find the value of r when y is 2.45 .

Write out the variation:

y r2

Change into a formula:

y = kr2

Sub. to work out k:

80 = k x 42

k = 5

So:

y = 5r2

And:

2.45 = 5r2

Working out r:

r = 0.7

Possible direct variation questions:

x p

t h2

s 3v

c i

g u3 g = ku3

c = ki

s = k3vt = kh2

x = kp

Inverse proportion

If one variable is inversely proportion to another (sometimes called inverse variation) their relationship is described by:

p 1/t p = k/t Again “Alpha” can be replaced by a constant “k” to give :

e.g. y is inversely proportional to the square root of r. If r is 9 when y is 10, find the value of r when y is 7.5 .

Write out the variation:

y 1/r

Change into a formula:

y = k/r

Sub. to work out k:

10 = k/9

k = 30So

: y = 30/r

And:

7.5 = 30/r

Working out r:

r = 16 (not 2)

Possible inverse variation questions:

x 1/p

t 1/h2

s 1/3v

c 1/i

g 1/u3 g = k/u3

c = k/i

s = k/3v

t = k/h2

x = k/p

Graphical solutions to equations

If an equation equals 0 then its solutions lie at the points where the graph of the equation crosses the x-axis.

e.g. Solve the following equation graphically:x2 + x – 6 = 0

All you do is plot the equation y = x2 + x – 6 and find where it crosses the x-axis (the line y=0)

y

x2-3

y = x2 + x – 6

There are two solutions tox2 + x – 6 = 0 x = - 3 and x =2

Graphical solutions to equations

If the equation does not equal zero :Draw the graphs for both sides of the equation andwhere they cross is where the solutions lie

e.g. Solve the following equation graphically:x2 – 2x – 11 = 9 – x

Plot the following equations and find where they cross:y = x2 – 2x – 20 y = 9 – x

There are 2 solutions to x2 – 2x – 11 = 9 – xx = - 4 and x = 5

y

x

y = x2 – 2x – 11

y = 9 – x

-4 5

If there is already a graph drawn and you are being asked to solve an equation using it, you must rearrange the equation until one side is the same as the equation of the graph. Then plot the other side of the equation to find the crossing points and solutions.

e.g. Solve the following equation using the graph that is given: x3 – 4x + 5 = 5x + 5

y

x

y = x3 – 8x + 7

Rearranging the equation x3 – 4x + 5 = 5x + 5 to get x3 – 8x + 7 :

Add 2 to both sidesx3 – 4x + 5 = 5x + 5

x3 – 4x + 7 = 5x + 7

x3 – 8x + 7 = x + 7

Take 4x from both sides

So we plot the equation y = x + 7 onto the graph to find the solutions

y

x

y = x3 – 8x + 7

-3 0 3

Solutions lie at –3, 0 and 3

y = x + 7

State the graphs you need to plot to solve the following equations describing how you will find your solutions:1. 3x2 + 4x – 2 = 02. 7x + 4 = x2 – 4x3. x4 + 5 = 04. 0 = 8x2 – 5x5. 2x = 96. 6x3 = 2x2 + 5If you have got the graph of y= 4x2 + 5x – 6 work out the other graph you need to draw to solve each of the following equations:1. 4x2 + 4x – 6 = 02. 4x2 + x - 2 = 73. 4x2 – 3x = 2x4. 3x2 = – 5

Solve this equation graphically:x3 + 8x2 + 3x = 2x2 – 2x