Standard Form, Ratio, Rates & Proportion

IGCSE – Chapter 1Number

Note 1: Standard Form

We use standard form when dealing with very large or very small numbers.

a x 10n is in standard form when 1 < a < 10 and n is a positive or negative number.

Note 1: Standard Form

Write the following in standard form.a.) 9000b.) 350c.) 0.006

= 9 x 1000 = 9 x 103

= 3.5 x 100 = 3.5 x 102

= 6 x = 6 x 10- 3

1000

1

d.) 83700e.) 0.00075

f.) 12.5 million

= 8.37 x 10000 = 8.37 x 104

= 7.5 x 10000

1= 7.5 x 10- 4

= 1.25 x 10 000 000 = 1.25 x 107

Make sure you use this notation and not calculator notation

Note 1: Standard Form

The speed of light is 300 000 km/s. Express this speed in cm/s in standard form.

Make sure you use this notation and not calculator notation

s

km 300000 xkm

m1000 xm

cm100

= 30000000000 s

cm

= 3 x 1010 scm

Note 1: Standard Form

Given that L = 2 , find the value of L in standard form when a = 4.5 x 1012 & k = 5 x 107

Make sure you use appropriate brackets on your calculator

k

a

= 2

= 600

7

12

x105

x105.4

= 6 x 102 IGCSE Ex 13 pg 13-14 oddEx 14 pg 14-15 odd

Note 2: Ratio and Proportion

The word ‘ratio’ is used to describe a fraction.

If the ratio of your height to your fathers height is 4:5, then you are of your fathers height.

5

4e.g.

e.g.Express the following ratios in the form 1 : na.) 2:5 b.) 7:8 c.) 33:990

1 : 1 : 1 : 30 2

57

8

e.g.Express the following ratios in the form n : 1a.) 2:5 b.) 3:300 c.) 65:875

: 1 5

2: 1

100

1: 1

175

13

Note 2: Ratio and Proportion

Divide $70 between John and Hamish in the ratio of 3:4

Consider that $70 has 7 equal parts(i.e. 3 + 4). Then John receives 3 parts and Hamish receives 4 parts.

John receives of $70 =

Hamish receives of $70 =

7

3

7

4

$30

$40

His friends each get of the brothers stamps

Note 2: Ratio and Proportion

A brother and sister share out their collection of 5000 stamps in the ratio 5:3. The brother then shares his stamps with two friends in the ratio 3:1:1, keeping the most for himself. How many stamps do each of his friends receive?

Brother receives of 5000 =

5

1

8

53125 stamps

IGCSE Ex 15 pg 15-16 odd

= 625 stamps

Note 2: Ratio and Proportion

Proportion – Finding a unit quantity

If a wire of length 5 metres costs $35, find the cost of a wire of length 75 cm

500 cm costs 3500 cents

1 cm costs

7

3

500

3500= 7 cents

75 cm costs 7 x 75 = 525 cents

= $5.25

of 3 ft is 7 ft

10 men days (3 ft)

Note 2: Ratio and Proportion

If it takes 6 men 4 days to dig a hole 3 feet deep, how long will it take 10 men to dig a hole 7 feet deep?

6 men 4 days (3 ft)

3

7

10

24

10 men x = 5 days

1 man 24 days (3 ft)

10

24

3

7

5

3

= 5.6 days IGCSE Ex 16 pg 17-18 odd

Note 3: Approximations & Estimation

Write the following correct to the nearest:

Whole Number

3 sf 2 dp

3.1210.5893.2559.896

0.0820

3

1

3100

3.120.589

3.269.90

0.0820

3.12

0.59

3.269.900.08

Note 3: Measurements & Bounds (Limits of accuracy)

Remember that measurements are approximate.

e.g. The length of a fabric is measured to 145 cm to the nearest cm.

The actual length is between 144.5 cm and 145.4999999…..

144.5 < length < 145.5

Lower bound (limit)

Upper bound (limit)

Note 3: Measurements & Bounds (Limits of accuracy)

Remember that measurements are approximate.

e.g. The weight of a butterfly is given as 0.032 g.

The actual weight is between and

< weight < Lower bound (limit)

Upper bound (limit)

0.0315 g 0.0325 g

0.0315 0.0325

IGCSE Ex 9 pg 9Ex 10 pg 10-11 oddEx 11 pg 11-12 odd

Note 4: Currency Exchange

An application of how we use proportion.

e.g. The following are exchange rates for NZD ($).

Country Exchange Rate

U.K. (pounds) £0.58= $1

Canada ($) $1.276 CAD = $1

Euro (euros) €0.785 = $1

Argentina (pesos) 0.897ARPO = $1

Convert $ 28.00 to euros Convert £500 to NZD $$1 = €0.785

$28 = €0.785 x 28$28 = €21.98

£0.58= $1£1=

58.0

1$

£500=$862.07

Note 5: Speed, distance & time

Great care must be taken with units in these problems.

e.g. How long is a train which passes a signal in twenty seconds at a speed of 108 km/hr?.

S

D

T

T = 20 s x s

hr

3600

T = 0.005556 hr D = S x T

D = 108 km/hr x 0.005556 hr D = 0.6 km

D = 600 m

Note 5: Speed, distance & time

Great care must be taken with units in these problems.

e.g. An earthworm of length 15cm is crawling along at 2 cm/s. An ant overtakes the worm in 5 seconds. How fast is the ant walking?.

S

D

T

How far does the earthworm travel in 5 seconds?

2 x 5 s = 10 cms

cm

The ant must overtake the length and distance travelled 10 cm + 15 cm = 25 cm (in 5 seconds)

The ant’s speed is s 5

cm 25 = 5s

cm IGCSE Ex 17 pg 18-19Ex 25 pg 29-30