8
Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1 = 5% of £1000 = 0.05 x £1000 = £50 Amount at end of Year 1 = £1050 Interest at end of Year 2 = 5% of £1050 = 0.05 x £1050 = £52.50 Amount at end of Year 2 = £1050 + £52.50 = £1102.50 and so on Method 1

Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x £1000 = £50 Amount at end of Year 1= £1050

Embed Size (px)

Citation preview

Page 1: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Compound Interest

Amount invested = £1000

Interest Rate = 5%

Interest at end of Year 1

= 5% of £1000

= 0.05 x£1000

= £50

Amount at end of Year 1

= £1050

Interest at end of Year 2

= 5% of £1050

= 0.05 x£1050

= £52.50

Amount at end of Year 2

= £1050 + £52.50

= £1102.50

and so on

Method 1

Page 2: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Compound Interest

Amount invested = £1000

Interest Rate = 5%

Amount at end of Year 1

= 105% of £1000

= 1.05 x£1000

= £1050

and so on

Method 2

Amount at end of Year 2

= 1.05 x£1050

= £1102.50

Page 3: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Example – Compound Interest

£1000 invested at 5% interest

End of Year n Amount A(£)

0 1000.00

1

2

3

4

5

1050.00

1102.50

1157.63

1215.51

1276.28

Page 4: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Compound Interest

Amount invested = £1000

Interest Rate = 5%

Method 3

Amount at end of Year n

= 1.05n x£1000

Amount at end of Year 2

Amount at end of Year 10

= 1.052 x£1000

= 1.0510 x£1000

= £1102.50

= £1628.89

Page 5: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

General Formulae

k, a and m positive a > 1

Exponential Growth

y = ka mx

A = 1.05n x£1000

Example – Compound Interest

y is A x is n

m = 1 k = 1000 a = 1.05

Can be written in other forms:

A = 1.10250.5n x£1000

m = 0.5

k = 1000

a = 1.1025

Page 6: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Example – Radioactive Decay

Plutonium has a half-life of 24 thousand years

Number of half-lives

Time (000s years)

Amount (g)

0 0 1000

1

2

3

4

5

24 500

48 250

72 125

96 62.5

120 31.25

Page 7: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

Example – Radioactive Decay of Plutonium

Decay functions

A = 1000 x 0.5n where n = no. of half lives

A = 1000 x 2-t/24 where t = time in thousands of years

A = 1000 x 2-n where n = no. of half lives

k and a positive a < 1

Exponential Decay

m positivey = ka mx

a > 1 m negative

A = 1000 x 2-0.0416t where t = time in thousands of years

Page 8: Compound Interest Amount invested = £1000 Interest Rate = 5% Interest at end of Year 1= 5% of £1000 = 0.05 x  £1000 = £50 Amount at end of Year 1= £1050

General Shape of Graphs

Exponential Growth

Exponential Decay

y

k

x

mx ka y

0

k positive m negative a > 1

y

k

x

mx ka y

0

k positive m positive a > 1