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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%
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
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
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
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
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
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
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