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## Compound Interest

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**2. **Compound Interest

**3. **Compound Interest
Ill start with a brief reminder about percentages and simple interest.

**4. **Compound Interest
Ill start with a brief reminder about percentages and simple interest.
Percentages always seem to cause trouble.

**5. **Compound Interest
Ill start with a brief reminder about percentages and simple interest.
Percentages always seem to cause trouble.
A number less than one can be expressed as a
fraction such as 3/47 or, as a decimal, 0.064, which will be a
number between 0 and 1.

**6. **A percentage is just the same fraction

**7. **A percentage is just the same fraction
expressed as a number between 0 and 100

**8. **A percentage is just the same fraction
expressed as a number between 0 and 100
which we get by by multiplying the fraction by 100.

**9. **A percentage is just the same fraction
expressed as a number between 0 and 100
which we get by by multiplying the fraction by 100.
In our example of (3/47)
this would be a percentage of
(3/47) x 100

**10. **A percentage is just the same fraction
expressed as a number between 0 and 100
which we get by by multiplying the fraction by 100.
In our example of (3/47)
this would be a percentage of
(3/47) x 100
= 300/47
= about 6.4%.
Note: Is there a way to use: (6 18/47)% or approximately 6.4% instead for the last line on this slide? (with the bolded items as the items to change)Note: Is there a way to use: (6 18/47)% or approximately 6.4% instead for the last line on this slide? (with the bolded items as the items to change)

**11. **A percentage is just the same fraction
expressed as a number between 0 and 100
which we get by by multiplying the fraction by 100.
In our example of (3/47)
this would be a percentage of
(3/47) x 100
= 300/47
= about 6.4%.
It is normal to use decimals, although values like 5% etc are also used.

**12. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500

**13. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500
= 0.064 x $500 = $32
Note: Does of equated with multiplication in this context need to be highlighted/explained?Note: Does of equated with multiplication in this context need to be highlighted/explained?

**14. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500
= 0.064 x $500 = $32
Simple interest involves the adding of a fixed % of a lump sum at regular intervals (usually one year). 6.4% p.a. (per annum) for example.

**15. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500
= 0.064 x $500 = $32
Simple interest involves the adding of a fixed % of a lump sum at regular intervals (usually one year). 6.4% p.a. (per annum) for example.
For our example, this means that at the end of each year $32 in interest is added.

**16. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500
= 0.064 x $500 = $32
Simple interest involves the adding of a fixed % of a lump sum at regular intervals (usually one year). 6.4% p.a. (per annum) for example.
For our example, this means that at the end of each year $32 in interest is added. After one year the total becomes $532.

**17. **E.g. 6.4% of $500
is just the fraction 6.4/100 (= 0.064) of $500
= 0.064 x $500 = $32
Simple interest involves the adding of a fixed % of a lump sum at regular intervals (usually one year). 6.4% p.a. (per annum) for example.
For our example, this means that at the end of each year $32 in interest is added. After one year the total becomes $532. After two years the interest would be another $32 and so the total becomes $564, etc.

**18. **Compound interest

**19. **Compound interest
This involves the interest being added at the end of each year and this new, larger amount being the new lump sum for the ensuing year.

**20. **Compound interest
This involves the interest being added at the end of each year and this new, larger amount being the new lump sum for the ensuing year.
This means that for our example, at the end of the second year the interest for that year would be 0.064 x $532 (the amount involved at the beginning of that year) x 0.064 = $34.05

**21. **Compound interest
This involves the interest being added at the end of each year and this new, larger amount being the new lump sum for the ensuing year.
This means that for our example, at the end of the second year the interest for that year would be 0.064 x $532 (the amount involved at the beginning of that year) x 0.064 = $34.05
We can write this as $532(1.064) or $500(1.064)(1.064) = $500(1.064)2.

**22. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years

**23. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500

**24. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500

**25. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500
$680.25 - $500

**26. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500
$680.25 - $500
$180.25

**27. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500
$680.25 - $500
$180.25
As a % of the original $500, this is
($180.25 / $500) x 100
Note: Can approximately equal to signs be used throughout (as above), since its not exact?Note: Can approximately equal to signs be used throughout (as above), since its not exact?

**28. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500
$680.25 - $500
$180.25
As a % of the original $500, this is
($180.25 / $500) x 100
36%

**29. **For another rate, say 8%, a similar calculation would give a final amount of
$500(1.08)4 after four years
and the total interest is
$500(1.084) $500
= $500(1.3605) - $500
$680.25 - $500
$180.25
As a % of the original $500, this is
($180.25 / $500) x 100
36%
For a simple interest calculation, the interest would have been 4 x 8% of $500 = 32% of $500 = $160 Ditto on the approximately equal signs as appropriate hereDitto on the approximately equal signs as appropriate here

**30. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
Addition: Please see aboveAddition: Please see above

**31. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
$P(1 + r/100)t

**32. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
$P(1 + r/100)t
This interest very rapidly gets large.

**33. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
$P(1 + r/100)t
This interest very rapidly gets large. If you borrow at a normal credit-card rate of 32% and do not pay off the amount owing for 5 years then the final amount owing would become $P(1 + (32/100))5 = $P(1.32)5

**34. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
$P(1 + r/100)t
This interest very rapidly gets large. If you borrow at a normal credit-card rate of 32% and do not pay off the amount owing for 5 years then the final amount owing would become $P(1 + (32/100))5 = $P(1.32)5
= $P(4.007) = $P + $3.007xP. The interest charge is over THREE times the original i.e. over 300% interest! Note: Super example!!!Note: Super example!!!

**35. **In general we have found the total amount owing from an initial amount $P at a rate of r% for a period of t years becomes
$P(1 + r/100)t
This interest very rapidly gets large. If you borrow at a normal credit-card rate of 32% and do not pay off the amount owing for 5 years then the final amount owing would become $P(1 + (32/100))5 = $P(1.32)5
= $P(4.007) = $P + $3.007xP. The interest charge is over THREE times the original i.e. over 300% interest!
(Simple interest would bring a charge of 4x0.32xP
= 1.28xP bad enough) Note: Super example!!!Note: Super example!!!