Angela C. Rosario

Mr. Thomas

IB Mathematics SL I (3)

24 March 2009

Practice IB Internal Assessment

Introduction: The purpose of interest is for a bank to pay an individual for the use of their money. Interest therefore represents one's return on the investment. To calculate n interest compoundings per year, one must utilize the formula:

. Alan invests $1000 at an interest rate of 12% per year. Copy and complete Table 1 which shows A, the value of the investment in dollars after t years, assuming that the interest is compounded yearly.

One must utilize the formula for Compound Interest in order to determine the answer. In the case of Alan, his principle (P) amount of money is $1000, and he collects interest at a rate (r) of 12% per year. To determine the value of the investment in dollars after t years in order to satisfy Table 1, the formula necessary is:

A= 1000 ( 1+ .12/1)(1)(t)

where P= $1000, r= 0.12, and n=1 because interest is being compounded annually or once a year, and t= the number of years the money is present in the account.

t=0, A= 1000( 1+ 0.12/1)(1)(0) = 1000(1+ 0.12) 0= 1000(1.12) 0= 1000(1)= 1000

t=1, A= 1000( 1+ 0.12/1)(1)(1) = 1000(1+ 0.12) 1= 1000(1.12) 1= 1000(1.12)= 1120

t=2, A= 1000( 1+ 0.12/1)(1)(2) = 1000(1+ 0.12) 2= 1000(1.12) 2= 1000(1.2544)= 1254.40

t=5, A= 1000( 1+ 0.12/1)(1)(5) = 1000(1+ 0.12) 5= 1000(1.12) 5˜ 1000(1.76234)˜ 1762.34

t=10, A= 1000( 1+ 0.12/1)(1)(10) = 1000(1+ 0.12) 10= 1000(1.12) 10˜ 1000(3.10585)˜ 3105.85

t=20, A= 1000( 1+ 0.12/1)(1)(20) = 1000(1+ 0.12) 20= 1000(1.12) 20˜ 1000(9.64629)˜ 9646.29

t=30, A= 1000( 1+ 0.12/1)(1)(30) = 1000(1+ 0.12) 30= 1000(1.12) 30˜ 1000(29959.92)˜ 29959.92

*Final answers rounded to the nearest hundredth in order to comply with the general money standard of cents.

Table 1 Alan's Investment of $1000 at an Interest Rate of 12% Per Year Compounded Yearly

t

0

2

5

0

20

30

A

000

120

254.40

762.34

3105.85

9646.29

29959.92

Table 1 represents Alan's investment of $1000 at an interest rate of 12% per year over a period of 1, 2, 5, 10, 20, and 30 years. As the data illustrates, it is evident that the longer his investment is in the account, the larger his final sum becomes. This effect can be seen with an extremely large sum of $29,959.92 being accumulated in the thirtieth year of investment, almost thirty times what was originally put into the investment account. Of the $29,959.92 that is in the account at the end of 30 years, $28,959.92 represents the interest earned with Alan's $1000 principal investment.

2. (a) Show that if interest of 12% per year is compounded monthly it is equivalent to an interest rate of approximately 12.68% per year if the interest is compounded yearly.

To calculate a 12% interest rate (r) with a monthly frequency, one must substitute 12 for n instead of 1 because there are 12 months in a year, resulting in the following formula:

A= P(1+0.12/12)(12)(t)

To calculate a 12.68% interest rate (r) with a yearly frequency, one must the value 1 for n, resulting in the following formula:

A= P(1+0.1268/1)(1)(t)

It is clear that 12% interest per year compounded monthly is approximately the same as 12.68% interest per year compounded annually. At first, each formula produces the same interest, however, as time goes on, the exactness between the two rates and frequencies becomes slightly skewed. This, however, is justifiable because it is impossible for these two formulas to yield the exact same results, which is why over time the values differ by less than 1%. Though there is some discrepancy between the two sums, the calculations of a 12.68% interest rate compounded yearly and a 12% interest rate compounded monthly are extremely close in value and will remain this similar even after a large time periods of time.

Mr. Thomas

IB Mathematics SL I (3)

24 March 2009

Practice IB Internal Assessment

Introduction: The purpose of interest is for a bank to pay an individual for the use of their money. Interest therefore represents one's return on the investment. To calculate n interest compoundings per year, one must utilize the formula:

. Alan invests $1000 at an interest rate of 12% per year. Copy and complete Table 1 which shows A, the value of the investment in dollars after t years, assuming that the interest is compounded yearly.

One must utilize the formula for Compound Interest in order to determine the answer. In the case of Alan, his principle (P) amount of money is $1000, and he collects interest at a rate (r) of 12% per year. To determine the value of the investment in dollars after t years in order to satisfy Table 1, the formula necessary is:

A= 1000 ( 1+ .12/1)(1)(t)

where P= $1000, r= 0.12, and n=1 because interest is being compounded annually or once a year, and t= the number of years the money is present in the account.

t=0, A= 1000( 1+ 0.12/1)(1)(0) = 1000(1+ 0.12) 0= 1000(1.12) 0= 1000(1)= 1000

t=1, A= 1000( 1+ 0.12/1)(1)(1) = 1000(1+ 0.12) 1= 1000(1.12) 1= 1000(1.12)= 1120

t=2, A= 1000( 1+ 0.12/1)(1)(2) = 1000(1+ 0.12) 2= 1000(1.12) 2= 1000(1.2544)= 1254.40

t=5, A= 1000( 1+ 0.12/1)(1)(5) = 1000(1+ 0.12) 5= 1000(1.12) 5˜ 1000(1.76234)˜ 1762.34

t=10, A= 1000( 1+ 0.12/1)(1)(10) = 1000(1+ 0.12) 10= 1000(1.12) 10˜ 1000(3.10585)˜ 3105.85

t=20, A= 1000( 1+ 0.12/1)(1)(20) = 1000(1+ 0.12) 20= 1000(1.12) 20˜ 1000(9.64629)˜ 9646.29

t=30, A= 1000( 1+ 0.12/1)(1)(30) = 1000(1+ 0.12) 30= 1000(1.12) 30˜ 1000(29959.92)˜ 29959.92

*Final answers rounded to the nearest hundredth in order to comply with the general money standard of cents.

Table 1 Alan's Investment of $1000 at an Interest Rate of 12% Per Year Compounded Yearly

t

0

2

5

0

20

30

A

000

120

254.40

762.34

3105.85

9646.29

29959.92

Table 1 represents Alan's investment of $1000 at an interest rate of 12% per year over a period of 1, 2, 5, 10, 20, and 30 years. As the data illustrates, it is evident that the longer his investment is in the account, the larger his final sum becomes. This effect can be seen with an extremely large sum of $29,959.92 being accumulated in the thirtieth year of investment, almost thirty times what was originally put into the investment account. Of the $29,959.92 that is in the account at the end of 30 years, $28,959.92 represents the interest earned with Alan's $1000 principal investment.

2. (a) Show that if interest of 12% per year is compounded monthly it is equivalent to an interest rate of approximately 12.68% per year if the interest is compounded yearly.

To calculate a 12% interest rate (r) with a monthly frequency, one must substitute 12 for n instead of 1 because there are 12 months in a year, resulting in the following formula:

A= P(1+0.12/12)(12)(t)

To calculate a 12.68% interest rate (r) with a yearly frequency, one must the value 1 for n, resulting in the following formula:

A= P(1+0.1268/1)(1)(t)

It is clear that 12% interest per year compounded monthly is approximately the same as 12.68% interest per year compounded annually. At first, each formula produces the same interest, however, as time goes on, the exactness between the two rates and frequencies becomes slightly skewed. This, however, is justifiable because it is impossible for these two formulas to yield the exact same results, which is why over time the values differ by less than 1%. Though there is some discrepancy between the two sums, the calculations of a 12.68% interest rate compounded yearly and a 12% interest rate compounded monthly are extremely close in value and will remain this similar even after a large time periods of time.