Simple Interest and Compound Interest (O Level E.Maths)
Question
An amount of $5,000 is deposited in a bank which paid a COMPOUND interest of 2.8% per annum. Find the total amount in the account at the end of 6 months.
Solution given by teacher
Total amount at the end of 6 months
= 5000 [ 1 + (2.8/100) ] ^ (6/12)
= $5069.52
Solution given by student
Total amount at the end of 6 months
= 5000 + 5000 x (2.8/100) x (6/12)
= $5070
Whose answer is correct ?
Hi,
First answer is correct.
See also:
http://qrc.depaul.edu/studyguide2009/notes/savings%20accounts/compound%20interest.htm
Cheers,
Wen Shih
Agree, first one is correct.
Second one means a different thing.
Why the answer looks similar, can be explained by expanding with binomial theorem. Since 2.8/100 is very much smaller than 1, higher powers have insignificant values.
However, the mathematical meaning is different for second one.
I agree also.
The teacher is basically counting 1.028^(½) × $5000.
The student is basically counting 1.014 x $5000.
The interest calculation for half a year should involve the square root of 1.028, not half of 2.8% interest.
My opinion is that the teacher is incorrect.
Using the knowledge of business finance (the future value of money), it is important to note how many time(s) the interest is / are being compounded /counted in a year.
(1) When the interest is compounded/ counted once a year the formula is P = A [ 1 + (i/100) ] ^ n
(2) When the interest is compounded / counted every month ie 12 times in a year, the formula is P = A [ 1 + ( { i / 12 } /100) ] ^ ( n x12 )
(3) When the interest is compounded / counted every 6 months ie 2 times in a year, the formula is P = A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 )
For the question, it is stated that the bank paid a compound interest of 2.8% per annum ie the interest is to be counted once at the end of every year.
So, there is no extra interest earned (since this is not yet one whole of the first year) on interest paid which is the concept of compound interest.
Hence, the simple interest is to be used for this question ie
Total amount at the end of 6 months
= 5000 + 5000 x (2.8/100) x (6/12)
= $5070
In addition, the amount calculated on a compound interest basis MUST always be GREATER
than the amount calculated on a simple interest basis.
It is to be noted that the teacher's answer of $5069.52 is less than $5070.
qn no say int compounded monthly or yearly
Originally posted by FireIce:qn no say int compounded monthly or yearly
When the question did not state how many times the interest is /are being compounded / counted in a year, it is always assumed that the interest will be compounded / counted once at the end of every year.
The teacher's way of calculating the answer shows a lack of the understanding of the concept of compound interest and the derivation of the compound interest formula and how the formula changes according to the number of times the interest is / are being compunded / counted in a year.
P = A [ 1 + ( i /100) ] ^ n P = A [ 1 + ( { i / 12 } /100) ] ^ ( n x12 )
P = A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 ) P = A [ 1 + ( { i / 365 } /100) ] ^ ( n x 365 )
By the way, no matter how many times the teacher assumed the interest is to be compounded / counted in a year, the formula cannot give the answer provided by the teacher.
Please take note of the relationship how the formula changes when how many times the interest is / are being compounded / counted in a year.
In Business Finance, students are taught to know the correct application of the compound interest formula and the assumptions of / restrictions when they use the formula to calculate the future value of money.
When the interest is counted at the begining of the year or at the end of the year, different compound interest formula will be used for the 2 situations.
There are cases whereby the compound interest formula that cannot be used directly.
Example
An amount of $5000 is deposited in a bank which paid a compound interest of 2.8% per annum and the interest is to be compounded every 6 months. Find the total amount in the account at the end of 27 months.
Solution
Total Amount at the end of 27 months
P = A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 ) + A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 ) x ( R / 100 ) x T
= [ 5000 [ 1 + ({ 2.8 / 2} / 100 ) ] ^ ( 2 x 2) ] + [ 5000 [ 1 + ({ 2.8 / 2} / 100 ) ] ^ ( 2 x 2) ] x ( 2.8 / 100 ) x ( 3 / 12 )
= $5322.94
Originally posted by Seowlah:When the question did not state how many times the interest is /are being compounded / counted in a year, it is always assumed that the interest will be compounded / counted once at the end of every year.
Oh I see, but for this kind of background assumption, maybe it is better to state earlier? If this background assumption is supposed to be part of the question, then of course you are quite right.
I think wen shih, eagle and myself were simply operating with a different background assumption. I personally got no background in banking and finance, what can I do? Probably the teacher should make the background assumption clear in the question.
Originally posted by Mad Hat:Oh I see, but for this kind of background assumption, maybe it is better to state earlier? If this background assumption is supposed to be part of the question, then of course you are quite right.
I think wen shih, eagle and myself were simply operating with a different background assumption. I personally got no background in banking and finance, what can I do? Probably the teacher should make the background assumption clear in the question.
The school teacher is unlikely to be familiar with the background assumption of the compound interest formula otherwise she will not have calculated the answer using the formula in that particular way ie she simply assumes that for simple interest formula, if the time period is x months, then substitute x / 12 in into T and so she also simply assumes that for compound interest formula, if the time period is x months, then substitute x / 12 into n.
What matters more for O level, is what is being tested in the syllabus.
The issue can be solved simply by looking at the O level syllabus
http://www.seab.gov.sg/oLevel/2013Syllabus/4016_2013.pdf
On page 11, the compound interest formula is there, which is what the teacher had applied correctly.
Many things taught at O Level are overly simplified. One good example from Physics would be Newton's 2nd Law, which at O Level, is grossly insufficient and would be marked wrong at A Level. But I guess the simplification is part of the process of slowly introducing students to the topic at hand.
It is NOT a case of oversimplication due to the "O" level syllabus.
May I refer you to the 2005 "O" level Paper 1 Q13(b)
$20,000 is invested in an account which pays 1% per month compound interest rate. Find the total amount in the account at the end of 3 months.
Using the teacher's incorrect logic,
Total amount at the end of 3 months
= 20000 [ 1 + ( 12 / 100) ] ^ ( 3 / 12 )
= $20 574. 75 which is incorrect
The correct answer is
Total amount at the end of 3 months
= 20000 [ 1 + ( 1 / 100) ] ^ 3
= $20 606.02
This is also the same answer given in the ten years series published by SAP, EPB and Dyna.
Typical "O" level Compound Interest Question
2011 "O" level Paper 2 Q(b)(ii)
Ben puts $1,000 savings into an account paying compound interest of 3.5 % per annum. Calculate the total amount of money in his account after 5 years.
Total amount after 5 years
P = A [ 1 + ( i /100) ] ^ n
= 1000 [ 1 + ( 3.5/ 100) ] ^ 5
= $1187.69
Note
(1) compound interest of 3.5% per annum means that the interest is to be compounded / counted ONCE in a year. The investment time period is 5 years ie interest will be compounded / counted 5 times during the 5 year investment period. Hence, n = 1 x 5 = 5
(2) interest is paid at 3.5 % per annum. Since the interest compounded / counted time period is 1 year. Hence i = 3.5
Not so typical "O" level Compound Interest Question
2005 "O" level Paper 1 Q13(b)
$20,000 is invested in an account which pays 1% per month compound interest rate.
Find the total amount in the account at the end of 3 months.
Total amount at the end of 3 months
= 20000 [ 1 + ( 1 / 100) ] ^ 3
= $20 606.02
Note
(1) 1% per month compound interest means that the interest is to be compounded / counted
ONCE per month. The investment time period is 3 months ie interest will be compounded /
counted 3 times during the 3 months investment period. Hence, n x 3 = 1 x 3 = 3
(2) interest is paid at 1 % per month. Since the interest compounded / counted time period
is per month. Hence i = 1
When the interest rate is given on a per annum basis and the interest is compounded /
counted per month
Modified question of 2005 "O" level Paper 1 Q13(b)
$20,000 is invested in an account which pays an interest rate of 12 % per annum and
the interest is to be compounded / counted every month.
Find the total amount in the account at the end of 24 months.
Total amount at the end of 24 months
= 20000 [ 1 + ( { 12 / 12 } / 100) ] ^ ( 2 x 12 )
= $25 394.69
Note
(1) Since the interest is to be compounded / counted every month, The interest rate
of 12% per annum needs to be converted into 12 / 12 =1% per month first
The investment time period is 24 months ie interest will be compounded /
counted 2 x 12 = 24 times during the 24 months investment period Hence, n x 12 = 2 x 12
= 24
(2) interest is paid at 12 / 12 = 1 % per month. Since the interest compounded / counted
time period is per month. Hence i / 12 = 12 / 12 = 1.
-
Originally posted by eagle:What matters more for O level, is what is being tested in the syllabus.
The issue can be solved simply by looking at the O level syllabus
http://www.seab.gov.sg/oLevel/2013Syllabus/4016_2013.pdf
On page 11, the compound interest formula is there, which is what the teacher had applied correctly.
Many things taught at O Level are overly simplified. One good example from Physics would be Newton's 2nd Law, which at O Level, is grossly insufficient and would be marked wrong at A Level. But I guess the simplification is part of the process of slowly introducing students to the topic at hand.
-
Back to the first question when the interest is to be compounded / counted monthly and
the interest rate is given on a per annum basis
Question
An amount of $5,000 is deposited in a bank which paid an interest of 2.8% per annum and
the interest is compounded / counted monthly.
Find the total amount in the account at the end of 6 months.
Total amount at the end of 6 months
= 5000 [ 1 + ( {2.8 / 12 } /100) ] ^ 6
= $5070.41
Note
(1) Since the interest is to be compounded / counted every month, The interest rate
of 2.8 % per annum needs to be converted into 2.8 / 12 % per month first
The investment time period is 6 months ie interest will be compounded /
counted 1 x 6 = 6 times during the 6 months investment period Hence, n x 6 = 1 x 6
= 6
(2) interest is paid at 2.8 / 12 % per month. Since the interest compounded / counted
time period is per month. Hence i / 12 = 2.8 / 12
I concur.
Edited my answer as above because I wasn't sure if what I remembered off my head was correct.
But in general: 1 + r = (1 + i / n)^n
where r is effective i/r, i is nominal i/r, n is no. of compounding periods a year.
If question does not specify if the compounding still applies to the sub-periods, it is taken as assumed thus by default.
Back to the first question when the interest is to be compounded / counted yearly and
the interest rate is given on a per annum basis and the time period of the investment is
in months and less than a year
Question
An amount of $5,000 is deposited in a bank which paid an interest of 2.8% per annum and
the interest is compounded / counted yearly.
Find the total amount in the account at the end of 6 months.
Total amount at the end of 6 months
= 5000 + 5000 x (2.8/100) x (6/12)
= $5070
Note
(1) The compound interest formula CANNOT be used for this question as
there is no extra interest earned (since this is not yet one whole of the first year)
on interest paid which is the concept of compound interest.
(2) The simple interest formula will be used for this question.
Back to the first question when the interest is to be compounded / counted yearly and
the interest rate is given on a per annum basis and the time period of the investment is
in months and more than a year
Question
An amount of $5,000 is deposited in a bank which paid an interest of 2.8% per annum and
the interest is compounded / counted yearly.
Find the total amount in the account at the end of 27 months.
Total amount in the account at the end of 27 months
P = A [ 1 + ( i /100) ] ^ n + A [ 1 + ( i /100) ] ^ n x ( R / 100 ) x T
= 5000 [ 1 + (2.8 / 100 ) ] ^ 2 + 5000 [ 1 + (2.8 / 100 ) ] ^ 2 x ( 2.8 / 100 ) x ( 3 / 12 )
= $5320.91
Note :
(1) The compound interest formula will be used for the 2 complete years of investment
and the remaining 3 months investment will earn simple interest.
Back to the first question when the interest is to be compounded / counted every 6 months
and the interest rate is given on a per annum basis and the time period of the investment is
in months and more than a year
Question
An amount of $5,000 is deposited in a bank which paid an interest of 2.8% per annum and
the interest is compounded / counted every 6 months.
Find the total amount in the account at the end of 27 months.
Total amount in the account at the end of 27 months
P = A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 ) + A [ 1 + ( { i / 2 } /100) ] ^ ( n x 2 ) x ( R / 100 ) x T
= [ 5000 [ 1 + ({ 2.8 / 2} / 100 ) ] ^ ( 2 x 2) ] + [ 5000 [ 1 + ({ 2.8 / 2} / 100 ) ] ^ ( 2 x 2) ]
x ( 2.8 / 100 ) x ( 3 / 12 )
= $5322.94
Note :
(1) The compound interest formula will be used for the 4 completed 6 months ie 2 years
of investment and the remaining 3 months investment will earn simple interest.