# 1 part $5000 at 6.6% interest monthly for 6years. 2nd part $1100 at 4.8% compounded annually for 6 years.

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The first part:

Principle , P =$5000 is invested for 6 years at the rate of 6.6% .The interest is on a monthly basis.You did not say whether it is compoundibg or simple interest. So, the monthly rate of interest is 6.6%= 6.6/12 %=0.55% = 0.0055 per dollar.

The amount after 6 years = P+P*nr, where P = principle, n number of months =6*12 = 72 and r is the rate of monthly interest per dollar. So, the amount including the simple interest for 72 months is given by:

Principle +interest on the principle = 5000+5000*72*0.0055=$(5000+1980) = $6980 including the simple interest. It is as good as the simple interest for 6 years.

The second part:

The investment is in compound interest annually. Pinciple ,P = $1100. The annual rate of interest = 4.8%. Therefore, the amount he gets with annual compounding is 1100*1.048^6=$1457.34

In total , for $( 5000+1100) = $6100 of investment , he gets $(6980+1457.34)= $8437.34.

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Had he invested the first sum also in monthly compounding interest, he would have got for $5000, an amount of 5000*1.0055^72=$7421.29. And his total amount of $6100 in 2 investments would have brought him an amount of $(7421.29+1466.29) = $8887.58 incuding the interest.

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Had he invested the first sum also in monthly compounding interest, he would have got for $5000, an amount of 5000*1.0055^72=$7421.29 ..........................(1)

Had he deposited a sum of $1100 every year for six 6 years for compounding annually at the rate of 4.8% (=0.048 per dollar), then he would have got $1100*(1.048^6+1.048^5+1.048^4+1.048^3+1.048^2+1.048^1)

=$1108*x{x^5+x^4+x^3+x^2+x^1), where x=1.048

=$1108*x(x^6-1)/(x-1),

=$1100{7.092623993}

= $7801.89 ..............................................(2)

So, the 2nd invetment brings : $7801.89-$7421.29 = $380.60 more.

Hope this helps. Any queries?

1. Since the 6.6% int is paid monthly, the net int paid per annum becomes 6.8033%. The present value of 5000$ becomes $ 7421.28 after 6 years.

2. $1100 becomes after 6 years is $1457.22

This information given in the question is not enough to know what is the criteria for comparison. For example the comparison could be on the basis of several criteria such as:

- Total interest amount paid.
- Total interest paid as simple percentage of the loan amount taken.
- Compound interest rate paid.

In the two alternatives the principal loan amounts are also different. There is no information available to determine which of the two alternative is better.

Further, is the comparison to be made from the viewpoint of borrower or the lender?

However, I suggest the most important aspect of attractiveness or otherwise of a loan is the the compound rate of interest taking a common period for compounding. Thus in the above question the two different loans can be compared by converting the interest rate of both the loans to compound interest rate compounded annually.

For the second loan annually compounded interest rate is already given as 4.8%.

In the first loan the interest rate is stated as 6.6% monthly. There is no mention of compounding. This implies that the interest amount at the rate of 6.6% is paid every month. The Principal amount in full is returned after 6 years. This amounts to 6.6 percent compounded monthly. We now need to convert this monthly compounding rate to annual compounding rate. This can be done by calculating the total interest per 100 for one full year. This is:

= 100*[1 + 6.6/(12*100)]^12 - 100= 100*(1 + 0.0055)^12 - 100

= 100*1.06803 - 100 = 106.803 - 100 = 6.803

Thus equivalent compound interest rate compounded yearly for the first loan is 6.803%. This is 1,417 times the rate of 4.8% for the second alternative.