# interest rate, compound amount and loans

Section 5.1

#2 Find the interest on each of these loans

$35,000 at 6% for 9 months

#4 Find the interest on each of these loans

$1875 at 5.3% for 7 months

#8 Find the interest on each of these loans

$8940 at 9%; loan made on May 7 and due September 19

#16 Find the future value of each of these loans

$3475 loan at 7.5% for 6 months

#18 Find the future value of each of these loans

$24,500 loan at 9.6% for 10 months

#22 Find the present value of each future amount

$48,000 for 8 months; money earns 5%

#26 The given Treasury bills were sold in August 2008 find:

a) The price of the T bill and

b) The actual interest rate paid by the Treasury

Six month $18,000 T-bill with discount rate 1.925%

#30 An accountant for a corporation forgot to pay the firm's income tax of $725,896.15 on time. The government charged a penalty of 9.8% interest for the 34 days the money was late. Find the total amount (tax and penalty) that was paid

#36 What is the time period of a $10,000 loan at 6.75% in which the total amount of interest paid was $618.75?

#44 WORK THE NEXT PROBLEM IN WHICH YOU ARE TO FIND THE ANNUAL SIMPLE INTEREST RATE. CONSIDER ANY FEES, DIVIDENDS OR PROFITS AS PART OF THE TOTAL INTEREST

Jerry Ryan borrowed $8000 for nine months at an interest rate of 7%. The bank also charges a $100 processing fee. What is the actual interest rate for this loan?

Section 5.2

#8 Find the compound amount of the following deposit

$1000 at 6% compounded annually for 10 years

#10 Find the compound amount of the following deposit

$15,000 at 4.6% compounded semiannually for 11 years

#14 Find the amount of interest earned by the following deposits

$22,000 at 5% compounded annually for 8 years

#18 Find the amount of interest earned by the following deposits

$27,630.35 at 4.6% compounded quarterly for 3.9 years

#22 Find the interest rate with annual compounding that makes the statement true

$9000 grows to $17,118 in 16 years

#24 Find the face value to the nearest dollar of the zero-coupon bond

10 year bond at 4.1%; price $13,328

#26 Find the face value to the nearest dollar of the zero-coupon bond

25 year bond at 4.4%; price $10,106

#32 Find the APY corresponding to the given nominal rates

4.7% compounded semiannually

#36 Find the present value of the given future amounts

$8500 at 6% compounded annually for 9 years

#44 If money can be invested at 6% compounded annually, which is larger, $10,000 now or $15,000 in 6 years? Use present value to decide

#46 A developer needs $80,000 to buy land. He is able to borrow the money at 10% per year compounded quarterly. How much will the interest amount to if he pays off the loan in 5 years?

#52 Two partners agree to invest equal amounts in their business. One will contribute $10,000 immediately. The other plans to contribute an equivalent amount in 3 years, when she expects to acquire a large sum of money. How much should she contribute at that time to match her partner's investment now, assuming an interest rate of 6% compounded semiannually?

#66 USE THE APPROACH BELOW TO FIND THE TIME IT WOULD TAKE FOR THE GENERAL LEVEL OF PRICES IN THE ECONOMY TO DOUBLE AT THE AVERAGE ANNUAL INFLATION RATES

THE QUESTION IS 4%

Suppose that the inflation rate is 3.5% (which means that the overall level of prices is rising 3.5% a year). How many years will it take for the overall level of prices to double?

We want to find the number of years it will take for $1 worth of goods or services to cost $2. Think of $1 as the present value and $2 as the future value, with an interest rate of 3.5% compounded annually.

Section 5.3

#4 Find the future value of the ordinary annuities with the given payments and interest rates

R = $20,000, 4.5% interest compounded annually for 12 years

#8 Find the future value of the ordinary annuities with the given payments and interest rates

R = $20,000, 6% interest compounded quarterly for 12 years

#10 Find the final amount rounded to the nearest dollar in the following retirement accounts, in which the rate of return on the account and the regular contribution change over time.

$500 per month invested at 5%, compounded monthly, for 20 years; then $1000 per month invested at 8%, compounded monthly for 20 years.

#14 Find the amount of each payment to be made into a sinking fund to accumulate the given amounts. Payments are made at the end of each period

$65,000; money earns 6% compounded semiannually for 4 ½ years

#20 Find the interest rate needed for the sinking fund to reach the required amount. Assume that the compounding period is the same as the payment period

$100,000 to be accumulated in 15 years; quarterly payments of $1200

#26 Find the future value of annuity due

Payments of $1050 for 8 years at 3.5% compounded annually

#28 Find the future value of each annuity due

Payments of $25,000 for 12 years at 6% compounded annually

#34 Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.

$12,000 annual payments for 6 years; interest rate 5.1%

#42 Hassi is paid on the first day of the month, and $80 is automatically deducted from his pay and deposited in a savings account. If the account pays 7.5% interest compounded monthly, how much will be in the account after 3 years and 9 months?

#44 Jasspreet Kaur deposits $2435 at the beginning of each semiannual period for 8 years in an account paying 6% compounded semiannually. She then leaves that money alone, with no further deposits, for an additional 5 years. Find the final amount on deposit after the entire 13 year period.

Section 5.4

#2 Find the present value of each ordinary annuity

Payments of $890 each year for 16 years at 6% compounded annually

#8 Find the amount necessary to fund the given withdrawals

Yearly withdrawals of $1200 for 14 years; interest rate is 5.6% compounded annually

#12 Find the payment made by the ordinary annuity with the given present values

$45,000 monthly payments for 11 years; interest rate is 5.3% compounded monthly

#16 Find the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years at each of the given interest rates

4% compounded annually

#20 Find the price a purchaser should be willing to pay for the given bond. Assume that the coupon interest is paid twice a year.

$20,000 bond with coupon rate 4.5% that matures in 8 years; current interest rate is 5.9%

#28 Find the payment necessary to amortize each of the given loans

$140,000; 12% compounded quarterly; 15 quarterly payments

#34 Find the monthly house payment necessary to amortize the given loan

$96,511 at 8.57% for 25 years

#40 USE THE FOLLOWING TABLE TO SOLVE THIS PROBLEM

PAYMENT AMT OF PAYMENT INTEREST FOR PERIOD PORTION TO PRINCIPAL PRINCIPAL END PERIOD

0 ---- --------- ---------- $1000.00

1 $88.85 $10.00 $78.85 921.15

2 88.85 9.21 79.64 841.51

3 88.85 8.42 80.43 761.08

4 88.85 7.61 81.24 679.84

5 88.85 6.80 82.05 597.79

6 88.85 5.98 82.87 514.92

7 88.85 5.15 83.70 431.22

8 88.85 4.31 84.54 346.68

9 88.85 3.47 85.38 261.30

10 88.85 2.61 86.24 175.06

11 88.85 1.75 87.10 87.96

12 88.84 .88 87.96 0

Using the above table how much of the 10th payment is used to reduce the debt?

#44 Find the cash value of the lottery jackpot to the nearest dollar. Yearly jackpot payments begin immediately (26 for mega millions and 30 for powerball). Assume the lottery can invest at the given interest rate.

Powerball; $207 million; 5.78% interest

#50 A student education loan has two repayment options. The standard plan repays the loan in 10 years with equal monthly payments. The extended plan allows from 12 to 30 years to repay the loan. A student borrows $35,000 at 7.43% compounded monthly.

Find the monthly payment and total interest paid under the standard plan

https://brainmass.com/business/business-math/interest-rate-compound-amount-and-loans-541042

#### Solution Preview

Please see the attached file for detailed solutions.

Section 5.1

#2 Find the interest on each of these loans

$35,000 at 6% for 9 months

This problem doesn't say that we're working with simple interest (as opposed to compound interest), but since the time span is only 9 months it makes sense that this is a simple interest-type of problem.

The formula to calculate the amount of simple interest earned by an investment or applied to a loan is this: , where represents the amount of interest earned, represents the amount of the principal (if it's an investment) or the total amount of the loan (if it's a loan), represents the interest rate being used (and must be expressed as a decimal to use it properly in the formula), and represents the amount of time that is being considered (and t MUST BE EXPRESSED IN YEARS before you plug it into the formula). So, in this problem the principal (which is the value of to plug into the formula is $35,000, the stated interest rate that's being used (which is the value of ) is 6% which is equivalent to 0.06 in decimal form and the amount of time involved (which is the value of to plug into the formula) is 9 months, which is equivalent to of a year. Plugging all of this into the formula for simple interest gives us our answer: . The total amount of interest in this problem is $1575.

#4 Find the interest on each of these loans

$1875 at 5.3% for 7 months

Using the same formula as in question #2 above, , but with , and we get , which should be rounded off to , since we're talking about money.

#8 Find the interest on each of these loans

$8940 at 9%; loan made on May 7 and due September 19

This problem is like questions #2 and #4 above in that we will use the same formula, , to calculate the amount of interest, but it's different in that we're NOT directly given the value of , the amount of time. We do know that and , and we now that the loan was taken out on May 7th and is due on September 19th, so we may calculate the value of as follows: there are 31 days in May and since this loan was taken out on the 7th of May it accumulates interest for the last

31 - 7 = 24 days of May, ALL 30 days of June, ALL 31 days of July, ALL 31 days of August and the first 19 days of September. That's a total of 24+30+31+31+19 = 135 days. Now, since this loan is active and generating interest for 135 days, and the value of needs to be expressed in years, 135 days is years, IF you're working with EXACT INTEREST (exact interest assumes that there are 365 days in a year, or 366 days in a leap year). Sometimes banks use "Banker's Interest", also known as "ordinary interest", which assumes that there are 360 days in the year (in the old days it was easier to work with 360 in the denominator than 365 AND, no surprise here, using banker's interest actually earns the banks MORE INTEREST!). If we use banker's interest, then years.

So we get two answers for the amount of interest, depending upon whether we are supposed to use Exact interest or Banker's interest:

With Exact interest we get: rounded off to the closest penny.

Using Banker's interest we get: rounded off to the closest penny. As you can now see, using Banker's interest earned the bank a little more than $4 extra in interest.

#16 Find the future value of each of these loans

$3475 loan at 7.5% for 6 months

The difference between this problem and the 3 previous problems is that in the 3 previous problems we were only asked to calculate the amount of interest applied to a loan after a certain amount of time passed. The future value of a loan, , is the TOTAL balance on the loan (that is, the total amount of money that you must repay) at a particular time. 3 things determine what you still owe on a loan: (1) the original amount of money that you borrowed, which is just , the principal of the loan; (2) the amount of interest added to the loan amount, which we may calculate from the formula, , like we did before; and, (3) whether or not you've made any payments along the way during the life of the loan (as you would if you had an installment loan, for example). This problem doesn't mention any installment type payments so in this problem we may ignore the effect of any payments along the way. The future value of this loan will just be the original principal plus any interest generated by the loan. As a formula this is what we're looking at: . We are told that , and years. Plugging this all into the above equation for gives us our answer:

rounded off to the nearest penny.

#18 Find the future value of each of these loans

$24,500 loan at 9.6% for 10 months

This problem is just like question #16 above, but with different values for , and . In the problem , and years. Plugging these values into the formula given in question #16 for future value gives us our answer:

.

#22 Find the present value of each future amount

$48,000 for 8 months; money earns 5%

Okay ... so for example, if you had invested $100 into a savings account on January 1, 2013 that paid you 10% interest for the year, then the $100 that you opened the account with would earn $10 of interest for that year ( ), so your savings account would be worth $110 on December 31, 2013. If you changed your mnd right after you opened that account and decided to withdraw all of your money and close the account, then you'd only get back the $100 that you had opened the account with because not enough time had passed from the time that you opened the account to the time 5 minutes later that you decided to close the account. Your $100 was still just $100. This is the present value of this account. If you let the money stay in the account for the full year we have already seen that the account would grow over the year to be worth $110 ... on January 1, 2013 (when you open the account) the $110 represents the future value of your $100, because that is what it will become if you leave it in the account. Clearly if you let the $100 stay in the account for more than one year it will continue to earn interest and so the future value of your $100 will continue to increase the longer you leave the account alone. So imagine the "future" you, on December 31, 2013 when you go to the bank and discover that you have $110 in that savings account ... if for some reason you had amnesia and could not remember how much money you had deposited in the account on January 1, 2013 you could work the equation "backwards" to figure out what P (which now represents the present value) must have been. Alternatively you could solve for P. This what we will do. Solving for P (the present value) gives us the equation: . In this problem the $48,000 is the future value (FV in our equation), the interest rate (r) is 5% (or 0.05 in decimal form) and the time (t) is of a year. Plugging these values into the formula for P gives us our answer: the Present Value in this problem is, rounded off to the closest penny. So, in order to have an account that will grow to being worth $48,000 after only 5 months earning 5% interest you must have opened the account with an initial deposit (the present value) of $47,020.41.

#26 The given Treasury bills were sold in August 2008 find:

a) The price of the T bill and

b) The actual interest rate paid by the Treasury

Six month $18,000 T-bill with discount rate 1.925%

a) The discount rate is the difference between what a Treasury bill is worth and what you actually paid for it. Since th discount rate in this problem is 1.925% and the value of the T-bill is $18,000, you must have paid $18,000 - (0.01925)($18,000) = $17,653.50.

b) Since you paid $17,653.50 for this T-bill and you will be getting back $18,000 in only 6 months the actual (annual) interest rate that you're earning is or about 3.926%.

#30 An accountant for a corporation forgot to pay the firm's income tax of $725,896.15 on time. The government charged a penalty of 9.8% interest for the 34 days the money was late. Find the total amount (tax and penalty) that was paid

Okay, so there's $725,896.15 on the table that the government believes is owed to it AND is past due. Anyone (or any corporation) who does not pay their taxes on time will be charged a penalty based on how many days they have missed the deadline ... the government basically looks at this like a loan. You have "their" money for a certain period of time, so you owe them interest on it. This is really just a straight application of the simple interest types of problems that we started out with. So, the answer will come from the formula that we used in question #16 and #18 above: rounded off to the nearest penny.

#36 What is the time period of a $10,000 loan at 6.75% in which the total amount of interest paid was $618.75?

We know that the amount of money that you must pay back on a loan is the original amount that you borrowed (the principal, or present value) plus any interest. We're told that the loan was for $10,000 and that the total interest was $618.75. Therefore the total amount that must be repaid (also known as the future value, or maturity amount) is: $10,000 + $618.75 = $10,618.75. Now since this is still a simple interest-type of problem, the formula: still applies. But this formula could be solved for to get: . Plugging the values of FV (= $10,618.75), P (= $10,000) ad r (= 0.0675) into this equation we can calculate t: years. Remembering that the unit on "t" is ALWAYS years in this equation, we can convert 0.9166667 ...

#### Solution Summary

The interest rates, compound amounts and loans are examined.

Interest, Annuity, Installment Loan, and Mortgage

1. Define simple interest and compound interest. Explain the difference between each.

2. Define annual percentage rate (APR). What effect does frequency of compounding have on the annual percentage yield (APY)? What condition would exist if the APR = APY?

3. What is an ordinary annuity? What is an annuity due? Which is more prevalent in apartment leases?

4. A very smart child understands annuities and wishes to invest part of his allowance each week to save up for college. The child's parents figure, they would kick in with matching funds, so each month, he deposits a total of $50. Let us suppose that the child continues from age 5 to age 20. Given a fixed APR of 5%, what might the child expect to have at age 20?

5. What is an example of an installment loan? What type of installment loan does not require the borrower to pay off the balance?

6. Explain the difference between a fixed rate and adjustable rate mortgage.

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