3. Interest
fee a borrower pays for
the temporary use of
the lender’s money
it is the “rent” paid
for borrowing money
4. Sam loans Danielle $500 for 100 days. Danielle
agrees to pay her $80 interest for the loan. How much
did Danielle pay Sam in total?
5. Sam loans Danielle $500 for 100 days. Danielle
agrees to pay her $80 interest for the loan. How much
did Danielle pay Sam in total?
Interest is added to the amount borrowed.
6. Sam loans Danielle $500 for 100 days. Danielle
agrees to pay her $80 interest for the loan. How much
did Danielle pay Sam in total?
Interest is added to the amount borrowed.
$500 + $80 = $580
7. Sam loans Danielle $500 for 100 days. Danielle
agrees to pay her $80 interest for the loan. How much
did Danielle pay Sam in total?
Interest is added to the amount borrowed.
$500 + $80 = $580
Danielle will pay Sam $580 at the end of 100 days.
8. Larry borrows $200 from Tom. Larry agrees to
repay the loan by giving Tom $250 in 1 year. How
much interest will Larry pay?
9. Larry borrows $200 from Tom. Larry agrees to
repay the loan by giving Tom $250 in 1 year. How
much interest will Larry pay?
The interest is the difference between what Larry borrows and
what Larry repays.
10. Larry borrows $200 from Tom. Larry agrees to
repay the loan by giving Tom $250 in 1 year. How
much interest will Larry pay?
The interest is the difference between what Larry borrows and
what Larry repays.
$250 - $200 = $50
11. Larry borrows $200 from Tom. Larry agrees to
repay the loan by giving Tom $250 in 1 year. How
much interest will Larry pay?
The interest is the difference between what Larry borrows and
what Larry repays.
$250 - $200 = $50
Larry will pay $50 in Interest.
13. Principal
The principal of a loan is the amount borrowed.
Danielle’s principal was $500.
Larry’s principal was $200.
14. Debtor Creditor
Debtor paying the debt
Creditor receiving the
debt payment
The amount of time for the loan is called the TERM.
15. Debtor Creditor
is the person who owes is the person to whom
someone money. the money is owed.
Danielle and Larry Sam and Tom
Debtor paying the debt
Creditor receiving the
debt payment
The amount of time for the loan is called the TERM.
17. Interest Rates
Interest Rates are expressed as percents, example 5%
Take the case of Tom and Larry. What interest
rate did Larry pay Tom?
Principal $200 and Interest was $50, so
$50/$200 equals 25%
Interest Rate for Larry was 25%.
20. Converting Percents to Decimal
Convert 25% to a decimal.
By dividing: 25% is the same as 25/100 which equals 0.25
21. Converting Percents to Decimal
Convert 25% to a decimal.
By dividing: 25% is the same as 25/100 which equals 0.25
By moving the decimal:.25.% = 0.25
22. Converting Percents to Decimal
Convert 25% to a decimal.
By dividing: 25% is the same as 25/100 which equals 0.25
By moving the decimal:.25.% = 0.25
Convert 18.25% to a decimal. Moving the decimal two places to the
left we see that 18.25% = 0.1825.
23. Converting Percents to Decimal
Convert 25% to a decimal.
By dividing: 25% is the same as 25/100 which equals 0.25
By moving the decimal:.25.% = 0.25
Convert 18.25% to a decimal. Moving the decimal two places to the
left we see that 18.25% = 0.1825.
Convert 5.79% to a decimal.Here, there aren’t two numbers to the
left of the decimal. Simply moving the decimal point two places to the
left would leave us with “0._579”. The blank space is obviously a
problem. We deal with it by placing a 0 in that position to “hold the
space.” So 5.79% = 0.0579.
26. Impact of Time
If the loan lasts longer than one year should the
debtor pay more for the increased time?
If you are the creditor you would want to receive more
compensation for the extra time.
27. Impact ot Time
Suppose that Ray loans Dianne $4,200 at a simple
interest rate of 81⁄2% for 3 years. How much interest
will Dianne pay?
28. Impact ot Time
Suppose that Ray loans Dianne $4,200 at a simple
interest rate of 81⁄2% for 3 years. How much interest
will Dianne pay?
We need to multiply the amount borrowed times the interest
rate, and then multiply by 3.
29. Impact ot Time
Suppose that Ray loans Dianne $4,200 at a simple
interest rate of 81⁄2% for 3 years. How much interest
will Dianne pay?
We need to multiply the amount borrowed times the interest
rate, and then multiply by 3.
Interest = (Amount Borrowed)(Interest Rate)(Time)
30. Impact ot Time
Suppose that Ray loans Dianne $4,200 at a simple
interest rate of 81⁄2% for 3 years. How much interest
will Dianne pay?
We need to multiply the amount borrowed times the interest
rate, and then multiply by 3.
Interest = (Amount Borrowed)(Interest Rate)(Time)
Interest = ($4,200)(0.085)(3)
31. Impact ot Time
Suppose that Ray loans Dianne $4,200 at a simple
interest rate of 81⁄2% for 3 years. How much interest
will Dianne pay?
We need to multiply the amount borrowed times the interest
rate, and then multiply by 3.
Interest = (Amount Borrowed)(Interest Rate)(Time)
Interest = ($4,200)(0.085)(3)
Interest = $1,071.00
34. Simple Interest
Formula
Interest = Principal * Rate(decimal) * Time(TERM of the loan)
I = PRT
35. Solve this problem
Heather borrows $18,500 at 57⁄8% simple interest for
2 years. How much interest will she pay?
36. Solve this problem
Heather borrows $18,500 at 57⁄8% simple interest for
2 years. How much interest will she pay?
The principal P = $18,500, the interest rate R = 0.05875,
and the time T = 2 years. So we begin with our formula:
37. Solve this problem
Heather borrows $18,500 at 57⁄8% simple interest for
2 years. How much interest will she pay?
The principal P = $18,500, the interest rate R = 0.05875,
and the time T = 2 years. So we begin with our formula:
I = PRT Replace each letter whose value we know:
38. Solve this problem
Heather borrows $18,500 at 57⁄8% simple interest for
2 years. How much interest will she pay?
The principal P = $18,500, the interest rate R = 0.05875,
and the time T = 2 years. So we begin with our formula:
I = PRT Replace each letter whose value we know:
I = ($18,500)(0.05875)(2)
39. Solve this problem
Heather borrows $18,500 at 57⁄8% simple interest for
2 years. How much interest will she pay?
The principal P = $18,500, the interest rate R = 0.05875,
and the time T = 2 years. So we begin with our formula:
I = PRT Replace each letter whose value we know:
I = ($18,500)(0.05875)(2)
I = $2,173.75
42. Other loans
Depositing money to a bank account means you are
actually loaning that money to the bank.
While it is in your account the bank has the use of it, and in fact does
use it (to make loans to other people.)
43. Other loans
Depositing money to a bank account means you are
actually loaning that money to the bank.
While it is in your account the bank has the use of it, and in fact does
use it (to make loans to other people.)
Different types of bank deposits:
44. Other loans
Depositing money to a bank account means you are
actually loaning that money to the bank.
While it is in your account the bank has the use of it, and in fact does
use it (to make loans to other people.)
Different types of bank deposits:
Demand accounts: checking and savings accounts
45. Other loans
Depositing money to a bank account means you are
actually loaning that money to the bank.
While it is in your account the bank has the use of it, and in fact does
use it (to make loans to other people.)
Different types of bank deposits:
Demand accounts: checking and savings accounts
Certificate of Deposit or CD stay with the bank for a
46. 2 year CD
Jake deposited $2,318.29 into a 2-year CD paying
5.17% simple interest per annum. How much will his
account be worth at the end of the term? Recall that
the phrase per annum simply means per year.
47. 2 year CD
Jake deposited $2,318.29 into a 2-year CD paying
5.17% simple interest per annum. How much will his
account be worth at the end of the term? Recall that
the phrase per annum simply means per year.
I = PRT I = ($2,318.29)(0.0517)(2)
48. 2 year CD
Jake deposited $2,318.29 into a 2-year CD paying
5.17% simple interest per annum. How much will his
account be worth at the end of the term? Recall that
the phrase per annum simply means per year.
I = PRT I = ($2,318.29)(0.0517)(2)
I = $239.71
49. 2 year CD
Jake deposited $2,318.29 into a 2-year CD paying
5.17% simple interest per annum. How much will his
account be worth at the end of the term? Recall that
the phrase per annum simply means per year.
I = PRT I = ($2,318.29)(0.0517)(2)
I = $239.71
The total is principal and interest
50. 2 year CD
Jake deposited $2,318.29 into a 2-year CD paying
5.17% simple interest per annum. How much will his
account be worth at the end of the term? Recall that
the phrase per annum simply means per year.
I = PRT I = ($2,318.29)(0.0517)(2)
I = $239.71
The total is principal and interest
$2,318.29 + $239.71 = $2,558.00.
55. 6 month problem
If Sarai borrows $5,000 for 6 months at 9% simple
interest, how much will she need to pay back?
I =PRT
I =($5,000)(0.09)(6/12)
I = ($5,000)(.09)(0.5)
I = $225
Sarai will pay back $5,000 + $225 = $5,225
56. Interest earned in 7 months
Zachary deposited $3,412.59 in a bank account
paying 51⁄4% simple interest for 7 months. How much
interest did he earn?
57. Interest earned in 7 months
Zachary deposited $3,412.59 in a bank account
paying 51⁄4% simple interest for 7 months. How much
interest did he earn?
I = PRT
58. Interest earned in 7 months
Zachary deposited $3,412.59 in a bank account
paying 51⁄4% simple interest for 7 months. How much
interest did he earn?
I = PRT
I =($3,412.59)(0.0525)(0.583333333)
59. Interest earned in 7 months
Zachary deposited $3,412.59 in a bank account
paying 51⁄4% simple interest for 7 months. How much
interest did he earn?
I = PRT
I =($3,412.59)(0.0525)(0.583333333)
I = $104.51
60. Interest earned in 7 months
Zachary deposited $3,412.59 in a bank account
paying 51⁄4% simple interest for 7 months. How much
interest did he earn?
I = PRT
I =($3,412.59)(0.0525)(0.583333333)
I = $104.51
Zachary earned $104.51 in interest.
61. Loans with Terms in Days—
The Exact Method
365 days: 366 days: 365.25 days:
62. Earnings from 90 day CD
Nick deposited $1,600 in a credit union CD with a
term of 90 days and a simple interest rate of 4.72%.
Find the value of his account at the end of its term.
63. Earnings from 90 day CD
Nick deposited $1,600 in a credit union CD with a
term of 90 days and a simple interest rate of 4.72%.
Find the value of his account at the end of its term.
I =PRT We divided by 365 instead of 12, since 90 days is 90⁄365 of a year
I =($1,600)(0.0472)(90/365)
I = $18.62
ending account value will be $1,600 + $18.62 = $1,618.62
64. Compare days
Calculate the simple interest due on a 120-day loan of
$1,000 at 8.6% simple interest in three different ways:
assuming there are 365, 366, or 365.25 days in the
year.
65. Compare days
Calculate the simple interest due on a 120-day loan of
$1,000 at 8.6% simple interest in three different ways:
assuming there are 365, 366, or 365.25 days in the
year.
I =PRT = ($1,000)(0.086)(120/365) = $28.27
66. Compare days
Calculate the simple interest due on a 120-day loan of
$1,000 at 8.6% simple interest in three different ways:
assuming there are 365, 366, or 365.25 days in the
year.
I =PRT = ($1,000)(0.086)(120/365) = $28.27
I = PRT = ($1,000)(0.086)(120/366) = $28.20
67. Compare days
Calculate the simple interest due on a 120-day loan of
$1,000 at 8.6% simple interest in three different ways:
assuming there are 365, 366, or 365.25 days in the
year.
I =PRT = ($1,000)(0.086)(120/365) = $28.27
I = PRT = ($1,000)(0.086)(120/366) = $28.20
I = PRT = ($1,000)(0.086)(120/365.25) = $28.26
68. 150 day note
Calculate the simple interest due on a 150-day loan of
$120,000 at 9.45% simple interest.
69. 150 day note
Calculate the simple interest due on a 150-day loan of
$120,000 at 9.45% simple interest.
Following the rules stated, we assume that interest should be
calculated using 365 days in the year.
70. 150 day note
Calculate the simple interest due on a 150-day loan of
$120,000 at 9.45% simple interest.
Following the rules stated, we assume that interest should be
calculated using 365 days in the year.
I = PRT = ($120,000)(0.0945)(150/365)= $4,660.27
72. Banker’s Rules
There is another commonly used approach to
calculating interest that, while not as true to the actual
calendar, can be much simpler. Under bankers’ rule
we assume that the year consists of 12 months having
30 days each, for a total of 360 days in the year.
73. Calculate Simple Interest
Calculate the simple interest due on a 120-day loan of
$10,000 at 8.6% simple interest using bankers rule.
360 days:
74. Calculate Simple Interest
Calculate the simple interest due on a 120-day loan of
$10,000 at 8.6% simple interest using bankers rule.
360 days:
I = PRT = ($1,000)(0.086)(120/360) = $28.67
76. Weekly Terms
Bridget borrows $2,000 for 13 weeks at 6% simple
interest. Find the total interest she will pay.
The only difference between this problem and the others is that, since the
term is in weeks, we divide by 52 (since there are 52 weeks per year).
77. Weekly Terms
Bridget borrows $2,000 for 13 weeks at 6% simple
interest. Find the total interest she will pay.
The only difference between this problem and the others is that, since the
term is in weeks, we divide by 52 (since there are 52 weeks per year).
I = PRT = ($2000)(0.06)(13/52) = $30
78. Weekly Terms
Bridget borrows $2,000 for 13 weeks at 6% simple
interest. Find the total interest she will pay.
The only difference between this problem and the others is that, since the
term is in weeks, we divide by 52 (since there are 52 weeks per year).
I = PRT = ($2000)(0.06)(13/52) = $30
So Bridget’s interest will total $30.
80. Determining Principal, Interest
Rates, and Time
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
81. Determining Principal, Interest
Rates, and Time
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Jim borrowed $500 from his brother-in-law, and agreed to pay back
$525 ninety days later. What rate of simple interest is Jim paying for
this loan, assuming that they agreed to calculate the interest with
bankers’ rule?
82. Determining Principal, Interest
Rates, and Time
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Jim borrowed $500 from his brother-in-law, and agreed to pay back
$525 ninety days later. What rate of simple interest is Jim paying for
this loan, assuming that they agreed to calculate the interest with
bankers’ rule?
Maria deposited $9,750 in a savings account that pays 5 1⁄4% simple
interest. How long will it take for her account to grow to $10,000?
83. Finding Principal
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
84. Finding Principal
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Plugging these values into the formula, we get:I = PRT
85. Finding Principal
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Plugging these values into the formula, we get:I = PRT
$1,000 = (P)(0.048)(1/12)
86. Finding Principal
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Plugging these values into the formula, we get:I = PRT
$1,000 = (P)(0.048)(1/12)
$1,000 = (P)(0.004)
87. Finding Principal
A retiree hopes to be able to generate $1,000 income per month from an
investment account that earns 4.8% simple interest. How much money
would he need in the account to achieve this goal?
Plugging these values into the formula, we get:I = PRT
$1,000 = (P)(0.048)(1/12)
$1,000 = (P)(0.004)
Now what? We need to get rid of the 0.004.
89. The Balance Principle
Equals
$1,000 (P)(0.004)
You can make any change you like to one side of an equation, as long as you make an
equivalent change to the other side.
$1000 = (P)(0.004)
0.004 0.004
$250,000 = P
Which is the same as
P = $250,000
90. Checking
A retiree hopes to be able to generate $1,000 income per
month from an investment account that earns 4.8% simple
interest. How much money would he need in the account to
achieve this goal?
91. Checking
A retiree hopes to be able to generate $1,000 income per
month from an investment account that earns 4.8% simple
interest. How much money would he need in the account to
achieve this goal?
Plug the P of $250,000 back into the formula and solve for I
I = ($250,000)(0.048)(1/12)
I = $1000 Compare this answer with the original
question, do they match?
92. Find Simple Interest Rate
Calculate the simple interest rate for a loan of $9,764.55 if the term is 125 days and the total
required to repay the loan is $10,000.
93. Find Simple Interest Rate
Calculate the simple interest rate for a loan of $9,764.55 if the term is 125 days and the total
required to repay the loan is $10,000.
First we need to find the interest by subtracting.
$10,000 = $9,764.55 = $235.45. Assume exact method 365 days.
Plugging values into the formula gives: I = PRT
$235.45 = ($9,764.55) R (125/365)
$235.45 = ($3,344.02397260) R Now, solve for R
To solve for R, we divide both sides by 3344.02397260
R = 0.0704091842
Moving the decimal two places to the right, we can state the rate as 7.04091842%.
We conclude that the interest rate is 7.04%.
95. Determining Rates
Complete this question:
Jim borrowed $500 from his brother-in-law, and agreed to pay back
$525 ninety days later. What rate of simple interest is Jim paying for
this loan, assuming that they agreed to calculate the interest with
bankers’ rule?
96. Answer Rate
Jim borrowed $500 from his brother-in-law, and agreed to pay
back $525 ninety days later. What rate of simple interest is Jim
paying for this loan, assuming that they agreed to calculate the
interest with bankers’ rule? (30 days per month, 360 days per year)
I = P R T $25 = $500 R (90/360) $25 = $500 x 90 R $25 =$125 R
360 $125 $12
0.2 = R move the decimal two places to the right and add %
20% = R or R = 20%
Check: I =$500 x 20% x 90/360 I = $25
97. Finding Time
Maria deposited $9,750 in a savings account that pays 51⁄4% simple
interest and wanted to know how long it would take for her account to
grow to $10,000. Calculate the growth. $10,000 - $9750 = $250. We
proceed as before: I = PRT
$250 = ($9,750)(0.0525)(T)
$250 = $511.875(T)
$250 $511.875(T)
____ = ______
$511.875 $511.875
T = 0.4884004884
T = 12 (0.4884004884) = 5.860805861 months
98. Problem with Rounding to 6
I = PRT
I = ($9,750)(.0525)(6/12)
I = $255.94
not the $250 Interest of the original question.
99. 365 days instead of months
Using the same logic for days as we did for months,
we multiply by 365 to get
T = 365(0.4884004884) = 178.2661783 days
which we would round to 178 days, following usual
rounding rules.
I = PRT
I = ($9750)(.0525)(178/365)
I = $249.63
100. Find Time
Suppose that you deposit $3,850 in an account
paying 4.65% simple interest. How long will it take to
earn $150 in interest?I = PRT
$150 = ($3,850)(0.0465)T
$150 = ($179.025)T
T = 0.837871806
This answer is in years. To convert it to more user-friendly terms, we
can multiply by 365 to get the answer in days:
T = (0.837871806)(365) = 306 days
101. More Examples
What amount of money must be invested in a 75-day certificate of deposit
paying 5.2% simple interest, using bankers’ rule, in order to earn $40?
I= PRT
$40 = (P)(0.052)(75/360)
$40 = (P)(0.010833333)
$40
___ = _________ (P)(0.010833333)
(0.010833333) (0.010833333) P = $3,692.31
102. Interest Rate
If Shay borrows $20,000 for 9 months and pays interest totaling
$1,129.56, find the rate of simple interest for this loan. I = PRT
$1,129.56 = ($20,000)(R)(9/12)
Text
$1,129.56 = $15,000(R)
____ $15,000(R)
$1,129.56 _____ R = 0.075304
$15,000 = $15,000
Moving the decimal two places to the right, we find that the rate of
simple interest is 7.5304%. Rounding to two decimal places gives that the
rate is 7.53%.
Read this example Sam loans Danielle $500 for 100 days. Danielle agrees to pay her $80 interest for the loan. How much did Danielle pay Sam in total?\n In other cases, the borrower and lender may agree on the amount borrowed and the amount to be repaid without explicitly stating the amount of interest. In those cases, we can determine the amount of interest by finding the difference between the two amounts (by subtracting.)\n
Read this example Sam loans Danielle $500 for 100 days. Danielle agrees to pay her $80 interest for the loan. How much did Danielle pay Sam in total?\n In other cases, the borrower and lender may agree on the amount borrowed and the amount to be repaid without explicitly stating the amount of interest. In those cases, we can determine the amount of interest by finding the difference between the two amounts (by subtracting.)\n
Read this example Sam loans Danielle $500 for 100 days. Danielle agrees to pay her $80 interest for the loan. How much did Danielle pay Sam in total?\n In other cases, the borrower and lender may agree on the amount borrowed and the amount to be repaid without explicitly stating the amount of interest. In those cases, we can determine the amount of interest by finding the difference between the two amounts (by subtracting.)\n
Read this example Sam loans Danielle $500 for 100 days. Danielle agrees to pay her $80 interest for the loan. How much did Danielle pay Sam in total?\n In other cases, the borrower and lender may agree on the amount borrowed and the amount to be repaid without explicitly stating the amount of interest. In those cases, we can determine the amount of interest by finding the difference between the two amounts (by subtracting.)\n
\n
\n
\n
\n
It is awkward to have to keep saying “the amount borrowed” over and over again, and so we give this amount a specific name, principal. Principal is the amount of money being borrowed.\n
A debtor is someone who owes someone else money. A creditor is someone to whom money is owed.\nIn Example 1.1.1 Sam is Danielle’s creditor and Danielle is Sam’s debtor. In Example 1.1.2 we would say that Tom is Larry’s creditor and Larry is Tom’s debtor.\nThe amount of time for which a loan is made is called its term. In Example 1.1.1 the term is 100 days. In Example 1.1.2 the term of the loan is 1 year.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
When we talk about percents, we usually are taking a percent of something. The mathematical operation that translates the “of” in that expression is multiplication. So, to find 25% of $1,000, we would multiply 25% times $1,000.\nHowever, if I simply multiply 25 times 1,000 on my calculator, I get 25,000, which is far too big and also does not agree with the answer of $250 which we know is correct. The reason for this discrepancy is that 25% is not the same as the number 25. The word percent comes from Latin, and means “out of 100.” So when we say “25%,” what we really mean is “25 out of 100”—or in other words 25/100.\nIf you divide 25/100 on a calculator, the result is 0.25. This process of converting a percent into its real mathematical meaning is often called converting the percent to a decimal.\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
so far we have been able to determine the interest, the principal, and the interest rate. How does time affect the interest amount?\nLet’s return again to Tom and Larry. Suppose that Larry returns to the original plan of borrowing $200, but instead of paying it back in 1 year, he offers to pay it back in 2 years. Should Larry pay twice the interest or some other amount of interest?\n
We have seen that to find interest we need to multiply the amount borrowed times the interest rate, and also that since this loan is for 3 years we then need to multiply that result by 3. Combining these into a single step, we get:\nInterest = (Amount Borrowed)(Interest Rate as a decimal)(Time) \nInterest = ($4,200)(0.085)(3) \nInterest =$1,071.00\n
We have seen that to find interest we need to multiply the amount borrowed times the interest rate, and also that since this loan is for 3 years we then need to multiply that result by 3. Combining these into a single step, we get:\nInterest = (Amount Borrowed)(Interest Rate as a decimal)(Time) \nInterest = ($4,200)(0.085)(3) \nInterest =$1,071.00\n
We have seen that to find interest we need to multiply the amount borrowed times the interest rate, and also that since this loan is for 3 years we then need to multiply that result by 3. Combining these into a single step, we get:\nInterest = (Amount Borrowed)(Interest Rate as a decimal)(Time) \nInterest = ($4,200)(0.085)(3) \nInterest =$1,071.00\n
We have seen that to find interest we need to multiply the amount borrowed times the interest rate, and also that since this loan is for 3 years we then need to multiply that result by 3. Combining these into a single step, we get:\nInterest = (Amount Borrowed)(Interest Rate as a decimal)(Time) \nInterest = ($4,200)(0.085)(3) \nInterest =$1,071.00\n
We have seen that to find interest we need to multiply the amount borrowed times the interest rate, and also that since this loan is for 3 years we then need to multiply that result by 3. Combining these into a single step, we get:\nInterest = (Amount Borrowed)(Interest Rate as a decimal)(Time) \nInterest = ($4,200)(0.085)(3) \nInterest =$1,071.00\n
unless it is clearly stated otherwise, interest rates are always assumed to be rates per year.\nOccasionally, you may see the Latin phrase per annum used with interest rates, meaning per year to emphasize that the rate is per year. \nI represents the amount of simple INTEREST for a loan P represents the amount of money borrowed (the PRINCIPAL) R represents the interest RATE (expressed as a decimal) and T represents the TERM of the loan\n
unless it is clearly stated otherwise, interest rates are always assumed to be rates per year.\nOccasionally, you may see the Latin phrase per annum used with interest rates, meaning per year to emphasize that the rate is per year. \nI represents the amount of simple INTEREST for a loan P represents the amount of money borrowed (the PRINCIPAL) R represents the interest RATE (expressed as a decimal) and T represents the TERM of the loan\n
\n
\n
\n
\n
\n
When you are depositing money to a bank account you are actually loaning that money to the bank. While it is in your account the bank has the use of it, and in fact does use it (to make loans to other people.)\nThere are many different types of bank deposits. Checking and savings accounts are familiar examples of ways in which we loan money to banks. These are sometimes referred to as demand accounts, because you can withdraw your money any time that you want (i.e., “on demand.”) Another common type of account is a certificate of deposit, or CD. When you deposit money into a CD, you agree to keep it on deposit at the bank for a fixed period of time. For this reason, CDs are often also referred to as term deposits or other similar names. CDs often offer better interest rates than checking or savings accounts, since with a CD the bank knows how long it will have the money, giving it more opportunity to take advantage of longer term loans on which it can collect higher interest rates.\n
When you are depositing money to a bank account you are actually loaning that money to the bank. While it is in your account the bank has the use of it, and in fact does use it (to make loans to other people.)\nThere are many different types of bank deposits. Checking and savings accounts are familiar examples of ways in which we loan money to banks. These are sometimes referred to as demand accounts, because you can withdraw your money any time that you want (i.e., “on demand.”) Another common type of account is a certificate of deposit, or CD. When you deposit money into a CD, you agree to keep it on deposit at the bank for a fixed period of time. For this reason, CDs are often also referred to as term deposits or other similar names. CDs often offer better interest rates than checking or savings accounts, since with a CD the bank knows how long it will have the money, giving it more opportunity to take advantage of longer term loans on which it can collect higher interest rates.\n
When you are depositing money to a bank account you are actually loaning that money to the bank. While it is in your account the bank has the use of it, and in fact does use it (to make loans to other people.)\nThere are many different types of bank deposits. Checking and savings accounts are familiar examples of ways in which we loan money to banks. These are sometimes referred to as demand accounts, because you can withdraw your money any time that you want (i.e., “on demand.”) Another common type of account is a certificate of deposit, or CD. When you deposit money into a CD, you agree to keep it on deposit at the bank for a fixed period of time. For this reason, CDs are often also referred to as term deposits or other similar names. CDs often offer better interest rates than checking or savings accounts, since with a CD the bank knows how long it will have the money, giving it more opportunity to take advantage of longer term loans on which it can collect higher interest rates.\n
When you are depositing money to a bank account you are actually loaning that money to the bank. While it is in your account the bank has the use of it, and in fact does use it (to make loans to other people.)\nThere are many different types of bank deposits. Checking and savings accounts are familiar examples of ways in which we loan money to banks. These are sometimes referred to as demand accounts, because you can withdraw your money any time that you want (i.e., “on demand.”) Another common type of account is a certificate of deposit, or CD. When you deposit money into a CD, you agree to keep it on deposit at the bank for a fixed period of time. For this reason, CDs are often also referred to as term deposits or other similar names. CDs often offer better interest rates than checking or savings accounts, since with a CD the bank knows how long it will have the money, giving it more opportunity to take advantage of longer term loans on which it can collect higher interest rates.\n
When you are depositing money to a bank account you are actually loaning that money to the bank. While it is in your account the bank has the use of it, and in fact does use it (to make loans to other people.)\nThere are many different types of bank deposits. Checking and savings accounts are familiar examples of ways in which we loan money to banks. These are sometimes referred to as demand accounts, because you can withdraw your money any time that you want (i.e., “on demand.”) Another common type of account is a certificate of deposit, or CD. When you deposit money into a CD, you agree to keep it on deposit at the bank for a fixed period of time. For this reason, CDs are often also referred to as term deposits or other similar names. CDs often offer better interest rates than checking or savings accounts, since with a CD the bank knows how long it will have the money, giving it more opportunity to take advantage of longer term loans on which it can collect higher interest rates.\n
At the end of the term, his account will contain both the principal and interest, so the total value of the account will be $2,318.29 +$239.71 = $2,558.00.\n
At the end of the term, his account will contain both the principal and interest, so the total value of the account will be $2,318.29 +$239.71 = $2,558.00.\n
At the end of the term, his account will contain both the principal and interest, so the total value of the account will be $2,318.29 +$239.71 = $2,558.00.\n
At the end of the term, his account will contain both the principal and interest, so the total value of the account will be $2,318.29 +$239.71 = $2,558.00.\n
At the end of the term, his account will contain both the principal and interest, so the total value of the account will be $2,318.29 +$239.71 = $2,558.00.\n
\n
So far, we’ve considered only loans whose terms are measured in whole years. While such terms are not uncommon, they are certainly not mandatory. A loan can extend for any period of time at all. When the interest rate is per year, and the term is also in years, we hardly even need to think about the units of time at all. When dealing with loans whose terms are not whole years, though, we have to take a bit more care with the units of time.\n
As before, we can calculate her interest by using I =PRT, plugging in P = $5,000 and R =0.09. T is a bit more complicated. We must be consistent with our units of time.\nThe term is 6 months, but we can’t just plug in T = 6, since the 9% interest rate is assumed to be a rate per year. Since the interest rate is per year, when we use it we must measure time in years. Plugging in T = 6 would mean 6 years, not 6 months.\nT should give the term of the loan in years. Since a year contains 12 months, 6 months is equal to 6⁄12 of a year, and so we plug in T = 6/12. (Another way of looking at this would be to say that since 6 months is half of a year, T = 1/2. Either way, we get the same result since 6/12 =1/2 = 0.5.)\n
We can use the simple interest formula once again, plugging in P = $3,412.59 and R = 0.0525. Since the term is expressed in months, we divide 7/12 to get T = 0.583333333.This example raises an issue. Since 7/12 does not come out evenly, can it be rounded? In general, the answer is no.\n In this text, rather than getting bogged down in determining how much rounding is too much, we will follow the general rule that up until the final answer numbers should be carried out to the full number of decimal places given by your calculator. In the example above, the value was shown out to nine decimal places. Your calculator may have more or fewer, but this will not be a problem. As long as you use the full precision of your calculator, any differences will be small enough to be lost in the final rounding.\nHowever, the dollar amount is rounded to significant figures. Round up if 5 or above.\n
We can use the simple interest formula once again, plugging in P = $3,412.59 and R = 0.0525. Since the term is expressed in months, we divide 7/12 to get T = 0.583333333.This example raises an issue. Since 7/12 does not come out evenly, can it be rounded? In general, the answer is no.\n In this text, rather than getting bogged down in determining how much rounding is too much, we will follow the general rule that up until the final answer numbers should be carried out to the full number of decimal places given by your calculator. In the example above, the value was shown out to nine decimal places. Your calculator may have more or fewer, but this will not be a problem. As long as you use the full precision of your calculator, any differences will be small enough to be lost in the final rounding.\nHowever, the dollar amount is rounded to significant figures. Round up if 5 or above.\n
We can use the simple interest formula once again, plugging in P = $3,412.59 and R = 0.0525. Since the term is expressed in months, we divide 7/12 to get T = 0.583333333.This example raises an issue. Since 7/12 does not come out evenly, can it be rounded? In general, the answer is no.\n In this text, rather than getting bogged down in determining how much rounding is too much, we will follow the general rule that up until the final answer numbers should be carried out to the full number of decimal places given by your calculator. In the example above, the value was shown out to nine decimal places. Your calculator may have more or fewer, but this will not be a problem. As long as you use the full precision of your calculator, any differences will be small enough to be lost in the final rounding.\nHowever, the dollar amount is rounded to significant figures. Round up if 5 or above.\n
We can use the simple interest formula once again, plugging in P = $3,412.59 and R = 0.0525. Since the term is expressed in months, we divide 7/12 to get T = 0.583333333.This example raises an issue. Since 7/12 does not come out evenly, can it be rounded? In general, the answer is no.\n In this text, rather than getting bogged down in determining how much rounding is too much, we will follow the general rule that up until the final answer numbers should be carried out to the full number of decimal places given by your calculator. In the example above, the value was shown out to nine decimal places. Your calculator may have more or fewer, but this will not be a problem. As long as you use the full precision of your calculator, any differences will be small enough to be lost in the final rounding.\nHowever, the dollar amount is rounded to significant figures. Round up if 5 or above.\n
We can use the simple interest formula once again, plugging in P = $3,412.59 and R = 0.0525. Since the term is expressed in months, we divide 7/12 to get T = 0.583333333.This example raises an issue. Since 7/12 does not come out evenly, can it be rounded? In general, the answer is no.\n In this text, rather than getting bogged down in determining how much rounding is too much, we will follow the general rule that up until the final answer numbers should be carried out to the full number of decimal places given by your calculator. In the example above, the value was shown out to nine decimal places. Your calculator may have more or fewer, but this will not be a problem. As long as you use the full precision of your calculator, any differences will be small enough to be lost in the final rounding.\nHowever, the dollar amount is rounded to significant figures. Round up if 5 or above.\n
After we have dealt with loans whose terms are measured in months, it’s not surprising that our next step is to consider loans with terms in days. The idea is the same, except that instead of dividing by 12 months, we divide by the number of days in the year. There are 365 days in a year except leap year. Unless told in the problem assume an ordinary year and use 365.\n
\n
\n
\n
\n
\n
Notice the how slight the difference is in the interest, 7 cents.\nUse 365 days unless the problem states differently. Use 365 days if the terms ordinary or simplified exact method, exact interest, are used in the problem.\n \n
Notice the how slight the difference is in the interest, 7 cents.\nUse 365 days unless the problem states differently. Use 365 days if the terms ordinary or simplified exact method, exact interest, are used in the problem.\n \n
Notice the how slight the difference is in the interest, 7 cents.\nUse 365 days unless the problem states differently. Use 365 days if the terms ordinary or simplified exact method, exact interest, are used in the problem.\n \n
Notice the how slight the difference is in the interest, 7 cents.\nUse 365 days unless the problem states differently. Use 365 days if the terms ordinary or simplified exact method, exact interest, are used in the problem.\n \n
\n
\n
\n
There is another commonly used approach to calculating interest that, while not as true to the actual calendar, can be much simpler. Under bankers’ rule we assume that the year consists of 12 months having 30 days each, for a total of 360 days in the year.\nBankers’ rule was adopted before modern calculators and computers were available. Financial calculations had to be done mainly with pencil-and-paper arithmetic. Bankers’ rule offers the desirable advantage that many numbers divide nicely into 360, while very few numbers divide nicely into 365. \n
You might suspect that the differences between bankers’ rule and the exact method leave an opportunity for sneaky banks to manipulate interest calculations to their benefit. After all, what prevents a bank from always choosing whichever rule works to its advantage (and thus to the customer’s disadvantage)? In practice, the method to be used will be specified either in a bank’s general policies, government regulations, or in the paperwork for any deposit or loan, and in any case, as we’ve seen above, the difference is slight. \n
You might suspect that the differences between bankers’ rule and the exact method leave an opportunity for sneaky banks to manipulate interest calculations to their benefit. After all, what prevents a bank from always choosing whichever rule works to its advantage (and thus to the customer’s disadvantage)? In practice, the method to be used will be specified either in a bank’s general policies, government regulations, or in the paperwork for any deposit or loan, and in any case, as we’ve seen above, the difference is slight. \n
\n
\n
\n
\n
\n
So far, we have developed the ability to calculate the amount of interest due when we know the principal, rate and time. However, situations may arise where we already know the amount of interest, and instead need to calculate one of the other quantities. For example, consider these situations:\n
So far, we have developed the ability to calculate the amount of interest due when we know the principal, rate and time. However, situations may arise where we already know the amount of interest, and instead need to calculate one of the other quantities. For example, consider these situations:\n
So far, we have developed the ability to calculate the amount of interest due when we know the principal, rate and time. However, situations may arise where we already know the amount of interest, and instead need to calculate one of the other quantities. For example, consider these situations:\n
\n
\n
\n
\n
\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
When we write an equation, we are making the claim that the things on the left side of the “=” sign have the exact same value as the things on the other side. We can visualize this by thinking of an equation as a balanced scale. The things on the left side of the equal sign are equal to the things on the right. If we imagine that we placed the contents of each side on a scale, it would balance.\nUsing this idea with our present situation, $1,000 = (P)(0.004), we’d have:\n\nwe can make any changes as long as we make them on each side and hold everything equal. I can multiple 2 on both sides to get $2000=(p)(0.008)\n
This is the original value for I, $1000.\n
This is the original value for I, $1000.\n
This is the original value for I, $1000.\n
This is the original value for I, $1000.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
It is not usually necessary to carry the final answer out to this many decimal places, though. There is no absolute rule about how many places to use, but in most situations two or at most three decimal places in the final percent is acceptable. For our purposes in this text, two decimal places will be fine, so we conclude that the interest rate is 7.04%.\n
So far, we have developed the ability to calculate the amount of interest due when we know the principal, rate and time. However, situations may arise where we already know the amount of interest, and instead need to calculate one of the other quantities. For example, consider these situations:\n
So far, we have developed the ability to calculate the amount of interest due when we know the principal, rate and time. However, situations may arise where we already know the amount of interest, and instead need to calculate one of the other quantities. For example, consider these situations:\n
\n
What does this answer mean? It does not make sense.\nT is TIME expressed in years, correct. This decimal number leads one to sense that this time is less than one year (It is less than a whole and if T is years, this must be less than one year).\nWhat measurement of time is less than one year? A month. It takes 12 months to make a year. Multiple 12 with the answer for 5.860805861 months. This answer is also unacceptable, though, since we don’t normally talk about messy deci- mal numbers of months either. We could round this to the nearest month, in which case we’d conclude that the term is six months.\n\n
\n
This is still a rounded answer, and in fact since we threw out 0.2661783 in the rounding,\nit may actually appear as though the rounding is at least as serious an issue as it was with the months.which is still not exactly $250, but it is quite a bit closer than before. The rounding is less of a problem here since 0.2661783 days is considerably less time than 0.14 months.\n37 cents discrepency\n
