math 1300: section 3-1 simple interest
DESCRIPTION
Lecture over Section 3-1 of Barnett's "Finite Mathematics."TRANSCRIPT
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Math 1300 Finite MathematicsSection 3.1 Simple Interest
Jason Aubrey
Department of MathematicsUniversity of Missouri
Jason Aubrey Math 1300 Finite Mathematics
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Definition (Simple Interest)
I = Prt
whereI = interest
P = principalr = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Definition (Simple Interest)
I = Prt
whereI = interestP = principal
r = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Definition (Simple Interest)
I = Prt
whereI = interestP = principalr = annual simple interest rate (written as a decimal)
t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Definition (Simple Interest)
I = Prt
whereI = interestP = principalr = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a $650 accountthat is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
I = 650(0.21)
(3
12
)= 34.13
Therefore $34.13 is owed.
Jason Aubrey Math 1300 Finite Mathematics
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Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a $650 accountthat is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
I = 650(0.21)
(3
12
)= 34.13
Therefore $34.13 is owed.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a $650 accountthat is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
I = 650(0.21)
(3
12
)= 34.13
Therefore $34.13 is owed.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a $650 accountthat is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
I = 650(0.21)
(3
12
)= 34.13
Therefore $34.13 is owed.
Jason Aubrey Math 1300 Finite Mathematics
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Example: A commercial for a loan company states, “You onlypay $0.16 a day for each $600 borrowed.” If you borrow $1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging?
You pay a fee of 3× $0.16 = $0.48 per day.This is a total fee of $0.48× 240 = $115.20 over the life ofthe loan.So, I = $115.20, P = $1800, and t = 240/360.
Jason Aubrey Math 1300 Finite Mathematics
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Example: A commercial for a loan company states, “You onlypay $0.16 a day for each $600 borrowed.” If you borrow $1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging?
You pay a fee of 3× $0.16 = $0.48 per day.
This is a total fee of $0.48× 240 = $115.20 over the life ofthe loan.So, I = $115.20, P = $1800, and t = 240/360.
Jason Aubrey Math 1300 Finite Mathematics
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Example: A commercial for a loan company states, “You onlypay $0.16 a day for each $600 borrowed.” If you borrow $1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging?
You pay a fee of 3× $0.16 = $0.48 per day.This is a total fee of $0.48× 240 = $115.20 over the life ofthe loan.
So, I = $115.20, P = $1800, and t = 240/360.
Jason Aubrey Math 1300 Finite Mathematics
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Example: A commercial for a loan company states, “You onlypay $0.16 a day for each $600 borrowed.” If you borrow $1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging?
You pay a fee of 3× $0.16 = $0.48 per day.This is a total fee of $0.48× 240 = $115.20 over the life ofthe loan.So, I = $115.20, P = $1800, and t = 240/360.
Jason Aubrey Math 1300 Finite Mathematics
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Now we apply our simple interest formula...
I = Prt
$115.20 = ($1, 800)r(
240360
)r = 0.096
So, you repay a total of $1,915.20 and the annual interest rateis 9.6%.
Jason Aubrey Math 1300 Finite Mathematics
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Now we apply our simple interest formula...
I = Prt
$115.20 = ($1, 800)r(
240360
)r = 0.096
So, you repay a total of $1,915.20 and the annual interest rateis 9.6%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Now we apply our simple interest formula...
I = Prt
$115.20 = ($1, 800)r(
240360
)
r = 0.096
So, you repay a total of $1,915.20 and the annual interest rateis 9.6%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Now we apply our simple interest formula...
I = Prt
$115.20 = ($1, 800)r(
240360
)r = 0.096
So, you repay a total of $1,915.20 and the annual interest rateis 9.6%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Now we apply our simple interest formula...
I = Prt
$115.20 = ($1, 800)r(
240360
)r = 0.096
So, you repay a total of $1,915.20 and the annual interest rateis 9.6%.
Jason Aubrey Math 1300 Finite Mathematics
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Theorem (Amount: Simple Interest)
A = P + Prt= P(1 + rt)
A = amount, or future value
P = principal, or present valuer = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
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Theorem (Amount: Simple Interest)
A = P + Prt= P(1 + rt)
A = amount, or future valueP = principal, or present value
r = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Theorem (Amount: Simple Interest)
A = P + Prt= P(1 + rt)
A = amount, or future valueP = principal, or present valuer = annual simple interest rate (written as a decimal)
t = time in years
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Theorem (Amount: Simple Interest)
A = P + Prt= P(1 + rt)
A = amount, or future valueP = principal, or present valuer = annual simple interest rate (written as a decimal)t = time in years
Jason Aubrey Math 1300 Finite Mathematics
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Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here
P = $989.37, A = $1, 000, and t = 1352
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
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Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here
P = $989.37, A = $1, 000, and t = 1352
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37,
A = $1, 000, and t = 1352
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000,
and t = 1352
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]
1.011 ≈ 1 + r1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044
or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = $989.37, A = $1, 000, and t = 13
52
A = P(1 + rt)
$1, 000 = $989.37[1 + r
(1352
)]1.011 ≈ 1 + r
1352
0.011 ≈ r1352
r ≈ 0.044 or 4.4%
Jason Aubrey Math 1300 Finite Mathematics
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Example: Many investment firms charge commissions ontransactions based on the amount of the transaction. Supposethat an investment firm charges commissions on stock tradesaccording to the following commission schedule:
Transaction Size CommissionUnder $3,000 $25+1.8% of principal
$3000-$10,000 $37 + 1.4% of principalOver $10,000 $107 + 0.7% of principal
Suppose an investor purchases 175 shares at $15.00 a share,holds the stock for 26 weeks, and then sells the stock for$17.25 per share. Find the annual interest rate earned by thisinvestment.
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the total cost of the purchase.
$15.00(175) = $2,625 - Principal$25 + $2,625(0.018) = $72.25 - Commission$2,625 + $72.25 = $2,697.25 - Total cost of purchase
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the total cost of the purchase.$15.00(175) = $2,625 - Principal
$25 + $2,625(0.018) = $72.25 - Commission$2,625 + $72.25 = $2,697.25 - Total cost of purchase
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the total cost of the purchase.$15.00(175) = $2,625 - Principal$25 + $2,625(0.018) = $72.25 - Commission
$2,625 + $72.25 = $2,697.25 - Total cost of purchase
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the total cost of the purchase.$15.00(175) = $2,625 - Principal$25 + $2,625(0.018) = $72.25 - Commission$2,625 + $72.25 = $2,697.25 - Total cost of purchase
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: Find the net revenue from the sale.
$17.25(175) = $3,018.75 - Principal$37 + ($3,018.75)(0.014) = $79.26 -Commission$3,018.75 - $79.26 = $2,939.49 - Net revenue from sale
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: Find the net revenue from the sale.$17.25(175) = $3,018.75 - Principal
$37 + ($3,018.75)(0.014) = $79.26 -Commission$3,018.75 - $79.26 = $2,939.49 - Net revenue from sale
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: Find the net revenue from the sale.$17.25(175) = $3,018.75 - Principal$37 + ($3,018.75)(0.014) = $79.26 -Commission
$3,018.75 - $79.26 = $2,939.49 - Net revenue from sale
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: Find the net revenue from the sale.$17.25(175) = $3,018.75 - Principal$37 + ($3,018.75)(0.014) = $79.26 -Commission$3,018.75 - $79.26 = $2,939.49 - Net revenue from sale
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25
,
and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49,
P = $2697.25
,
and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49, P = $2697.25,
and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)
2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Step 3: Calculate annual interest rate:Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))
r = 0.1796
Jason Aubrey Math 1300 Finite Mathematics
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Example: Suppose that after buying a new car you decide tosell your old car to a friend. You accept a 270-day note for$3,500 at 10% simple interest as payment. (Both principal andinterest will be paid at the end of 270 days.) Sixty days lateryou find that you need the money and sell the note to a thirdparty for $3,550. What annual interest rate will the third partyrecieve for the investment? (Express the answer as apercentage, correct to three decimal places).
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the amount that will be paid at the end of 270days to the holder of the note:
A = P(1 + rt)
= ($3, 500)
[1 + (0.1)
(270360
)]= $3, 762.50
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the amount that will be paid at the end of 270days to the holder of the note:
A = P(1 + rt)
= ($3, 500)
[1 + (0.1)
(270360
)]= $3, 762.50
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the amount that will be paid at the end of 270days to the holder of the note:
A = P(1 + rt)
= ($3, 500)
[1 + (0.1)
(270360
)]
= $3, 762.50
Jason Aubrey Math 1300 Finite Mathematics
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Step 1: Find the amount that will be paid at the end of 270days to the holder of the note:
A = P(1 + rt)
= ($3, 500)
[1 + (0.1)
(270360
)]= $3, 762.50
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
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Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50,
P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550,
t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]
r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262
or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Step 2: For the third party, we are to find the annual rate ofinterest r required to make $3,550 grow to $3,762.50 in 210days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550, t = 210360
.
A = P(1 + rt)
$3, 762.50 = $3, 550[1 + r
(210360
)]r = 0.10262 or 10.262%.
Jason Aubrey Math 1300 Finite Mathematics
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Example: Many tax preparation firms offer their clients a refundanticipation loan (RAL). For a fee, the firm will give the client hisrefund when the return is filed. The loan is repaid when the IRSsends the refund directly to the firm. Thus, the RAL fee isequivalent to the interest charge for the loan. The schedulebelow is from a major RAL lender.
RAL Amount RAL Fee0-$500 $29.00
$501-$1,000 $39.00$1,001-$1,500 $49.00$1,501-$2,000 $69.00$2,001-$2,500 $89.00
A client recieves a $480 RAL which is repaid in 25 days. Whatis the annual interest rate for this loan?
Jason Aubrey Math 1300 Finite Mathematics
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First, P = $480.
So according to the schedule, the amountcharged is I = $29.00. t = 25
360
.
We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
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First, P = $480. So according to the schedule, the amountcharged is I = $29.00.
t = 25360
.
We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
university-logo
First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25
360 .
We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
university-logo
First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25
360 . We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
university-logo
First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25
360 . We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
university-logo
First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25
360 . We now compute r :
I = Prt
29 = (480)r(
25360
)
r = 0.87
Jason Aubrey Math 1300 Finite Mathematics
university-logo
First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25
360 . We now compute r :
I = Prt
29 = (480)r(
25360
)r = 0.87
Jason Aubrey Math 1300 Finite Mathematics