math 1300: section 3-1 simple interest

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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

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I = Prt

whereI = interestP = principal

r = annual simple interest rate (written as a decimal)t = time in years

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I = Prt

whereI = interestP = principalr = annual simple interest rate (written as a decimal)

t = time in years

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I = Prt

whereI = interestP = principalr = annual simple interest rate (written as a decimal)t = time in years

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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)

)= 34.13

Therefore $34.13 is owed.

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I = Prt

I = 650(0.21)

)= 34.13

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I = Prt

I = 650(0.21)

)= 34.13

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I = Prt

I = 650(0.21)

)= 34.13

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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.

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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.

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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.

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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.

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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%.

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I = Prt

$115.20 = ($1, 800)r(

240360

)r = 0.096

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I = Prt

$115.20 = ($1, 800)r(

240360

r = 0.096

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I = Prt

$115.20 = ($1, 800)r(

240360

)r = 0.096

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I = Prt

$115.20 = ($1, 800)r(

240360

)r = 0.096

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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

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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

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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

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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

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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

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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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

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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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

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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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

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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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

A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

1.011 ≈ 1 + r1352

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044

or 4.4%

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A = P(1 + rt)

$1, 000 = $989.37[1 + r

)]1.011 ≈ 1 + r

0.011 ≈ r1352

r ≈ 0.044 or 4.4%

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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A = P(1 + rt)

2939.49 = 2697.25(1 + r(.5))

r = 0.1796

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A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))

r = 0.1796

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A = P(1 + rt)2939.49 = 2697.25(1 + r(.5))

r = 0.1796

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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).

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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

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A = P(1 + rt)

= ($3, 500)

[1 + (0.1)

(270360

)]= $3, 762.50

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A = P(1 + rt)

= ($3, 500)

[1 + (0.1)

(270360

= $3, 762.50

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A = P(1 + rt)

= ($3, 500)

[1 + (0.1)

(270360

)]= $3, 762.50

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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%.

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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%.

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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%.

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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%.

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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%.

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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%.

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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%.

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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%.

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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?

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First, P = $480.

So according to the schedule, the amountcharged is I = $29.00. t = 25

We now compute r :

I = Prt

29 = (480)r(

)r = 0.87

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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(

)r = 0.87

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First, P = $480. So according to the schedule, the amountcharged is I = $29.00. t = 25

We now compute r :

I = Prt

29 = (480)r(

)r = 0.87

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360 . We now compute r :

I = Prt

29 = (480)r(

)r = 0.87

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I = Prt

29 = (480)r(

)r = 0.87

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I = Prt

29 = (480)r(

r = 0.87

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I = Prt

29 = (480)r(

)r = 0.87

math 1300: section 3-1 simple interest

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