Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

521

Solve the system using elimination.

Alice is currently 20 years old. To determine Mel’s age, substitute y = 20 into

either equation of the system and solve for x.

Mel is currently 45 years old, 25 years older than Alice.

Calculating Interest

Simple, compound, and continuously compounding

23.9 How much simple interest accrues on a loan of $1,200 at an annual rate of

4.5% over a three-year period?

The formula for simple interest is i = prt, where i is interest, p is the principal,

r is the annual interest rate expressed as a decimal, and t is the length of time

(in years) the principal accrues interest. Substitute p = 1,200, r = 0.045, and t = 3

into the formula to calculate i.

The total accrued interest is $162.

23.10 If you wish to borrow $3,000 from a friend who will charge you simple interest

at an annual rate of 7%, and you cannot exceed $3,300 in total debt. What

is the maximum length of time you have to repay the debt and the accrued

interest? Round the answer to the nearest whole day.

You intend to borrow $3,000 but your total debt cannot exceed $3,300.

Therefore, the maximum total interest you can afford to pay is

$3,300 – $3,000 = $300. Substitute i = 300, p = 3,000, and r = 0.07 into the

simple interest formula to calculate the number of years t it would take to

accrue $300 in interest.

Multiply

the second

equation by –1, add

the equations, and

solve for y.

Make sure

you answer the

question posed by

the word problem.

In this case, you’re

supposed to gure

out how much older

Mel is than Alice, so

subtract their ages:

x – y = 45 – 20 = 25.

The principal

is the dollar

amount that earns

interest. In this

case, the principal

is $1,200.

To change

4.5% into a

decimal, move the

decimal point two

places to the left:

0.045.

Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

522

To convert t into days, multiply by 365.

365(1.42857142857) ≈ 521.428571429 ≈ 521

The maximum length of time you can afford to borrow the money is 521 days.

23.11 Describe the difference between simple and compound interest.

Simple interest accrues only on the principal, whereas compound interest

transfers interest accrued into principal each time it is compounded. Consider

a savings account that pays only simple interest. No matter what length of time

the money remains in the account, interest is earned only on the principal

deposited initially. However, a compound interest account would earn interest

on all the money in the account.

23.12 If $500 is deposited into a savings account with a 3.75% annual interest rate

compounded monthly, what is the balance of the account ten years later?

Round the answer to the hundredths place.

The formula for compound interest is , where b is the balance,

p is the principal, r is the annual interest rate expressed as a decimal, n is the

number of times the interest is compounded in one year, and t is the length of

time (in years) interest is earned. Substitute p = 500, r = 0.0375, n = 12, and

t = 10 into the formula to calculate b.

The balance of the account is $727.07.

Because

r is an

annual

interest rate,

t is measured in

years. There are

365 days in a (non

leap) year, so

multiply t by

365.

Including

the interest

you’ve already

earned on your

initial investment.

Each time it

compounds, all the

interest you’ve earned

is added to your

initial investment,

and you start

earning interest

on that.

The balance

is the original

deposit (principal)

plus all the interest

it earned. In other

words, the balance

is the total amount

of money in your

account.

Interest

is compounded

monthly, a total of

n = 12 times per year.

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