Chapter 18

Using Established Formulas

A formula is a rule that describes a situation that happens consistently or exists without variation. One of the first formulas that people learn is that for the area of a rectangle — just multiply the length by the width. The trick to using formulas is to understand what the different symbols represent and then to be able to apply the mathematical rules correctly.

The Problems You'll Work On

The majority of the problems in this chapter involve simply determining which formula to use, where to use it, and applying the following techniques:

What to Watch Out For

Whether you struggle remembering formulas or are a formula whiz, be sure you don't overlook the following:

Getting Interested in Interest Problems

766–769 Solve each using the simple interest formula, I = Prt.

766. How much interest is earned if you invest $20,000 at 2.5% for ten years?

767. If you're lending a friend $4,000 at 4% simple interest for two years, what is the total amount you'll be paid back?

768. You're buying a television from an appliance store for $3,600. They're charging you 11% simple interest for three years. How much will your monthly payments be?

769. You have $10,000 to invest. You want to put it in an account that earns 4% simple interest. How long will it take for your investment to total $13,400 (so you can buy that boat)?

Heating It Up with Temperature Problems

770–773 Solve the problems using the temperature formulas:

770. What temperature in degrees Fahrenheit corresponds to 37 degrees Celsius?

771. What temperature in degrees Celsius corresponds to 59 degrees Fahrenheit?

772. What temperature in degrees Celsius corresponds to 212 degrees Fahrenheit?

773. What temperature in degrees Fahrenheit corresponds to −40 degrees Celsius?

Adding up Natural Numbers

774–777 Solve the problems using S_n = for the sum of the first n natural numbers.

774. What is the sum of the numbers from 1 through 50?

775. If the sum of the first n numbers is 5,050, then what are the numbers?

776. What is the sum of the natural numbers 40 through 60?

777. A theater has seating that begins with 36 seats in the front row and increases by one seat each row through the 30th row back. How many seats are in the theater?

Going the Distance with the Distance Formula

778–785 Solve the problems using the distance, rate, and time formula, d = rt.

778. How far do you travel if you're driving at 55 mph for six hours?

779. How fast were you driving if you traveled 450 miles in 7 hours, 30 minutes?

780. How long does it take to travel 1,050 miles if you're averaging 60 miles per hour?

781. What was your average speed if you left home at 8:00 a.m., drove 150 miles, stopped for an hour, drove another 200 miles, and arrived at your destination at 4:00 p.m.?

782. Hank left home at 8:00 a.m. traveling at 50 mph. Helen found his briefcase on the table and left at 8:30 trying to catch up with Hank. She was traveling at 60 mph. What time did Helen catch up with Hank?

783. A bus left Chicago at 6:00 a.m. traveling due east at 40 mph. A second bus left Chicago at 7:00 traveling due west at 55 mph. At what time are the two buses 800 miles apart?

784. Claire and Charlie decided to walk around the lake. Claire started in a clockwise direction walking 4 mph, and Charlie started at the same time in a counterclockwise direction walking 5 mph. After half an hour, Charlie took a 15-minute break. If the lake is 10 miles around, how long did it take for them to meet?

785. Bill and Will drove from Peoria to their home in Missouri in separate vehicles. It took Bill one hour longer than Will, because Bill drives an average of 10 mph slower than Will. If it took Bill five hours, then how far was the trip in miles?

Getting the Inside Scoop with Sums of Interior Angles

786–789 Solve the problems using the formula for the sum of the measures of all the interior angles of a polygon with n sides: A = 180(n − 2).

786. What is the sum of the measures of all the angles in a hexagon (six-sided)?

787. What is the sum of the measures of all the angles in a decagon?

788. If the sum of the measures of the angles in a polygon is 1,080°, then what is the polygon?

789. If the sum of the measures of the angles in a regular (all sides equal) polygon is 1,800 degrees, then what is the measure of just one of the angles in the polygon?

Averaging Out the Numbers

790–793 Solve the problems using the formula for the average of n numbers, A = .

790. Stephan got scores of 81, 67, 93, and 99 on his exams. What is his average?

791. Stephanie got three scores of 9, two scores of 10, four scores of 7, and one score of 8 on her quizzes. What is her average quiz score?

792. Joel has test scores of 85, 87, 93, and 100 on his first exams. What does he need on the last exam to have an average of at least 90?

793. Tomas has an average of 91 on his first three exams. What does he have to earn on the last exam to bring the average up to at least a 93?

Summing the Squares of Numbers

794–795 Solve the problems using the formula for the sum of n squares, S = .

794. What is the sum of the numbers 1, 4, 9, 16, 25, and 36?

795. What is the sum of the squares of the numbers from 1 through 15?

Finding the Terms of a Sequence

796–799 Solve the problems using the formula for the nth term in an arithmetic sequence, a_n = a₁ + (n − 1)d.

796. What is the tenth term in an arithmetic sequence whose first term is 1, and where the difference between the terms is 4?

797. What is the 100th term in an arithmetic sequence whose first term is −6, and where the difference between the terms is 2?

798. What is the 40th term in the arithmetic sequence that starts: 3, 7, 11, 15, . . . ?

799. What is the 20th term in the arithmetic sequence that starts: 100, 97, 94, 91, . . . ?

Adding the Terms in an Arithmetic Sequence

800–801 Solve the problems using the formula for the sum of the first n terms in an arithmetic sequence, S_n = (a₁ + a_n).

800. What is the sum of the even numbers from 40 through 60?

801. What is the sum of the first 40 numbers in the arithmetic sequence starting: 2, 5, 8, 11, . . . ?

Using the Formula for the Sum of Cubes

802–803 Solve the problems using the formula for the sum of the first n cubes, S_n = .

802. What is the sum of the numbers 1, 8, 27, 64, 125, 216, and 343?

803. What is the sum of the cubes of the numbers from 5 through 20?

Compounding the Problems Involving Compound Interest

804–807 Solve the problems using the compound interest formula A = P , where A is the total accumulated amount, P is the principal (deposit), r is the interest rate written as a decimal, n is the number of times per year the money is compounded, and t is the number of years.

804. What is the total amount of money that you accumulate after ten years if you deposit $5,000 in an account earning 3% interest compounded quarterly?

805. How much interest do you earn on $40,000 deposited for five years in an account earning 5%, compounded monthly?

806. What is the total amount of money you accumulate in an account earning 6% interest compounded daily if you deposit $20,000 and leave it in the account for 15 years?

807. A member of Columbus's crew deposited $1 in the Bank of the West Indies in 1492. The deposit was earning 2% interest compounded quarterly. You were contacted in 2012, because you are a descendant of this crew member, and the account has been abandoned. They're going to send you a check for the amount in the account. How much money will you get?