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Intermediate Algebra
Tutorial 9:
Formulas and Problem Solving

 Learning Objectives

 After completing this tutorial, you should be able to: Solve a formula for a given variable. Use Polya's four step process to solve word problems involving formulas.

 Introduction

 In this tutorial we will be solving problems using formulas to help us.   In Tutorial 8: An Introduction to Problem Solving, we had to create our own equations based on the given information. However, all of the problems we will be solving in this tutorial will involve using a formula to get to our final answer.  We will be looking at such formulas as compound interest, distance/rate, volume of a sphere, and so on.  Since we are still problem solving, I will use Polya’s four steps to Problem Solving as introduced in Tutorial 8: An Introduction to Problem Solving to step us through the word problems in this tutorial. After going through this tutorial, you will be an old pro at solving word problems involving formulas.

 Formulas

 A formula is an equation that involves two or more variables that have a specific relationship with each other.

 Solving a Formula  for a Specified Variable

 Basically, you want to get the variable you are solving for alone on one side and everything else on the other side (including variables you are not solving for) using INVERSE operations.   Even though there is more than one variable in a formula, you solve for a specific variable using the exact same steps that you do with an equation in one variable, as shown in Tutorial 7: Linear Equations in One Variable.   It is really easy to get overwhelmed when there is more than one variable involved.  Sometimes your head feels like it is spinning when you see all of those variables.  Isn’t math suppose to be about numbers?  Well, just remember that a variable represents a number, so if you need to move it to the other side of the equation you use inverse operations, just like you would do with a number.

 Example 1:   Solve the equation P = 2L + 2W for L.

 Do you recognize this formula? This happens to be the formula for the perimeter of a rectangle, where P = perimeter, L = length, and W = width. In this problem, we need to solve for L.  This means we need to get L on one side and EVERYTHING ELSE on the other side using inverse operations. Let’s solve this formula for L:

 *Inverse of add 2W is sub. 2W   *Inverse of mult. by 2 is div. by 2   *Formula solved for L

 Example 2:   Solve the equation   for r.

 Do you recognize this formula? This happens to be the formula for the circumference of a circle, where C = circumference,  = pi,  and r = radius. In this problem, we need to solve for r.  This means we need to get r on one side and EVERYTHING ELSE on the other side using inverse operations. Let’s solve this formula for r:

 *Inverse of mult. by 2pi is div. by 2pi *Formula solved for r

 Example 3:   Solve the equation 5x + 4y = 11 for y.

 This is an equation for a line.  In later tutorials, we will go over graphing lines on a two dimensional graph. This is a little sneek peak at working with a linear equation in two variables.  When graphing such an equation, a lot of times, we have it written in what is called the slope-intercept form, which boils down to solving the equation for y.   Lo and behold, that is what we are doing here (sneaky how we are slipping it in now, huh?).  So if you get this concept down now, you will be an old pro at it by the time we reach graphs. In this problem, we need to solve for y.  This means we need to get y on one side and EVERYTHING ELSE on the other side using inverse operations. Let’s solve this formula for y:

 *Inverse of add 5x is sub. 5x   *Inverse of mult. by 4 is div. by 4 *Formula solved for y

 Example 4:   Solve the equation V = lwh for h.

 Do you recognize this formula? This happens to be the formula for the volume of a rectangular solid, where V = volume, l = length, w = width, and h = height. In this problem, we need to solve for h.  This means we need to get h on one side and EVERYTHING ELSE on the other side using inverse operations. Let’s solve this formula for h:

 *Inverse of mult. by lw is div. by lw   *Formula solved for h

 Polya’s  Four-step Process  for Problem Solving (revisited)

As mentioned above, since we are still problem solving, we will use the exact same four step process we used in Tutorial 8: Introduction to Problem Solving.  To refresh your memory, here they are again:

Step 1:

 Compound Interest

Compound interest is a type of word problem that involves something that we all can relate to - MONEY, MONEY, MONEY!!!

 Example 5:   You recently received a windfall of \$7000, and being the smart responsible person that you are, you invested it in an account paying an annual percentage rate of 8%.  Find the amount in the account after 9 years if the account is compounded quarterly.

 Step 1:  Understand the problem.

 Make sure that you read the question carefully several times.  Since we are looking for compound interest, we will need the compound interest formula:  The variables in this formula represent the following: A = ending amount in the account P = principal (starting amount) r = annual rate of interest t = time in years n = number of times compounded per year

 Step 2:  Devise a plan (translate).

 In this problem,  A = ? = this is the variable we are looking for P = 7000 r = 8% = .08 t = 9 n = 4   (There are 4 quarters in a year) Plugging the values into the formula we get:

 Step 3:  Carry out the plan (solve).

 *.08/4 = .02 and 4(9) = 36 *Add inside the (   ) *Raise 1.02 to the exponent of 36 *Multiply

 Step 4:  Look back (check and interpret).

 If you take \$7000 and compound it quarterly for 9 years, you do end up with 14279.211.   FINAL ANSWER:  The compound amount is \$14279.21.

 Distance

 Example 6:   It takes you 4.5 hours to drive from your home to your favorite weekend get away, which is 315 miles away.  What is your average speed?

 Step 1:  Understand the problem.

 Make sure that you read the question carefully several times.  Since we are looking for speed, we can use the distance/rate formula: d = rt The variables in this formula represent the following: d = distance r = rate t = time

 Step 2:  Devise a plan (translate).

 In this problem,  d = 315 r = ? = this is the variable we are looking for t = 4.5 d = rt Plugging the values into the formula we get:

 Step 3:  Carry out the plan (solve).

 *Inverse of mult. by 4.5 is div. by 4.5

 Step 4:  Look back (check and interpret).

 If you go at a rate of 70 miles per hour for 4.5 hours, you would travel 315 miles.   FINAL ANSWER:  The average speed is 70 mph.

 Volume of a Sphere

 Example 7:   The diameter of a beach ball was found to be 18 inches.  What is the volume of this beach ball?

 Step 1:  Understand the problem.

 Make sure that you read the question carefully several times.  Since we are looking for the volume of a sphere,  we can use this formula: The variables in this formula represent the following: V = volume of a sphere r = radius

 Step 2:  Devise a plan (translate).

 In this problem,  V = ? = this is the variable we are looking for r = 9   (radius is half the diameter, so r = 18/2 = 9) Plugging the values into the formula we get:

 Step 3:  Carry out the plan (solve).

 *Cube 9 *Multiply

 Step 4:  Look back (check and interpret).

 If you put in 9 for the radius, it does compute to have a volume of .   FINAL ANSWER:  The volume of the beach ball is cubic inches.

 Area of a Rectangle

 Example 8:   One bag of fertilizer will cover 500 square feet of lawn.  Your rectangular lawn is 70 feet by 50 feet.  How many bags of fertilizer will you need to cover it?

 Step 1:  Understand the problem.

 Make sure that you read the question carefully several times.  Since we are needing to find the area of a rectangle,  we can use this formula: A = LW The variables in this formula represent the following: A = Area of a rectangle L = length W = width

 Step 2:  Devise a plan (translate).

 In this problem,  A = ? = this is the variable we are looking for L = 70 W = 50 A = LW Plugging the values into the formula we get:

 Step 3:  Carry out the plan (solve).

 First, find the area of the lawn:

 *Multiply

 For every 500 square feet, you need 1 bag of fertilizer.  So, we need to see how many times 500 sq. feet goes into 3500 sq. feet to find the number of bags of fertilizer needed.

 *Divide

 Step 4:  Look back (check and interpret).

 If you take 70 times 50 you do get 3500, so the area checks out.  And if you take 3500 and divide it by 500 you do get 7, so the number of bags needed checks out.    FINAL ANSWER:  7 bags of fertilizer.

 Practice Problems

 These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 Practice Problems 1a - 1b:  Solve each equation for the specified variable.

 1a.  I = PRT;   for T (answer/discussion to 1a)

 1b.  3x - 7y = 2;   for y (answer/discussion to 1b)

 Practice Problems 2a - 2c:  Solve the following word problems.

 2a.  A principle of \$50,000 is invested into a CD paying an annual percentage rate of 7.5%.  Find the amount in the account after 10 years if the account is compounded monthly. (answer/discussion to 2a)

 2b.  In Nebraska on I-80, the speed limit is 75 mph.  How long would it take you to travel 525 miles in Nebraska on  I-80 if you went the speed limit the whole time? (answer/discussion to 2b)

 2c. Sally is training for the Olympics.  She likes to run around  a circular track that has a diameter of 60 yards, 20 times during a workout.  How many yards does she run during her workout? (answer/discussion to 2c)

 Need Extra Help on These Topics?

The following are webpages that can assist you in the topics that were covered on this page:

 This webpage helps you with solving formulas for a specified variable. This webpage involves a combination of problem solving ideas from both Tutorial 8: An Introduction to Problem Solving and Tutorial 9: Formulas and Problem Solving.  It includes problem solving using formulas.

for some more suggestions.

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June 22, 2003