Beginning Algebra
Tutorial 17: Further Problem Solving

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Learning Objectives

After completing this tutorial, you should be able to:

Use Polya's four step process to solve word problems involving geometry concepts, distance, mixtures and interest.

Introduction

In this tutorial we will be solving problems involving geometry concepts, distance, mixtures and interest. Since we are still problem solving, I will use Polya’s four steps to Problem Solving as introduced in Tutorial 15: Introduction to Problem Solving to step us through the problems in this tutorial. After finishing this tutorial, you will be able to answer those tricky word problems. Let's see how you do on these problems.

Tutorial

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 15: Introduction to Problem Solving. To refresh your memory, here they are again:

Step 1: Understand the problem.

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

Step 4: Look back (check and interpret).

Rectangle Problem

The following formula will come in handy for solving example 1:

Perimeter of a rectangle = 2(length) + 2(width)

Example 1: In a blueprint of a rectangular room, the length is 1 inch more than 3 times the width. Find the dimensions if the perimeter is to be 26 inches.

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for the length and width of the rectangle. Since length can be written in terms of width, we will let

w = width

length is 1 inch more than 3 times the width:

1 + 3w = length

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

*Remove ( ) by using dist. prop.
*Combine like terms

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8

Step 4: Look back (check and interpret).

If width is 3, then length, which is 1 inch more than 3 times the width would have to be 10. The perimeter of a rectangle with width of 3 inches and length of 10 inches does come out to be 26.

FINAL ANSWER:

Width is 3 inches.

Length is 10 inches.

Distance Problem

The following formula will come in handy for solving example 2:

Distance = Rate * Time

Example 2: 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).

Plugging the values into the formula we get:

example 2a

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.

Mixture Problem

Example 3: How much 20% alcohol solution and 50% alcohol solution must be mixed to get 12 gallons of 30%?

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for the amount of 20% solution and 50% solution needed to get 12 gallons of 30%.

x = number of gallons of the 20%.

Since the two mixtures together need to be 12 gallons, then we can take the total (12) and subtract from it the “given” number of gallons (x):

12 - x = number of gallons of the 50%.

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

*Remove ( ) by using dist. prop.

*Combine like terms

*Inv. of add. 60 is sub. 60

*Inv. of mult. by -3 is div. by -3

Step 4: Look back (check and interpret).

If there are 8 gallons of the 20%, then there would have to be 12 - 8 = 4 gallons of the 50% solution.

If you have 8 gallons of 20% solution and 4 gallons of 50% solution you do get 12 gallons of 30% alcohol solution.

FINAL ANSWER:

8 gallons of the 20% solution.
4 gallons of the 50% solution.

Simple Interest Rate Problem

Example 4: An investor with $70,000 decides to place part of her money in corporate bonds paying 12% per year and the rest in a certificate of deposit paying 8% per year. If she wishes to obtain an overall return of $6300 per year, how much should she place in each investment?

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for how much she invested in EACH account.

x = amount invested in 12%

Since the two accounts together need to be $70000, then we can take the total (70000) and subtract from it the “given” number in the 12% account (x):

70,000 - x = amount invested in 8%

Note that you could have reverse those, the problem would still work out the same.

Step 2: Devise a plan (translate).

12% return plus 8% return results in 6300

.12x + .08(70000 - x) = 6300

Step 3: Carry out the plan (solve).

*Remove ( ) by using dist. prop.

*Combine like terms

*Inv. of add. 5600 is sub. 5600

*Inv. of mult. by .04 is div. by .04

Step 4: Look back (check and interpret).

If she invested $17500 in corporate bonds, then she would have to invest $70000 - $17500 = $52000 in the certificate of deposit.

If you take 12% of $17500 and add it to 8% of $52500 you do get $6300.

FINAL ANSWER:

She invested $17500 at 12% and $52500 at 8%.

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 - 1d: Solve the word problem.

1a. A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet.
(answer/discussion to 1a)

1b. 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 1b)

1c. How much 25% antifreeze and 50% antifreeze should be combined to give 40 liters of 30% antifreeze?
(answer/discussion to 1c)

1d. You recently came into $20,000 (lucky you!) and you want to place part of your money in a savings account paying 7% per year and part in a certificate of deposit paying 9% per year. If you wish to obtain an overall return of $1700 per year, how much would you place in each investment?
(answer/discussion to 1d)

Need Extra Help on these Topics?

The following is a webpage that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/mixture.htm
This webpage goes over mixture problems.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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