Beginning Algebra
Tutorial 17:
Further Problem Solving
 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)
|
| 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. |
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
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*Inv. of add. 2 is sub. 2
*Inv. of mult. by 8 is div. by 8
|
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.
|
| 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? |
| 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
|
| Plugging the values into the formula we get:
|
 |
*Inverse of mult. by 4.5 is
div. by 4.5
|
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.
|
| Example
3: How much 20% alcohol solution and 50% alcohol
solution must be mixed to get 12 gallons of 30%? |
| 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%.
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*Inv. of add. 60 is sub. 60
*Inv. of mult. by -3 is div. by
-3
|
| 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.
|
| 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? |
| 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.
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*Inv. of add. 5600 is sub. 5600
*Inv. of mult. by .04 is div.
by .04
|
| 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?
|
All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 22, 2003 by Kim Seward. |
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to the THEA Math Help Page)
