Learning Objectives
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
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).
Perimeter of a rectangle = 2(length) + 2(width)
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
*Inv. of add. 2 is sub. 2
*Inv. of mult. by 8 is div. by 8
FINAL ANSWER:
Length is 10 inches.
Distance = Rate * Time
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
FINAL ANSWER:
The average speed is 70 mph.
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%.
*Combine like terms
*Inv. of add. 60 is sub. 60
*Inv. of mult. by -3 is div. by -3
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.
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.
12% return plus 8% return results in 6300
.12x + .08(70000 - x) = 6300
*Combine like terms
*Inv. of add. 5600 is sub. 5600
*Inv. of mult. by .04 is div. by .04
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
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?
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.
Last revised on July 27, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.