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# MATHS Word Problems BOOK by Immergut, Brita.

“To the Reader Many people are afraid of word problems. Why is that”?

Maybe it’s because they remember that they had previous trouble with word problems. Or they think that they can’t understand word problems because word problems are “difficult.” Or they don’t know how to unravel the problem to find out what the real question is. Or they simply don’t know where to start.

In this book, you will learn how to overcome those difficulties. You will be asked to read the problems slowly and to first concentrate on the words rather than on the numbers. Then you will learn how to break down the problem into smaller segments and to use a simple table to list the known numbers presented in the problem and the unknown number (usually x) that you are asked to figure out what it stands for. The solution for the problem usually involves the use of an equation and,
for those of you who are a bit hazy about equations, you will
find a short refresher in the Appendix.
The problems in this book mostly deal with situations from daily life: percents and discounts; interest (simple and compound); mixing of liquids and mixing of solids; ratios and pro-
portions; and measurements in the English (customary) and the metric system and how to convert from one to the other.

## MATHS Word Problems

12 Master Math: Solving Word Problems
There will also be problems dealing with the motion of cars,
boats, and people at different speeds and how quickly work
gets done. Then we will move on to statistics and probability
problems: averages, graphs, probabilities, and odds. There will
be rolling of dice, tossing of pennies, and drawing of playing
cards.

## MATHS Word Problems

Finally, you will learn how to solve word problems involving geometrical figures, such as triangles, polygons, circles, and cylinders. Some problems will deal with plane geometry, others with solid geometry, trigonometry, and analytic geom etry. Each chapter contains not only worked-out problems but also plenty of practice problems. The answers to the practice problems are at the end of the book.

I hope that when you are finished with this book you will feel like one of my former students did who told me: “Before I took your course I cried because I couldn’t solve the word prob-
lems and now I cry because I am so happy that I can solve them.”‘

In order to solve mathematical word problems we often need to use equations. In this chapter, you will learn how to set
up simple equations to solve different kinds of word problems.
For example, we will cut up a length of board or rope into shorter
and longer pieces and, given the known total length and other
facts, we will calculate the lengths of the pieces cut from it. In
other examples we will calculate the ages of two children once
we know how many years they are apart and what the sum of
their ages is. We will also look at situations where one person
weighs more or less than another and calculate each person’s
weight from the information given in the problem.

Then, we will learn the mathematical symbols for inequality-
ties, that is, situations where something is greater than or smaller

then something else and also how to solve problems in which
we are told that something is at most so big or that something
costs at least so much.
Finally, we will tackle word problems involving all kinds of
numbers: positive and negative integers, including zero; odd
and even integers; and consecutive integers.
The last example will show you how to solve a problem that
requires the use of a quadratic equation.

#### MATHS Word Problems

14 Master Math: Solving Word Problems
(Note: If you want to brush up on your skills for solving
equations, see the Appendix.)

Length Problems

Example:
Cut a 10-foot (ft.) long piece of wood into two pieces so that
one piece is 2 ft. longer than the other. To solve this problem
you have two choices:
By using algebra:
Call one piece x, then the other piece is x + 2.
Write an equation: x+x+2=10
Solve the equation: 2x=8
x=4
x +2=6
Total = 10

Or by using arithmetic:
10 – 2 = 8 Take away the 2 ft. from the whole piece.
8 + 2 = 4 Divide the piece by 2.
4 + 2 = 6 Add the 2 ft. to one of the pieces.
Total = 10
The pieces were 4 ft. and 6 ft.
10 ft. Read the problem again to check all the facts.
Example:
A length of board was 10 inches shorter than another length.
Together the boards were 20 inches. How long were the boards?

Simple Equation Problems 15

##### MATHS Word Problems

Call the long piece x and the short piece x – 10.
x+x-10=20
2x = 30
x = 15
15 – 10 = 5
The pieces were 5 and 15 in.
Reread the problem. Is it true that the pieces equal 20 in.
when put together? Is one piece 10 in. shorter than the other?
Practice Problems:
1.1 Solve the previous problem by calling the short
piece of boardx.
1.2 A12-ft. ropeiscut into threepiecesso that the second piece is 1 ft. longer than the first and the
third piece is 1 ft. longer than the second. How long
are the pieces?
1.3 A 9-ft. board is cut into two pieces so that one piece
is twice the other. How long are the pieces?
1.4 An 80-in. board is to be cut into three pieces so
that one piece is twice another and the third piece
is 10 in. more than the second. Find the length of
each piece.
1.5 Two ropes are together 275 yards long. One rope is
50% longer than the other. How long are the ropes?
Age Problems

Example:
Leah is 2 years older than Tracy. Together the girls are 10 years
old. How old are they?

16 Master Math: Solving Word Problems
Call Leah’s age x. Then Tracy’s age is x – 2.
x+x-2 = 10
2x = 12
x=6
x – 2 = 4 Leah is 6 years old and Tracy is 4.
Do you recognize this problem as essentially the same as
the first length problem?
Practice Problems:
1.6 Elsa is 7 years younger than Thor. The sum of their
ages is 35. How old are they?
1.7 Kristina’s grandmother is 12 times as old as Kristina.
Together they are 91 years. How old is the
grandmother?
1.8 The sum of the ages of Jessica, her mother, and her
grandmother is 100 years. The grandma is twice as
old as the mother. Jessica’s mother is 28 years older
than Jessica. How old is grandma?
1.9 The sum of Eric’s age and Lucas’s age is 65. Two
times Eric’s age is the same as three times Lucas’s
age. Find the ages of the men.
1.10 Michael’s age is multiplied by 7. Then 9 is added.
The result is 93. How old is Michael?
Use of the Words “More Than”

and “Less Than”

Beware of keywords! Many people believe that if they see
“more than” in a word problem they must add and with “less
than” they must subtract. That is not always the case. Look at
the following examples:

Simple Equation Problems 17

Examples:
a) Sue weighs 5 pounds more than Amanda. If Amanda
weighs 103 pounds, how much does Sue weigh?
b) Sue weighs 5 pounds less than Amanda. If Sue weighs
103 pounds, how much does Amanda weigh?
c) A school has an enrollment of 2381 students. This
is 53 students less than last year. What was the
enrollment last year?
d) A school had an enrollment of 2381 students last
year. This is 53 students less than this year. What is
the enrollment this year?
In each of these examples should you add or subtract?
Solutions:
a) Who weighs more? Sue. So we add 5 pounds to
Amanda’s 103 pounds.
b) Who weighs more? Amanda. So we add 5 pounds
to Sue’s 103 pounds
c) Is the enrollment larger this year? No, it is smaller
by 53 students. We add 53 students to 2381.
d) Last year’s enrollment was smaller than this year’s.