1.09
Introduction
to Word Problems
from Basic Algebra: One Step at
a Time © 2002
P.
57–62
Dr. Robert J. Rapalje
Seminole Community College
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
“What
good is equation solving?” That’s
a common question at this point! An
even better question is “What good is math?”!
Math and likewise equation solving are seldom ends in themselves,
but rather they are tools or stepping stones to something else.
Math is a tool for the sciences, business, and technology, for
example, and it is not really very useful at all until you need it to
solve a problem in real life. Unfortunately,
real life problems do not usually come with equations.
Rather, you must take the situation in sentences (words) and
translate it into mathematics (equations).
In
this sense, word problems (applications) are like a foreign language.
(Do you identify with that statement?)
There is a vocabulary of words to be translated from English into
math symbols (or equation!). Moreover,
there is a definite sentence structure, even as there is when
translating from one language to another.
You must first know the vocabulary.
The following is an English to math dictionary that may help you
get started.
ENGLISH
to MATH DICTIONARY
English
Mathematics (operation)
“Decreased by”
“–” (be careful
of the order)
“Difference”
“–” (be careful of the order)
“Equals”
“=”
“Increased
by” “+”
“Is” “=”
“Less
than”**
“–” (be careful of the order)
“More
than”**
“+”
“Of”
“X”, “·”, “times”
“Perimeter”
“Sum of the sides”
“Product”
“X”, “·”, “times”
“Quotient”
“÷”
“Such
that” “Get ready––here comes
the equation!”
“Sum” “+”
“Times”
“X”, “·”, “times”
“Twice”
“Two times” or “2·_____”
**NOTE:
“Less than” and “more than” may also refer to
inequalities, which will be explained in
Section 1.11.
Question #1: If
a jar contains 15 marbles, and 3 marbles are taken out of the jar, how many marbles are left in the
jar? _____
What operation
did you use? _____________.
Answer:
12 marbles,
and the operation was subtraction.
Question
#2: What
if the jar contains 15 marbles,
and 5 are taken out. How
many marbles are left? _______.
Answer:
10 marbles.
Question
#3: What
if the jar contains 15 marbles,
and x (an unknown number) marbles are removed.
Now what operation is used, and how many marbles are left?
______________________.
Answer:
The
operation is subtraction.
There
will be “15 ‘take away’ x” marbles
left. That is, there will be “15 – x”
marbles.
Question
#4: Suppose
a board that is 15 feet long is
cut so that one piece is 3 feet
long. How long is the other
piece? _____
What operation
was used? __________.
Answer:
12 feet,
and the operation was subtraction.
Question
#5: What
if the board was 15 feet long,
and the piece cut off is 5 feet
long. How long is the other piece? _______.
Answer:
10 feet.
Question
#6: What
if the board is 15 feet long,
and a piece of board that is x feet
long is cut off. How long is
the other piece? _____. What operation was used?
__________.
Answer:
Subtraction,
and the other piece is “15 – x”
feet.
Question
#7: Suppose
you have $15,000 to invest in
two separate investments (or accounts).
If you place $3,000 in
one account, how much is in the other account? _______.
Answer:
$12,000.
Question
#8: If
you have a total of $15,000,
and you place $5,000 in one
account, how much is in the other account? _______.
Answer:
$10,000.
Question
#9: Now,
if you have a total of $15,000,
and you place an unknown x
dollars in one account, how much is in the other account?
Answer:
15,000 – x.
It
may seem that these are three different problems: a marble problem, a
board problem, and an investment problem.
However, if you can see above the specifics involved, there is
really just one type of problem. The
word problems of this chapter will be like this.
Try to see above the specifics, and recognize the broad categories
of problems.
Before
attempting the real applications, it will be helpful to practice with a
few examples and exercises in vocabulary––that is, translating from
English words to math symbols.
EXAMPLES:
Let x represent the unknown number, and translate into math
symbols.
ANSWERS
1.
Three more than an unknown number
x + 3
or 3 + x
2. Three
less than an unknown number
x – 3
(Not 3 – x !!)
3.
The sum of an unknown number and eight
x + 8
4.
Twice an unknown number
2x
5.
The sum of 4 and twice an unknown number
4 + 2x
or 2x + 4
6. Four
times an unknown number, plus 5
4x + 5
7.
Four times an unknown number, less 5
4x – 5
8.
Five less than four times an unknown number
4x – 5
9.
Five more than four times an unknown number
4x + 5
10.
The product of a number and four more than the number
x(x
+ 4)
11.
The product of a number and four less than the number
x(x – 4)
12.
If there are a total of 20 marbles in a dish,
20 – x
and someone takes x of them, how many are left?
13.
If there are a total of 20 coins, in dimes and quarters in a box,
20 – x
and x are dimes, how many quarters are in the box?
14.
If a person has $20 and he/she spends $x, how much is left?
20 – x
EXERCISES:
Translate each of the following statements into math symbols.
Let x represent the unknown number.
1. 6 more than x
______________
2. 15 more than x
______________
3. 6 less than x
______________
4.
15 less than x
______________
5.
4 increased by n
______________
6.
n increased by 4
______________
7.
12 decreased by n ______________
8.
n decreased by 12 ______________
9.
x less 8
______________
10.
8 less x
______________
11.
5 less than x
______________
12.
x less than 5
______________
13.
Twice an unknown number ______________
14.
Twice an unknown number, less 5 ______________
15.
5 more than twice an unknown number
______________
16.
5 less than twice an unknown number
______________
17.
8 less than twice an unknown number
______________
18.
Four times an unknown number less 3
______________
19.
Four times an unknown number increased by 3
______________
20.
Four times an unknown number decreased by 3
______________
21.
Five less than four times an unknown number
______________
22.
Ten decreased by five times an unknown number
______________
23.
If you have $50 and you spend $x, how much is left?
______________
24.
If there are 50 marbles in a box, and you take out
x marbles, how many are left?___________
25.
There are a total of 50 coins, some nickels and the rest
are dimes. If there
are x nickels, how many dimes are there?
_______________
26.
There are two accounts totaling $50,000 with $ x in the
first account. How
much money is in the second account?
_______________
27.
A piece of string is 50 centimeters in length.
It is cut into two pieces, such that the first piece is x centimeters in length.
How long is the second piece? __________
28.
There are a total of 38 coins, some nickels and the rest dimes.
If there are x dimes, how many nickels are there?
______________
29.
If you spend $ x out of $200, how much is left?
______________
30.
If you have $ x, and you spend $200, how much is left?_____________
31.
If you have $ x, and you spend $35, how much is left?
_____________
32.
If you spend $ x out of $35,
how much is left?
______________
ANSWERS 1.09
p. 61 –
62:
1.
x+6;
2. x+15;
3. x–6;
4. x–15; 5. 4+n;
6. n+4; 7. 12–n; 8. n–12;
9. x–8; 10. 8–x;
11. x–5;
12. 5–x;
13. 2x;
14. 2x–5; 15. 2x+5; 16. 2x–5; 17.
2x–8; 18. 4x–3; 19. 4x+3; 20. 4x–3;
21. 4x–5; 22. 10–5x; 23. 50–x; 24. 50–x; 25. 50–x; 26.
50000–x; 27. 50–x; 28. 38–x; 29.
200–x; 30. x–200; 31. x–35; 32. 35–x.
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