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Solving Word Problems

Word Problems
Many of the math problems on tests are word problems.

A word problem can include any kind of math, including simple arithmetic, fractions, decimals, percentages, even algebra and geometry.
The hardest part of any word problem is translating English into math.

When you read a problem, you can frequently translate it word for word from English statements into mathematical statements.

At other times, however, a key word in the word problem hints at the mathematical operation to be performed.

Here are the translation rules:
EQUALS key words: is, are, has

English - Math
 

Bob is 18 years old.
B = 18

There are 7 hats.
H = 7

Judi has 5 books.
J = 5

ADDITION key words: sum; more, greater, or older than; total; altogether

The sum of two numbers is 10.
X + Y = 10

Karen has $5 more than Sam.
K = 5 + S

The base is 3'' greater than the height.
B = 3 + H

Judi is 2 years older than Tony.
J = 2 + T

The total of three numbers is 25.
A + B + C = 25

How much do Joan and Tom have altogether?
J + T = ?

SUBTRACTION key words: difference, less or younger than, remain, left over

The difference between two numbers is 17.
X + Y = 17

Mike has 5 less cats than twice the number Jan has.
M = 2J - 5

Jay is 2 years younger than Brett.
J = B - 2

After Carol ate 3 apples, R apples remained.
R = A - 3

MULTIPLICATION key words: of, product, times

20% of Matthew's baseball caps
.20 × M

Half of the boys
1/2 × B

The product of two numbers is 12
A × B = 12

DIVISION key word: per

15 drops per teaspoon


22 miles per gallon


Distance Formula: Distance = Rate × Time

The key words are movement words like: plane, train, boat, car, walk, run, climb, swim

How far did the plane travel in 4 hours if it averaged 300 miles per hour?
D = 300 × 4
D = 1200 miles

Ben walked 20 miles in 4 hours. What was his average speed?
20 = r × 4
5 miles per hour = r

Solving a Word Problem Using the Translation Table
Remember the problem at the beginning of this chapter about the jellybeans?

Q: Juan ate 1/3 of the jellybeans. Maria then ate 3/4 of the remaining jellybeans, which left 10 jellybeans.
How many jellybeans were there to begin with?
A: We solved it by working backwards. Now let's solve it using our translation rules.

Assume Juan started with J jellybeans. Eating 1/3 of them means eating 1/3 × J jellybeans. Maria ate a fraction of the remaining jellybeans, which means we must subtract to find out how many are left:
J - 1/3 ×
J = 2/3 ×
J. Maria then ate 3/4, leaving 1/4 of the 2/3 ×
J jellybeans, or 1/4 × 2/3 ×
J jellybeans.
Multiplying out 1/4 × 2/3 ×
J gives 1/6
J as the number of jellybeans left. The problem states that there were 10 jellybeans left, meaning that we set 1/6 ×
J equal to 10:


Solving this equation for J gives J = 60.

Thus, the right answer is a (the same answer we got when we worked backwards). As you can see, both methodsworking backwards and translating from English to mathwork. You should use whichever method is more comfortable for you.

Practice Word Problems
Practice using the translation table above on problems that just require you to work with basic arithmetic. 

1. Joan went shopping with $100 and returned home with only $18.42. How much money did she spend? a. $81.58 b. $72.68 c. $72.58 d. $71.68 e. $71.58
2. Mark invited ten friends to a party. Each friend brought 3 guests. How many people came to the party, excluding Mark? a. 3 b. 10 c. 30 d. 40 e. 41
3. The office secretary can type 80 words per minute on his word processor. How many minutes will it take him to type a report containing 760 words? a. 8 b. 8 1/2 c. 9 d. 9 1/2 e. 10
4. Mr. Wallace is writing a budget request to upgrade his personal computer system. He wants to purchase 4 mb of RAM, which will cost $100, two new software programs at $350 each, a tape backup system for $249, and an additional tape for $25. What is the total amount Mr. Wallace should write on his budget request? a. $724 b. $974 c. $1049 d. $1064 e. $1074