Dealing with Word Problems Study Guide (page 4)
Introduction to Dealing with Word Problems
A mathematician is a blind man in a dark room looking for a black cat, which isn't there.
—CHARLES R. DARWIN, English naturalist (1809–1882)
Word problems abound both on math tests and in everyday life. This lesson will show you some straightforward approaches to making word problems easier. The practice problems in this lesson incorporate the various kinds of math you have already studied in this book.
A word problem tells a story. It may also present a situation in terms of numbers or unknowns or both. (An unknown, also called a variable, is a letter of the alphabet that is used to represent an unknown number.) Typically, the last sentence of the word problem asks you to answer a question. Here's an example: Last week, Jason earned $57, and Karen earned $82. How much more money did Karen earn than Jason? Word problems involve all the concepts covered in this book:
- Arithmetic (whole numbers, fractions, decimals)
- Ratios and proportions
- Probability and counting
Doing all the problems in these two chapters is a good way to review what you have learned in the previous lessons.
Steps to Solving a Word Problem
While some simple word problems can be solved by common sense or intuition, most require a multistep approach as follows:
- Read a word problem in chunks rather than straight through from beginning to end. As you read each chunk, stop to think about what it means. Make notes, write an equation, label an accompanying diagram, or draw a picture to represent that chunk. You may even want to underline important information in a chunk. Repeat the process with each chunk. Reading a word problem in chunks rather than straight through prevents the problem from becoming overwhelming, and you won't have to read it again to answer it.
- When you get to the actual question, it. This will keep you more focused as you solve the problem.
- If it's a multiple-choice question, glance at the answer choices for clues. If they're fractions, you probably should do your work in fractions; if they're decimals, you should probably work in decimals; and so on.
- Make a plan of attack to help you solve the problem. That is, figure out what information you already have and how you're going to use it to develop a solution.
- When you get your answer, reread the circled question to make sure you've answered it. This helps you avoid the careless mistake of answering the wrong question. Test writers love to set set traps: Multiple questions often include answers that reflect the most common mistakes test takers make.
- Check your work after you get an answer. In a multiple-choice test, test takers get a false sense of security when they get an answer that matches one of the given answers. But even if you're not taking a multiple-choice test, you should always check your work if you have time. Here are a few suggestions:
- Ask yourself if your answer is reasonable, if it makes sense.
- Plug your answer back into the problem to make sure the problem holds together.
- Do the question a second time, but use a different method.
If a multiple-choice question stumps you, try one of the backdoor approaches, working backward or nice numbers, explained in the next lesson.
Translating Word Problems
The hardest part of any word problem is translating from 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. The translation rules follow this explanation.
When reading a word problem, cross out all the unnecessary words that do not directly pertain to the math. They only serve as a distraction, and your process will be more visible when they're removed.
EQUALS Key words: is, are, has
|Bob is 18 years old.||B = 18|
|There are 7 hats.||h = 7|
|Judi has 5 books.||J = 5|
ADD 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|
|Al threw the ball 8 feet further than Mark.||A = 8 +M|
|The total of three numbers is 25.||a + b + c = 25|
|How much do Joan and Tom have altogether?||J + T = ?|
SUBTRACT Key words: difference; fewer, less, or younger than; remain; left over
|The difference between two numbers is 17.||x – y = 17|
|Jay is 2 years younger than Brett.||J = B – 2 (NOT 2 – B)|
|After Carol ate 3 apples, r apples remained.||r = a – 3|
|Mike has 5 fewer cats than twice the number Jan has.||M = 2J – 5|
MULTIPLY Key words: of, product, times, twice
|25% of Matthew's baseball caps||0.25 × m, or 0.25m|
|Half of the boys|
|The product of two numbers is 12.||a × b = 12, or ab = 12|
|Murray has twice as many cards as Tina.||M = 2T|
|[Notice that it isn't necessary to write the times symbol (×) when multiplying by an unknown.|
DIVIDE Key word: per
|15 blips per 2 bloops|
|60 miles per hour|
|22 miles per gallon|
Converting Decimals and Fractions into Time: Remember, decimals are based on units of 10, while seconds and minutes are based on units of 60, so be careful when converting information in word problems! When faced with time as a decimal or fraction, multiply the fraction or decimal portion by 60 to convert it into seconds.
DISTANCE FORMULA: DISTANCE = RATE × TIME
Look for key words like plane, train, boat, car, walk, run, climb, swim, travel, and move.
How far did the plane travel in 4 hours if it averaged 300 miles per hour?
d = 300 × 4
d = 1,200 miles
Ben walked 20 miles in 4 hours. What was his average speed?
20 = r × 4
5 miles per hour = r
Using the Translation Rules
Here's an example of how to solve a word problem using the translation table.
Example: Carlos ate of the jelly beans. Maria then ate of the remaining jelly beans, which left 10 jelly beans. How many jelly beans were there to begin with?
Here's how we marked up the question and took notes as we read it. Notice how we used abbreviations to cut down on the amount of writing. Instead of writing the names of the people who ate jelly beans, we used only the first letter of each name; we wrote the letter j instead of the longer word, jelly bean.
Example: Carlos ate of the jelly beans. Maria then ate of the remaining jelly beans, which left 10 jelly beans. jelly beans were there
M = remaining
The following straightforward approach assumes a knowledge of fractions and elementary algebra. With the previous lessons under your belt, you should have no problem using this method. However, the same problem is presented in the next lesson, but it is solved by a backdoor approach, working backward, which does not involve algebra.
What we know:
- Carlos and Maria each ate jelly beans.
- Carlos ate of the jelly beans.
- Maria then ate of the jelly beans that Carlos left.
- Afterward, there were 10 jelly beans.
The question itself: How many jelly beans were there to begin with?
Plan of attack:
- Find out how many jelly beans Carlos and Maria each ate.
- Add 10, the number of jelly beans that were finally left, to get the number of jelly beans they started with.
Let's assume there were j jelly beans when Carlos started eating them. Carlos ate of them, or jelly beans (of means multiply). Since Maria ate a fraction of the remaining jelly beans, we must subtract to find out how many Carlos left for her: j – = j. Maria then ate of the j. jelly beans Carlos left her, or × j. jelly beans, which is .Altogether, Carlos and Maria ate + jelly beans, or jelly beans. Add the number of jelly beans they both ate ()to the 10 leftover jelly beans to get the number of jelly beans they started with, and solve the equation:
10 = j –
60 = j
Thus, there were 60 jelly beans to begin with.
Check: We can most easily do this by plugging 60 back into the original problem and seeing if the whole thing makes sense.
Carlos ate of 60 jelly beans. Maria then ate of the remaining jelly beans, which left 10 jelly beans. How many jelly beans were there to begin with?
Carlos ate of 60 jelly beans, or 20 jelly beans (×60=20).That left 40 jelly beans for Maria (60 – 20 = 40). She then ate of them, or 30 jelly beans ( × 40 = 30). That left 10 jelly beans (40 – 30=10), which agrees with the problem.
The next time you walk into a clothing or department store, bring a notepad and look around for a discount sign of a percentage taken off the regular price of a product. First, write down the full price of the item. Then, create a word problem that asks what dollar amount you'd save if you bought the item with that percentage discount. After you create the word problem, try solving it.
Dealing with Word Problems Sample Question
Four years ago, the sum of the ages of four friends was 42 years. If their ages were consecutive numbers, what is the current age of the oldest friend?
Solutions to Sample Question
Here's how to mark up the problem:
Four years ago, the sum of the ages of four friends was 42 years. If their ages were consecutive numbers, what is the
What we know:
- Four friends are involved.
- Four years ago, the sum (which means add) of their ages was 42.
- Their ages are consecutive (that means numbers in sequence, like 4, 5, 6, etc.).
The question itself:
How old is the oldest friend now?
Plan of attack:
Use algebra or trial and error to find out how old the friends were four years ago. After finding their ages, add them up to make sure they total 42. Then add 4 to the oldest to find his current age.
Let the consecutive ages of the four friends four years ago be represented by: f, f + 1, f + 2, and f + 3. Since their sum was 42 years, write and solve an equation to add their ages:
f + f + 1+ f + 2 + f + 3 = 42
4f + 6 = 42
4f = 36
f = 9
Since f represents the age of the youngest friend four years ago, the youngest friend is currently 13 years old (9 + 4 = 13). Since she is 13, the ages of the four friends are currently 13, 14, 15, and 16. Thus, the oldest friend is currently 16.
Add up the friends' ages of four years ago to make sure the total is 42: 9 + 10 + 11+ 12 = 42. Check the rest of your arithmetic to make sure it's correct.
Find practice problems and solutions for these concepts at Dealing with Word Problems Practice Questions.
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