In this section, we decide which operation to perform in order to solve a problem.
Don’t forget the key words and phrases that help indicate which operation to use.
Some of these are listed below and were introduced earlier in the chapter. Also included are several words and phrases that translate to the symbol “ =”.
Addition
sum difference product quotient equals
plus minus times divide is equal to
added to subtract multiply shared equally is/was
more than less than multiply by among yields
increased by decreased by of divided by
total less double/triple divided into
The following problem-solving steps may be helpful to you:
Problem-Solving Steps
1. UNDERSTAND the problem. Some ways of doing this are to read and reread the problem, construct a drawing, and look for key words to identify an operation.
2. TRANSLATE the problem. That is, write the problem in short form using words, and then translate to numbers and symbols.
3. SOLVE the problem. It is helpful to estimate the solution by rounding. Then carry out the indicated operation from step 2.
4. INTERPRET the results. Check the proposed solution in the stated problem and state your conclusions. Write your results with the correct units attached.
Example 1
Calculating the Length of a RiverThe Hudson River in New York State is 306 miles long. The Snake River in the northwestern United States is 732 miles longer than the Hudson River. How long is the Snake River? (Source: U.S. Department of the Interior)
Solution:
1. UNDERSTAND. Read and reread the problem, and then draw a picture.
Notice that we are told that Snake River is 732 miles longer than the Hudson River. The phrase “longer than” means that we add.
306 miles
Bank of America Transamerica Pyramid
Objectives
Solve Problems by Adding, Subtracting, Multiplying, or Dividing Whole Numbers.
Solve Problems That Require More Than One Operation.
PRACTICE 1
The Bank of America Building is the second-tallest building in San Francisco, California, at 779 feet. The tallest building in San Francisco is the
Transamerica Pyramid, which is 74 feet taller than the Bank of America Building. How tall is the Transamerica Pyramid?
(Source: The World Almanac)
Answer 1. 853 ft Continued on next page
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2. TRANSLATE.
In words:
Translate:
3. SOLVE: Let’s see if our answer is reasonable by also estimating. We will estimate each addend to the nearest hundred.
700
the Hudson River longer than
732 miles is
Snake River
Example 2
Filling a Shipping OrderHow many cases can be filled with 9900 cans of jalapeños if each case holds 48 cans? How many cans will be left over? Will there be enough cases to fill an or-der for 200 cases?
Solution:
1. UNDERSTAND. Read and reread the problem. Draw a picture to help visual-ize the situation.
Since each case holds 48 cans, we want to know how many 48s there are in 9900. We find this by dividing.
2. TRANSLATE.
In words:
Translate:
3. SOLVE: Let’s estimate a reasonable solution before we actually divide. Since 9900 rounded to the nearest thousand is 10,000 and 48 rounded to the nearest ten is 50, Now find the exact quotient.
206 R 12 Number of cases
T Number of cases
rounds to rounds to
exact estimate
PRACTICE 2
Four friends bought a lottery ticket and won $65,000. If each person is to receive the same amount of money, how much does each person receive?
Answer 2. $16,250
9900 cans
48 cans 48 cans 48 cans
4. INTERPRET. Check your work. The answer is reasonable since 1038 is close to our estimated answer of 1000. State your conclusion: The Snake River is 1038 miles long.
Work Practice 1
S E C T I O N 1 . 8 I AN INTRODUCTION TO PROBLEM SOLVING
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4. INTERPRET. Check your work. The answer is reasonable since 206 R 12 is close to our estimate of 200. State your conclusion: 206 cases will be filled, with 12 cans left over. There will be enough cases to fill an order for 200 cases.
Work Practice 2
Answers
3. $1700 4. $53,400
PRACTICE 3
The director of the learning lab also needs to include in the budget a line for 425 blank CDs at a cost of $4 each. What is this total cost for the blank CDs?
From the phrase “total cost,” we might decide to solve this problem by adding.
This would work, but repeated addition, or multiplication, would save time.
2. TRANSLATE.
In words:
Translate:
3. SOLVE: Once again, let’s estimate a reasonable solution.
200
4. INTERPRET. Check your work. State your conclusion: The total cost of the DVD players is $6171.
Work Practice 3
$ 187 $ 187
33 DVD Players
$ 187
Example 4
Calculating a Public School Teacher’s SalaryIn 2008, the average salary of a public school teacher in California was $64,424.
For the same year, the average salary for a public school teacher in Iowa was
$17,760 less than this. What was the average public school teacher’s salary in Iowa? (Source: National Education Association)
Solution:
1. UNDERSTAND. Read and reread the problem. Notice that we are told that the Iowa salary is $17,760 less than the California salary. The phrase “less than”
indicates subtraction.
PRACTICE 4
In 2008, the average salary for a public school teacher in Alaska was $56,758. For the same year, the average salary for a public school teacher in Hawaii was
$3358 less than this. What was the average public school teacher’s salary in Hawaii?
(Source: National Education Association)
Continued on next page
Example 3
Calculating Budget CostsThe director of a learning lab at a local community college is working on next year’s budget. Thirty-three new DVD players are needed at a cost of $187 each.
What is the total cost of these DVD players?
Solution:
1. UNDERSTAND. Read and reread the problem, and then draw a diagram.
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2. TRANSLATE. Remember that order matters when subtracting, so be careful when translating.
In words:
Translate:
3. SOLVE. This time, instead of estimating, let’s check by adding.
Check:
4. INTERPRET. Check your work. The check is above. State your conclusion:
The average Iowa teacher’s salary in 2008 was $46,664.
Work Practice 4