WORD PROBLEMS

Key Words

The following key words help you understand the relationships between the pieces of information given and give you clues as to how the problem should be solved.

Add

• Addition: as in The team needed the addition of three new players …

• Sum: as in The sum of 5, 6, and 8 …

• Total: as in The total of the last two games …

• Plus: as in Three chairs plus five chairs …

• Increase: as in Her pay was increased by $30 …

Subtract

• Difference: as in What is the difference between 8 and 5 …

• Fewer: as in There were ten fewer girls than boys …

• Remainder: as in What is the remainder when … or How many are left when …

• Less: as in A number is six less than another number …

• Reduced: as in His allowance was reduced by $5 …

• Decreased: as in What number decreased by 7 is 5 …

• Minus: as in Seven minus some number is …

Multiply

• Product: as in The product of 3 and 6 is …

• Of: as in One‐half of the people in the room …

• Times: as in Six times as many men as women …

• At: as in The cost of five yards of material at $9 a yard is …

• Total: As in If you spend $20 per week on gas, what is the total for a two‐week period …

• Twice: as in Twice the value of some number … (multiplying by 2)

Divide

• Quotient: as in The final quotient is …

• Divided by: as in Some number divided by 5 is …

• Divided into: as in The coins were divided into groups of …

• Ratio: as in What is the ratio of …

• Half: as in Half of the cards were … (dividing by 2)

As you practice working word problems, you will discover more key words and phrases that give you insight into the solving process.

Example 1

Jack bowled four games for a total score of 500. What was his average score for a game?

• Step 1: Find and underline or circle what the question is asking.

What was his average score for a game?

• Step 2: Focus on and pull out important information.

four games for a total score of 500

• Step 3: Set up the work that is needed.

500 ÷ 4

(The total divided by the number of games gives the average.)

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Jack's average score for a game is 125.

• Step 6: Check to make sure that your answer is reasonable.

Since four games of 125 total 500, your answer is reasonable and correct.

Example 2

Judy scored 85, 90, and 95 on her last three algebra tests. What was her average score for these tests?

• Step 1: Find and underline or circle what the question is asking.

What was her average score for these tests?

• Step 2: Focus on and pull out important information.

85, 90, and 95 on three tests

• Step 3: Set up the work that is needed.

(85 + 90 + 95) ÷ 3 =

(The total divided by the number of scores gives the average.)

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Judy's average test score for these tests was 90.

• Step 6: Check to make sure that your answer is reasonable.

Since her scores were 85, 90, and 95, the average should be halfway between 85 and 95. So 90 is a reasonable answer.

Example 3

Frances goes to the market and buys two boxes of cereal at $4 each, three bottles of milk at $2 each, and two cans of soup at $1 each. How much change will Frances get from a $20 bill?

• Step 1: Find and underline or circle what the question is asking.

How much change will Frances get from a $20 bill?

• Step 2: Focus on and pull out important information.

two boxes at $4 each

three bottles at $2 each

two cans at $1 each

$20 bill used

• Step 3: Set up the work that is needed.

2 × 4 =

3 × 2 =

2 × 1 =

$20 – ? =

• Step 4: Do the necessary work or computation carefully.

2 × 4 = 8

3 × 2 = 6

2 × 1 = 2

8 + 6 + 2 = 16

20 – 16 = 4

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Frances will get $4 change.

• Step 6: Check to make sure that your answer is reasonable.

Since the total expenses were $16, then $4 change from a $20 bill is reasonable.

Example 4

Sarah can purchase a television for $275 cash or for $100 as a down payment and ten monthly payments of $30 each. How much money can Sarah save by paying cash for the television?

• Step 1: Find and underline or circle what the question is asking.

How much money can Sarah save by paying cash for the television?

• Step 2: Focus on and pull out important information.

cash $275

$100 down plus ten payments of $30

• Step 3: Set up the work that is needed.

100 + (10 × 30) =

? – 275 =

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Sarah can save $125 by paying cash.

Step 6: Check to make sure that your answer is reasonable.

Ten payments of $30 each is $300 plus a $100 down payment gives $400. This is $125 more than $275, so the answer is reasonable.

Example 5

If apples sell for $3.25 per dozen, how many apples can Maria buy for $13.00?

• Step 1: Find and underline or circle what the question is asking.

How many apples can Maria buy for $13.00?

• Step 2: Focus on and pull out important information.

$3.25 per dozen

$13.00

• Step 3: Set up the work that is needed.

$13.00 ÷ 3.25 = ? dozen

= ? apples

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Maria can buy 48 apples for $13.00.

• Step 6: Check to make sure that your answer is reasonable.

Since Maria could buy 12 apples for about $3, then it is reasonable that she could buy 48 apples for about $13.

Example 6

Sequoia Junior High School has a student enrollment of 2,000. If 30% of the students are seventh graders, how many seventh graders are enrolled at the school?

• Step 1: Find and underline or circle what the question is asking.

How many seventh graders are enrolled at the school?

• Step 2: Focus on and pull out important information.

2,000 students

30% are seventh graders

• Step 3: Set up the work that is needed.

30% of 2,000 =

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

There are 600 seventh graders enrolled at Sequoia Junior High.

• Step 6: Check to make sure that your answer is reasonable.

Since 30% of 1,000 is 300, then 30% of 2,000 is 600. The answer is reasonable.

Example 7

Jim Chamberlain, the center for the West Hills Basketball Stars, makes 75% of his free throws. If Jim attempts 80 free throws in a season, how many of his free throws does he make?

• Step 1: Find and underline or circle what the question is asking.

How many of his free throws does he make?

• Step 2: Focus on and pull out important information.

makes 75%

shoots 80 free throws

• Step 3: Set up the work that is needed.

75% of 80 =

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Jim makes 60 free throws.

• Step 6: Check to make sure that your answer is reasonable.

Since 50% of his 80 free throws would be 40, then it's reasonable for 75% to be 60.

Example 8

Each week, John spends $50 of his income on entertainment. If John earns $200 a week, what percent of his income is spent on entertainment?

• Step 1: Find and underline or circle what the question is asking.

What percent of his income is spent on entertainment?

• Step 2: Focus on and pull out important information.

$50 on entertainment

$200 income

• Step 3: Set up the work that is needed.

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

John spends 25% of his income on entertainment.

• Step 6: Check to make sure that your answer is reasonable.

Since 50 is half, or 50% of 100, then 50 would reasonably be 25% of 200.

Example 9

The Gomez family spends 30% of its income for food. If the family spent $6,000 for food last year, what was the family income for last year?

• Step 1: Find and underline or circle what the question is asking.

What was the family income for last year?

• Step 2: Focus on and pull out important information.

30% of income for food

$6,000 spent for food last year

• Step 3: Set up the work that is needed.

30% of income = 6,000

30% x = 6,000

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

The Gomez family income for last year was $20,000.

• Step 6: Check to make sure that your answer is reasonable.

Since 30% of the family income is spent for food, and 30% of $20,000 is $6,000, then the answer is reasonable.

Example 10

A miniature piano keyboard is  inches wide. If each key is  inches wide, how many keys are there?

• Step 1: Find and underline or circle what the question is asking.

How many keys are there?

• Step 2: Focus on and pull out important information.

 inches wide

 inch keys

• Step 3: Set up the work that is needed.

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

There are 11 keys on the miniature keyboard.

• Step 6: Check to make sure that your answer is reasonable.

Since 16 divided by 1 is 16, and 16 divided by 2 is 8, the answer of 11 is reasonable.

Example 11

The low temperature on Big Bear mountain was 30 degrees on Monday, 20 degrees on Tuesday, ‐10 degrees on Wednesday, and 15 degrees on Thursday. If you total the changes in low temperature from each day to the next, what is the total number of degrees change?

• Step 1: Find and underline or circle what the question is asking.

What is the total number of degrees change?

• Step 2: Focus on and pull out important information.

30 on Monday

20 on Tuesday

–10 on Wednesday

15 on Thursday

• Step 3: Set up the work that is needed.

30 to 20 =

20 to –10 =

–10 to 15 =

• Step 4: Do the necessary work or computation carefully.

30 to 20 = –10

20 to –10 = –30

–10 to 15 = 25

total change = –10 + –30 + 25

= –15

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

The low temperature changed –15 degrees during the days Monday through Thursday.

• Step 6: Check to make sure that your answer is reasonable.

Since the low temperature dropped 10, dropped 30, and rose 25, the total of –15 is correct and reasonable.

Example 12

The Silverado Flash, a solar‐powered land vehicle, travels at a maximum speed of 97 miles per hour. At this rate, how far will the Silverado Flash travel in 15 hours?

An important formula to remember is d = rt, or distance equals rate times time.

• Step 1: Find and underline or circle what the question is asking.

How far will the Silverado Flash travel in 15 hours?

• Step 2: Focus on and pull out important information.

97 miles per hour

15 hours

• Step 3: Set up the work that is needed.

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

The Silverado Flash will travel 1,455 miles in 15 hours.

• Step 6: Check to make sure that your answer is reasonable.

At 100 miles per hour for 15 hours, the Silverado Flash would have traveled 1,500 miles. So the answer of 1,455 miles is reasonable.

Example 13

Asaf can run around the track, 440 yards, in 65 seconds. At this same rate, how far could Asaf run in 195 seconds?

• Step 1: Find and underline or circle what the question is asking.

How far could Asaf run in 195 seconds?

• Step 2: Focus on and pull out important information.

440 yards in 65 seconds

195 seconds

• Step 3: Set up the work that is needed.

You could set up the proportion

or simply divide 195 by 65 and multiply that result by 440.

(195 ÷ 65) × 440 = ?

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Asaf could run 1,320 yards in 195 seconds.

• Step 6: Check to make sure that your answer is reasonable.

✕Powered by ExploreadsIf Asaf ran 400 yards every 60 seconds, or one minute, and since 195 seconds is just over three minutes, then 3 times 400, or 1,200, would be a good approximation. So 1,320 yards is a reasonable answer.

Example 14

A researcher tagged 100 frogs in a nearby pond. One week later she took a sample and only 5 out of 20 frogs were tagged. Using this method, how many frogs would she approximate are in the pond?

• Step 1: Find and underline or circle what the question is asking.

How many frogs would she approximate are in the pond?

• Step 2: Focus on and pull out important information.

100 tagged

5 out of 20 tagged

• Step 3: Set up the work that is needed.

You could set up the proportion.

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

She would approximate that there are 400 frogs in the pond.

• Step 6: Check to make sure that your answer is reasonable.

Since she tagged 100, and 5 out of 20, or 1 out of 4, came out tagged, then it is reasonable to have a total of 400 frogs in the pond.

Example 15

Julia has $300 in the bank. She works at a bakery and makes $40 per day. If she deposits all of her earnings in the bank and does not make any withdrawals, how many days of work will it take for her to have $740 in the bank?

• Step 1: Find and underline or circle what the question is asking.

How many days of work will it take for her to have $740 in the bank?

• Step 2: Focus on and pull out important information.

$300 in the bank

$40 per day

$740 total

• Step 3: Set up the work that is needed.

Let d stand for the number of days. Then 40 d is the amount of money earned in d days, and 40 d + 300 is the amount of money she would have in the bank at the end of d days. So

40 d + 300 = 740

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

Julia would have to work 11 days to have $740 in the bank.

• Step 6: Check to make sure that your answer is reasonable.

At $40 per day, if she worked 11 days, she would have $440. Add this to the $300 she already had, and the total of $740 is correct and reasonable.

Example 16

A train is 50 miles from Seattle. It is traveling away from Seattle at a speed of 60 miles per hour. In how many hours will the train be 290 miles from Seattle?

• Step 1: Find and underline or circle what the question is asking.

In how many hours will the train be 290 miles from Seattle?

• Step 2: Focus on and pull out important information.

50 miles from Seattle

60 miles per hour

290 miles

• Step 3: Set up the work that is needed.

Let h stand for the number of hours. Then 60 h is the distance traveled in h hours, and 60 h + 50 is the distance from Seattle in h hours. So

60 h + 50 = 290

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

It will take the train four hours to be 290 miles from Seattle.

• Step 6: Check to make sure that your answer is reasonable.

At 60 miles per hour, in four hours the train will travel 240 miles. since it is starting 50 miles from Seattle, the total would be 290 miles.

Example 17

A number n is increased by 25, and the outcome is 57. What is the value of the number n?

• Step 1: Find and underline or circle what the question is asking.

What is the value of the number n ?

• Step 2: Focus on and pull out important information.

n is increased by 25

outcome is 57

• Step 3: Set up the work that is needed.

n + 25 = 57

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

The value of the number is 32.

• Step 6: Check to make sure that your answer is reasonable.

Since 32 plus 25 equals 57, the answer checks and is reasonable.

Example 18

The length of a Lambo Speed Wagon is 75 centimeters less than the length of a Corvette. The Lambo is 410 centimeters long. How long is a Corvette?

• Step 1: Find and underline or circle what the question is asking.

How long is a Corvette?

• Step 2: Focus on and pull out important information.

Lambo is 75 centimeters less

Lambo is 410 centimeters

• Step 3: Set up the work that is needed.

Let c represent the length of a Corvette. Then c –75 is the length of a Lambo. So

c – 75 = 410

• Step 4: Do the necessary work or computation carefully.

• Step 5: Put your answer into a sentence to make sure that you answered the question being asked.

The length of a Corvette is 485 centimeters.

• Step 6: Check to make sure that your answer is reasonable.

Since 485 – 75 is 410, the answer checks and is reasonable.