To review these concepts, go to Fractions, Percents,and Decimals in Word Problems Study Guide.

### Practice

- When Zoe's birthday party is over, of the cake remains. If Morgan eats of the cake, how much of the cake will be left?
- Sue has ounces of almond extract. If she divides it equally over four batches of fudge, how many ounces of almond extract are in each batch of fudge?
- Yves uses yards of string to tie a bundle of magazines. If he has six bundles to tie, how many yards of string does he need?
- Every student in Sayda's class must bring in square feet of fabric for an art project. If there are 24 students in the class, and her teacher brings in an additional square feet of fabric, how many total square feet of fabric will be used for the project?
- Patrick spends three-fourths of an hour studying for his math test and half an hour studying for his science test. How many total minutes did Patrick spend studying?
- The average rainfall per day for November in Sunnydale was 1.304 inches. How many total inches of rain fell in Sunnydale in November?
- Find 3.27 more than the product of 9.3 less than 12.007 and the difference between 3.33 and 0.1.
- A jar contains 0.346 liters of water. Every hour, 0.106 liters are added to the jar and 0.055 liters evaporate. How much water is in the jar after five hours?
- Elle makes $43.75 per day at her part-time job. If she wants to buy a stereo that costs $600, how many full days must she work?
- A multivitamin contains 0.015 grams of vitamin E. If a jar holds 36 vitamins, how many grams of vitamin E are in 7.5 jars of vitamins?

**Directions:** Use the following information to answer questions 11–13:

Dan buys a music album that is 120 minutes long.

- If Dan listens to 45% of the album, how many minutes of the album has he heard?
- If Dan listens to 48 minutes of the album, what percent of the album has he heard?
- If Dan has heard all but 15 minutes of the album, what percent of the album has he heard?
- Aiden hauled 9,800 pounds of sand on Monday. If he hauls 8,700 pounds on Tuesday, how much less, by percent, did he haul on Tuesday? Round your answer to the nearest tenth of one percent.
- Judi swims 25 laps every day. If she swims 12% more laps today, how many laps does she swim today?
- Kellyann was 35 inches tall when she was three years old. If she is 65 inches tall now, by what percent has her height increased? Round your answer to the nearest tenth of one percent.
- Jeffrey could bench-press 70 pounds in the seventh grade. A year later, he could press 20% more. As a ninth grader, he can press 25% more than he could as an eighth grader. How much can Jeffrey bench-press now?

### Solutions

- of a cake is shown here:
*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.- We are looking for 3.27 more than a number, which is ( ) + 3.27. That number is the product of two numbers: ( × ) + 3.27. The first of the two numbers is 9.3 less than 12.007, which is 12.007 – 9.3: [(12.007 – 9.3) × ] + 3.27. The other number is the difference between 3.33 and 0.1, which is 3.33 – 0.1. The number sentence is now [(12.007 – 9.3) × (3.33 – 0.1)] + 3.27. Do the subtraction in the innermost parentheses first: 12.007 – 9.3 = 2.707 and 3.33 – 0.1 = 3.23. The number sentence is now (2.707 × 3.23) + 3.27. Do the multiplication next: 2.707 × 3.23 = 8.74361. Finally, add 8.74361 and 3.27: 8.74361 + 3.27 = 12.01361.
- We can use a table to solve this problem. Each hour, we add 0.106 and subtract 0.055, and then carry the new volume of water into the next row of the table:
- We can use a table to solve this problem. Multiply the number of days worked by $43.75, increasing the number of days until the total earned reaches $600. We can tell that she must work more than just a few days, so we skip ahead to day five and then skip ahead to day ten. Once we see that the total is beginning to approach $600, we increase the number of days by one:
*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.*Read the entire word problem*.

If we remove one of those eighths, are left:

We are given the amount of almond extract that Sue has and the number of batches of fudge that she makes.

*Identify the question being asked*

We are looking for how much almond extract is in each batch.

*Underline the keywords*.

The keyword divides signals division.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

The keyword each also appears in this problem, but the word divides clearly tells us that we need to use division.

*Write number sentences for each operation*.

*Solve the number sentences and decide which answer is reasonable*.

The reciprocal of 4 is , so multiply by :

ounces

*Check your work*.

We solved this problem using division, so we must use multiplication to check our work. Multiply the number of ounces of almond extract in each batch by the number of batches, and that product should give us the total amount of almond extract: ounces.

We are given the amount of string Yves needs to tie one bundle of magazines and the number of bundles he has to tie.

*Identify the question being asked*.

We are looking for how many yards of string he needs.

*Underline the keywords*.

There are no keywords in this problem.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

We're given the number of yards of string needed to tie one bundle, so we will have to multiply that value by 6 to find the amount needed to tie six bundles.

*Write number sentences for each operation*.

× 6

*Solve the number sentences and decide which answer is reasonable*.

yards

*Check your work*.

We solved this problem using multiplication, so we must use division to check our work. Divide the total number of yards by the number of yards needed to tie one bundle. This should give us the number of bundles Yves tied: bundles.

We are given the amount of fabric each student must bring and the amount of fabric the teacher is bringing.

*Identify the question being asked*.

We are looking for the total square feet of fabric.

*Underline the keywords*.

The keyword every can signal multiplication or division, and the keyword additional signals addition.

*Cross out extra information and translate words into numbers*

There is no extra information in this problem.

*List the possible operations*.

If one student must bring square feet of fabric, then to find how many square feet 24 students must bring, we have to multiply by 24. Once we have that answer, we can add to it the amount of fabric the teacher brings.

*Write number sentences for each operation*.

× 24

*Solve the number sentences and decide which answer is reasonable*.

= 21 square feet

*Write number sentences for each operation*.

The students bring 21 square feet of fabric. Their teacher brings an additional square feet, so we must add that to 21 to find the total: + 21

*Solve the number sentences and decide which answer is reasonable*.

square feet

*Check your work*.

We solved this problem using multiplication and addition, so we must use subtraction and division to check our work. Subtract the number of square feet of fabric brought by the teacher from the total. This should give us the amount of fabric brought by the students: = 21 square feet. Divide 21 square feet by the number of children in the class. This should give us the number of square feet each student brought: square feet.

We are given the amount of time Patrick spent studying for his math test and for his science test.

*Identify the question being asked*.

We are looking for the total time, in minutes, spent studying.

*Underline the keywords*.

The keyword *total* signals addition.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem, but three-fourths of an hour and half an hour must be translated into fractions. Peter spent hours studying for his math test and hours studying for his science test. Also, we must express our answer in minutes, not hours. Remember, there are 60 minutes in an hour.

*List the possible operations*.

We must add the two fractions to find the total number of hours Patrick spent studying, and then multiply that sum by 60 to find the total number of minutes he spent studying.

*Write number sentences for each operation*.

*Solve the number sentences and decide which answer is reasonable*.

Convert halves to fourths and add:

hours

*Write number sentences for each operation*.

Multiply the number of hours by 60 to find the number of minutes Patrick spent studying:

× 60

*Solve the number sentences and decide which answer is reasonable*.

minutes

*Check your work*.

We solved this problem using addition and multiplication, so we must use division and subtraction to check our work. Divide the total number of minutes by 60 to convert the time spent studying from minutes to hours: . Subtract the number of hours Patrick spent studying for his math test, and we should be left with the number of hours he spent studying for his science test: hour.

We are given the average rainfall per day.

*Identify the question being asked*.

We are looking for the total rainfall for the month of November.

*Underline the keywords*.

The keywords *average* and *per* can signal multiplication or division.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem, but there are 30 days in the month of November. Replace the word *November* with "30 days."

*List the possible operations*.

Since we are given the amount of rainfall (on average) for one day and we are looking for the rainfall for 30 days, we must multiply.

*Write number sentences for each operation*.

1.304 × 30

*Solve the number sentences and decide which answer is reasonable*.

1.304 × 30 = 39.12 inches

*Check your work*.

We solved this problem using multiplication, so we will use division to check our answer. Divide the total rainfall by 30, the number of days in November. This should equal the average rainfall per day: = 1.304 inches.

After five hours, there are 0.601 liters in the jar.

After 13 days, Elle does not have enough for the stereo, so she must work 14 days to have enough money to buy the stereo.

We are given the number of grams of vitamin E in one vitamin, the number of vitamins in a jar, and the number of jars.

*Identify the question being asked*.

We are looking for the total number of grams of vitamin E.

*Underline the keywords*.

There are no keywords in this problem.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

Since we are given the number of grams in one vitamin, we must multiply to find the number of grams in 36 vitamins, or one jar of vitamins. We must multiply again to find the number of grams of vitamin E in 7.5 jars.

*Write number sentences for each operation*.

First, find the number of grams in one jar of 36 vitamins:

0.015 × 36

*Solve the number sentences and decide which answer is reasonable*.

0.015 × 36 = 0.54 grams

*Write number sentences for each operation*.

Now find the number of grams in 7.5 jars of vitamins:

0.54 × 7.5

*Solve the number sentences and decide which answer is reasonable*.

0.54 × 7.5 = 4.05 grams

*Check your work*.

We solved this problem using multiplication twice, so we will use division twice to check our answer. Divide the total number of grams of vitamin E by 7.5 to find the number of grams in one jar: = 0.54 grams. Divide this number by 36, the number of vitamins in one jar, and this should give us the number of grams in one multivitamin: = 0.015 grams. Our answer is correct.

We are given the length of the album in the directions, and we are given the percent of the album to which Dan has listened.

*Identify the question being asked*.

We are looking for the number of minutes of the album Dan has heard.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain a % symbol, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

Dan listens to 45% of a 120-minute album, so we must find 45% of 120.

*Write number sentences for each operation*.

Convert 45% to a decimal and multiply it by the length of the album:

120 × 0.45

*Solve the number sentences and decide which answer is reasonable*.

120 × 0.45 = 54 minutes

*Check your work*.

We multiplied to find our answer, so we will divide to check our work. Since Dan listened to 54 minutes of the album, which is 45% of the album, divide 54 by 0.45 to find the full length of the album: = 120 minutes.

We are given the length of the album in the directions, and we are given the number of minutes to which Dan has listened.

*Identify the question being asked*.

We are looking for the percent of the album Dan has heard.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain the word percent, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

Dan listens to 48 minutes of a 120-minute album, so we must find what percent 48 is of 120.

*Write number sentences for each operation*.

Divide 48 by 120 to find what percent 48 is of 120:

*Solve the number sentences and decide which answer is reasonable*.

= 0.4 = 40%

*Check your work*.

We divided to find our answer, so we will multiply to check our work. If 48 minutes is 40% of the album, then 120 multiplied by 40%, or 0.4, should equal 48: 120 × 0.4 = 48 minutes. Our answer is correct.

We are given the length of the album in the directions, and we are given the number of minutes to which Dan has not listened. We are looking for the percent of the album Dan has heard.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain the word percent, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

Dan has heard all but 15 minutes of the album, so we must first subtract 15 from the length of the album, and then find what percent that number is of the total number of minutes.

*Write number sentences for each operation*.

First, subtract 15 from 120 to find the number of minutes Dan has heard:

120 – 15

*Solve the number sentences and decide which answer is reasonable*.

120 – 15 = 105

*Write number sentences for each operation*.

Divide 105 by the total length of the album, 120, to find the percent of the album that Dan has heard:

*Solve the number sentences and decide which answer is reasonable*.

= 0.875 = 87.5%

*Check your work*.

We subtracted and divided to find our answer, so we will multiply and add to check our work. If 105 minutes is 87.5% of the album, then 120 multiplied by 87.5%, or 0.875, should equal 105: 120 × 0.875 = 105 minutes. The number of minutes Dan has heard, 105 minutes, plus the number of minutes he has not heard, 15, should equal the total length of the album: 105 + 15 = 120 minutes, the total length of the album.

We are given the number of pounds Aiden hauls on Monday and on Tuesday.

*Identify the question being asked*.

We are looking for the percent decrease in the amount he hauled from Monday to Tuesday.

*Underline the keywords*.

The keyword less often signals subtraction.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

This problem asks how much less Aiden hauls on Tuesday than Monday by percent, which means that we must find the percent decrease from 9,800 to 8,700. The percent decrease is found by taking the difference between the original number and the new number and dividing by the original number.

*Write number sentences for each operation*.

The original number is 9,800 pounds, and the new number is 8,700. Plug these values into the formula:

*Solve the number sentences and decide which answer is reasonable*.

, which ≈ 0.1122, or 11.2% rounded to the nearest tenth of one percent.

*Check your work*.

If 9,800 to 8,700 ≈ an 11.2% decrease, then increasing 8,700 by 11.2% should give us approximately 9,800. Instead of multiplying 8,700 by 0.112 and adding 8,700, we will simply multiply 8,700 by 1.112: 8,700 × 1.112 = 9,674.4, which ≈ 9,800.

We are given the number of laps Judi swims every day and the percentage increase in the number of laps she swam today.

*Identify the question being asked*.

We are looking for the number of laps she swam today.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain a % symbol, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

If Judi swam 12% more laps today, then the number of laps she swam increased by 12%. We will use a percent increase formula. Once we find how many more laps she swam today, we can add it to 25 to find the number of laps she swam.

*Write number sentences for each operation*.

If a number is increased by some percent, we can multiply that number by the percent to find by how much the number was increased:

25 × 0.12

*Solve the number sentences and decide which answer is reasonable*.

25 × 0.12 = 3 laps. Judi swam three more laps today.

*Write number sentences for each operation*.

Add the number of additional laps she swam to the usual number of laps she swims:

3 + 25

*Solve the number sentences and decide which answer is reasonable*.

3 + 25 = 28 laps

*Check your work*.

Since 28 is a 12% increase, divide 28 by 1.12 to find the number of laps Judi usually swims: = 25 laps.

We are given Kellyann's height when she was three years old and her height now.

*Identify the question being asked*.

We are looking for the percent by which her height has increased.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain the word percent, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

We are given the age three years old, but we don't need the number 3 to solve this problem. Kellyann's age is given there to show how her height has increased.

*List the possible operations*.

The problem asks us to find by what percent her height has increased. The percent increase is found by taking the difference between the original number and the new number and dividing by the original number.

*Write number sentences for each operation*.

The original number is 35 inches, and the new number is 65 inches. Plug these values into the formula:

*Solve the number sentences and decide which answer is reasonable*.

, which ≈ 0.8571, or 85.7% rounded to the nearest tenth of one percent.

*Check your work*.

If 35 to 65 ≈ an 85.7% increase, then multiplying 35 by 185.7%, or 1.857, should give us approximately 65: 35 × 1.857 = 64.995, which ≈ 65.

We are given how much Jeffrey could bench-press in seventh grade, and his percent increases in that number from seventh grade to eighth grade and from eighth grade to ninth grade.

*Identify the question being asked*.

We are looking for how much Jeffrey can bench-press now.

*Underline the keywords*.

There are no keywords in this problem, but the problem does contain the % symbol, so we will likely have to use a percent formula.

*Cross out extra information and translate words into numbers*.

There is no extra information in this problem.

*List the possible operations*.

We need to find how much Jeffrey can bench-press as a ninth grader. We are given how much he can bench-press in seventh grade and the percentage increase in that weight to the amount he can bench-press in eighth grade. We must find how much he can bench-press in eighth grade, and then use that to find how much he can bench-press in ninth grade.

*Write number sentences for each operation*.

Convert 20% to a decimal. Before multiplying, add 1 to that decimal, so that after we multiply it by the number of pounds Jeffrey could bench-press in seventh grade, we won't need to add in order to find how many pounds Jeffrey could bench-press in eighth grade:

70 × 1.20

*Solve the number sentences and decide which answer is reasonable*.

70 × 1.20 = 84 pounds

*Write number sentences for each operation*.

Jeffrey can bench-press 25% more as a ninth grader. Convert 20% to a decimal, and again, add 1 before multiplying. This will give us the weight Jeffrey can bench-press as a ninth grader:

84 × 1.25

*Solve the number sentences and decide which answer is reasonable*.

84 × 1.25 = 105 pounds

*Check your work*.

Since we multiplied twice to find our answer, we will divide twice to check our work. Divide the weight Jeffrey could bench-press in ninth grade by 1 plus the percentage increase from eighth grade. This should give us how much Jeffrey could bench-press in eighth grade: = 84 pounds. Divide this weight by 1 plus the percentage increase from seventh grade. This should give us how much Jeffrey could bench-press in seventh grade: = 70 pounds. Our answer is correct.

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