PERCENTS: SUBJECT REVIEW
9. 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 num- ber of days by one:
Number of days worked Pay per day Total earned
1 ≈$43.75 $43.75 5 ≈$43.75 $218.75 10 ≈$43.75 $437.50 11 ≈$43.75 $481.25 12 ≈$43.75 $525.00 13 ≈$43.75 $568.75 14 ≈$43.75 $612.50
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.
10. Read the entire word problem.
We are given the number of grams of vitamin E in one vitamin, the num- ber 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 multi- ply 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.
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 divi- sion 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: 47..0055 = 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: 03.564 = 0.015 grams. Our answer is correct.
11. Read the entire word problem.
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: 05.445= 120 minutes. 12. Read the entire word problem.
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:
14280
Solve the number sentences and decide which answer is reasonable.
14280 = 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. 13. Read the entire word problem.
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 num- ber 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:
112005
Solve the number sentences and decide which answer is reasonable.
112005 = 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 min- utes. The number of minutes Dan has heard, 105 minutes, plus the num- ber 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.
14. Read the entire word problem.
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 Mon- day 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 dif- ference between the original number and the new number and divid- ing 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.
(9,8090,8–080,700)= 91,,810000, 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.
15. Read the entire word problem.
We are given the number of laps Judi swims every day and the per- centage 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 num- ber 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.
Check your work.
Since 28 is a 12% increase, divide 28 by 1.12 to find the number of laps Judi usually swims: 12.182= 25 laps.
16. Read the entire word problem.
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 orig- inal 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:
(653–535)
Solve the number sentences and decide which answer is reasonable.
(653–535) = 3305, 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. 17. Read the entire word problem.
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.
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 per- centage 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: 11.0255 = 84 pounds. Divide this weight by 1 plus the percentage increase from sev- enth grade. This should give us how much Jeffrey could bench-press in seventh grade: 18.42= 70 pounds. Our answer is correct.