 Read the entire word problem.
Identify the question being asked.
We are looking for the difference between two numbers.
Underline the keywords and words that indicate formulas.
The keyword difference signals subtraction.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We need to subtract 37 from 56.
Write number sentences for each operation.
56 – 37
Solve the number sentences and decide which answer is reasonable.
56 – 37 = 19
Check your work.
Since we used subtraction to solve this problem, we can use addition to check the answer. 19 + 37 = 56, so this answer is correct.
 Read the entire word problem.
We are given the number of times a press can print a newspaper in an hour.
Identify the question being asked.
We are looking for the number of newspapers that six presses can print in an hour.
Underline the keywords and words that indicate formulas.
The keyword each can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We need to either divide 250 by 6 or multiply 250 by 6. Since we are given the number of newspapers printed by one press, and we are looking for the number printed by six presses, we must multiply.
Write number sentences for each operation.
(250)(6)
Solve the number sentences and decide which answer is reasonable.
(250)(6) = 1,500 newspapers
Check your work.
Since we used multiplication to solve this problem, we can use division to check the answer. The number of newspapers divided by the number of newspapers printed by each press should give us the number of presses: = 6 presses.
 Read the entire word problem.
We are given the number of rooms and the number of floors in a hotel.
Identify the question being asked.
We are looking for the number of rooms on each floor of the hotel.
Underline the keywords and words that indicate formulas.
The keyword each can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We need to either divide 360 by 24 or multiply 360 by 24. Since we are given the total number of rooms on 24 floors and we are looking for the number of rooms on one floor, we must divide.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
= 24 rooms
Check your work.
Since we used division to solve this problem, we can use multiplication to check the answer. The number of rooms on each floor multiplied by the number of floors should give us the number of rooms in the hotel: (24)(15) = 360 rooms.
 Read the entire word problem.
We are given the number of dresses Terra has and the number of dresses Souyma has.
Identify the question being asked.
We are looking for the number of dresses the two have altogether.
Underline the keywords and words that indicate formulas.
The keyword altogether signals addition.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We need to add the number of dresses Terra has to the number of dresses Souyma has.
Write number sentences for each operation.
18 + 15
Solve the number sentences and decide which answer is reasonable.
18 + 15 = 33 dresses
Check your work.
Since we used addition to solve this problem, we can use subtraction to check the answer. The total number of dresses minus the number of dresses Terra has should equal the number of dresses Souyma has: 33 – 18 = 15 dresses.
 Read the entire word problem.
We are given the number of times a hummingbird beats its wings in a second.
Identify the question being asked.
We are looking for the number of times it beats its wings in a minute.
Underline the keywords and words that indicate formulas.
The keyword per can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem. There are 60 seconds in a minute, so this question is asking us how many times a hummingbird beats its wings in 60 seconds.
List the possible operations.
Since we are given the number of times a hummingbird beats its wings in one second and we are looking for the number of times it beats its wings in 60 seconds, we must multiply.
Write number sentences for each operation.
(53)(60)
Solve the number sentences and decide which answer is reasonable.
(53)(60) = 3,180 beats
Check your work.
Since we used multiplication to solve this problem, we can use division to check the answer. The total number of beats in a minute divided by the number of seconds in a minute should give us back the number of beats per second: = 53 beats.
 Read the entire word problem.
We are given the number of milliliters of iodine and the number of milliliters of water needed for a science experiment.
Identify the question being asked.
We are looking for the number of milliliters of iodine needed for 15 experiments.
Underline the keywords and words that indicate formulas.
There are no keywords in this problem.
Cross out extra information and translate words into numbers.
We do not need the number of milliliters of water used in each experiment, so that information can be crossed out.
List the possible operations.
Since we are given the number of milliliters of iodine used in one experiment and we are looking for the number of milliliters used in 15 experiments, we must multiply.
Write number sentences for each operation.
(5)(15)
Solve the number sentences and decide which answer is reasonable.
(5)(15) = 75 milliliters
Check your work.
Since we used multiplication to solve this problem, we can use division to check the answer. The total number of milliliters divided by the number of experiments should equal the number of milliliters of iodine needed for one experiment: = 5 milliliters.
 Read the entire word problem.
We are given the cost of skate rental, the current number of pairs of skates, and the number of skates that are ordered.
Identify the question being asked.
We are looking for the total number of skates.
Underline the keywords and words that indicate formulas.
The keyword total signals addition.
Cross out extra information and translate words into numbers.
We do not need the cost of skate rental, so that information can be crossed out.
List the possible operations.
We are given the current number of pairs of skates and the number of pairs of skates ordered, so we must add to find the total.
Write number sentences for each operation.
85 + 35
Solve the number sentences and decide which answer is reasonable.
85 + 35 = 120 pairs of skates
Check your work.
Since we used addition to solve this problem, we can use subtraction to check the answer. The total number of skates minus the number of skates ordered should equal the original number of skates: 120 – 35 = 85 pairs of skates.
 The following diagram shows 38 people waiting outside a theater:
17 of them are let in, so cross out 17 of them:
We can see now that there are 21 people still waiting outside.
 The following diagram shows 18 people at a party:
Six more people arrive, so add six guests to the diagram:
We can see now that there are 24 people at the party.
 We can enter these names into a table to find the order of their birthdays. Svetlana's birthday comes before Veronica's and after Cara's, so place Cara's name above Svetlana's name and Veronica's name:
Cara's birthday is after Rachel's, so we can place Rachel at the top of the table:
We don't know where to place Susie's birthday, but we know it is after Cara's, which means it can be no higher than third. Rachel's party must be first.
 Build a table with columns for the number of booths, the number of tables, the total people from booths, the total people from tables, and the overall total. The overall total is equal to the number of booths multiplied by 4 plus the number of tables multiplied by 5. The total number of booths and tables should always equal 12:
If there are seven booths and five tables, then there are 53 people seated in the restaurant.
 Draw a Venn diagram. We must draw it as shown here, so that there is an area where just the water park and amusement park overlap, an area where just the water park and the kiddie park overlap, an area where just the amusement park and the kiddie park overlap, and an area where all three overlap. We are given the number of people who visit only one park, so we can label those areas of the diagram. The unknown areas, the areas that represent people who visited multiple parks, are labeled with variables:
If 620 visited the park, and 77 + 187 + 151 = 415 of them visited only one park, then 620 – 415 = 205 visited more than one park, which means that a + b + c + d = 205. We know that 204 people visited the water park, which means that 77 + a + b + d = 204. We also know that 370 people visited the amusement park, which means that 187 + a + d + c = 370. In the same way, 295 people visited the kiddie park, which means that 151 + b + d + c = 295. We can combine these equations to find the values of a, b, c, and d. If 77 + a + b + d = 204, then a + b + d = 127. We know that a + b + c + d = 205. Subtract the first equation from the second:
There are 78 people who visited the kiddie park and the amusement park. 187 + a + d + c = 370, so a + d + c = 183. Subtract this equation from a + b + c + d = 205:
There are 22 people who visited the water park and the kiddie park. Now that we know how many people visited the water park and the kiddie park (22) and how many people visited the kiddie park and the amusement park (78), we can find the number of people who visited all three parks, since we know 295 people visited the kiddie park: 151 + 22 + 78 + d = 295, 251 + d = 295, d = 44. There are 44 people who visited all three parks.
 Read the entire word problem.
Identify the question being asked.
We are looking for ten fewer than twelve.
Underline the keywords and words that indicate formulas.
The keywords fewer than signal subtraction.
Cross out extra information and translate words into numbers.
Rewrite ten as "10" and twelve as "12."
List the possible operations.
The keywords fewer than are a backward phrase; 12 is the number from which we are subtracting. We must find 12 – 10.
Write number sentences for each operation.
12 – 10
Solve the number sentences and decide which answer is reasonable.
12 – 10 = 2
Check your work.
Since we used subtraction to solve this problem, we can use addition to check the answer. The difference, 2, plus the subtrahend, 10, should equal the minuend, 12: 2 + 10 = 12.
 Read the entire word problem.
We are given the number of fish Alimi catches and how many more fish he catches than Larry.
Identify the question being asked.
We are looking for the number of fish Larry catches.
Underline the keywords and words that indicate formulas.
The keywords more than usually signal addition, but in this problem, we already have the number of fish Alimi catches. We are looking for the number of fish Larry catches, which is less than the number of fish Alimi catches.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since Alimi catches more fish than Larry, we must subtract the difference in the number of fish each catches from Alimi's total to find how many fish Larry catches.
Write number sentences for each operation.
23 – 12
Solve the number sentences and decide which answer is reasonable.
23 – 12 = 11 fish
Check your work.
Since we used subtraction to solve this problem, we can use addition to check the answer. Since Alimi catches 12 more fish than Larry, 12 plus the number of fish Larry catches should equal the number of fish Alimi catches: 12 + 11 = 23 fish.
 Read the entire word problem.
We are given the number of books Hector buys at the convention and the number of books he has now.
Identify the question being asked.
We are looking for the number of books he had before the convention.
Underline the keywords and words that indicate formulas.
The keyword adds usually signals addition, but in this problem, we already have the number of books Hector has in his collection now. We are looking for the number of books he had before he added 23 to his collection.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since we already have the total number of books after Hector added 23 books, we must "undo" that addition and subtract to find how many books he had originally.
Write number sentences for each operation.
289 – 23
Solve the number sentences and decide which answer is reasonable.
289 – 23 = 266 books
Check your work.
Since we used subtraction to solve this problem, we can use addition to check the answer. The number of books Hector had before the convention plus the number of books he bought at the convention should equal the number of books he has now: 266 + 23 = 289 books.
 Read the entire word problem.
We are given the number of ounces in the fountain, the number of ounces in each cup, and the number of cups.
Identify the question being asked.
We are looking for the number of ounces that are left in the fountain.
Underline the keywords and words that indicate formulas.
The keyword left signals subtraction, but before we can subtract, we must determine how many ounces to subtract.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
One cup of water contains 8 ounces, and there are 45 cups. We must multiply to find the total number of ounces removed from the fountain. Once we have that value, we can subtract it from the original number of ounces in the fountain.
Write number sentences for each operation.
Find the number of ounces removed from the fountain:
(45)(8)
Solve the number sentences and decide which answer is reasonable.
(45)(8) = 360 ounces
Write number sentences for each operation.
Now subtract that amount from the original number of ounces in the fountain:
640 – 360
Solve the number sentences and decide which answer is reasonable.
640 – 360 = 280 ounces
Check your work.
Since we used multiplication and subtraction to solve this problem, we can use addition and division to check the answer. Divide the number of ounces in 45 cups, 360, by 45 to find the amount of water in one cup: = 8 ounces. Add the amount of water removed from the fountain, 360 ounces, to the amount of water remaining in the fountain, 280 ounces, to check that it is equal to the original volume of the fountain: 360 + 280 = 640 ounces.
 Read the entire word problem.
We are given the number of pounds in a knapsack, the number of pounds in a duffel bag, and the numbers of knapsacks and duffel bags.
Identify the question being asked.
We are looking for the total weight of the baggage.
Underline the keywords and words that indicate formulas.
The keyword each can signal multiplication or division, and the keyword appears twice in the problem. The keyword add 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 knapsack weighs 12 pounds, then we must multiply to find the weight of 12 knapsacks. In the same way, if one duffel bag weighs 20 pounds, then we must multiply to find the weight of five duffel bags. Once we have those weights, we can add them to find the total weight of the baggage.
Write number sentences for each operation.
Find the weight of the knapsacks:
(12)(12)
Solve the number sentences and decide which answer is reasonable.
(12)(12) = 144 pounds
Write number sentences for each operation.
Next, find the weight of the duffel bags:
(20)(5)
Solve the number sentences and decide which answer is reasonable.
(20)(5) = 100 pounds
Write number sentences for each operation.
Finally, add the weight of the knapsacks to the weight of the duffel bags to find the total weight:
144 + 100
Solve the number sentences and decide which answer is reasonable.
144 + 100 = 244 pounds
Check your work.
Retrace your steps. Divide the total weight of the knapsacks, 144 pounds, by the number of knapsacks: = 12 pounds per knapsack, which is what was given. Divide the total weight of the duffel bags, 100 pounds, by the number of duffel bags: = 20 pounds per bag, which is what was given.
 Read the entire word problem.
We are given the number of people one bus holds, the number of trips made by one bus, and the number of buses.
Identify the question being asked.
We are looking for the total number of people served in five days.
Underline the keywords and words that indicate formulas.
The keyword per can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
If one bus holds 22 people, then to find the number of people served by eight trips, we must multiply. Once we have that product, we can multiply it by nine, the number of buses the tour company has. This will give us the number of people served in one day. To find the number of people served in five days, we will have to multiply again.
Write number sentences for each operation.
First, find the number of people served by eight trips:
(22)(8)
Solve the number sentences and decide which answer is reasonable.
(22)(8) = 176 people
Write number sentences for each operation.
Next, find the number of people served by nine buses:
(176)(9)
Solve the number sentences and decide which answer is reasonable.
(176)(9) = 1,584 people
Write number sentences for each operation.
Since 1,584 people are served in one day, multiply that by 5 to find the number of people served in five days:
(1,584)(5)
Solve the number sentences and decide which answer is reasonable.
(1,584)(5) = 7,920 people
Check your work.
We can use division to check our answer. Divide the number of people served in five days by 5 to find the number of people served in one day: = 1,584. Divide by 9 the number of people served in one day by nine buses to find the number of people served by one bus: = 176. Divide by 8 the number of people served by 8 trips to find the number of people served by one trip: = 22, the given number of people the tour bus holds.
 Read the entire word problem.
Identify the question being asked.
We are looking for the sum of six fewer than eleven and nineteen.
Underline the keywords and words that indicate formulas.
The keyword sum signals addition, and the keyword phrase fewer than signals subtraction.
Cross out extra information and translate words into numbers.
Rewrite six as "6," eleven as "11," nineteen as "19."
List the possible operations.
We are looking for the sum of 19 and a number. That number is 6 fewer than 11, so we must subtract 6 from 11 before adding.
Write number sentences for each operation.
11 – 6
Solve the number sentences and decide which answer is reasonable.
11 – 6 = 5
Write number sentences for each operation.
Now, find the sum of 5 and 19:
5 + 19
Solve the number sentences and decide which answer is reasonable.
5 + 19 = 24
Check your work.
We can use subtraction to check our answer. Subtract 19 from 24, and you should be left with a number that is 6 fewer than 11. 24 – 19 = 5, as is 11 – 6.
 Read the entire word problem.
Identify the question being asked.
We are looking for the difference of two numbers, the first of which is the product of eight and fourteen, and the second of which is the sum of twelve and a number that is four more than nine.
Underline the keywords and words that indicate formulas.
The keyword product signals multiplication, the keyword minus signals subtraction, the keyword sum signals addition, and the keyword phrase more than signals addition.
Cross out extra information and translate words into numbers.
Rewrite eight as "8," fourteen as "14," twelve as "12," four as "4," and nine as "9."
List the possible operations.
We are looking for the difference between the product of 8 and 14 and a sum, but before we can find the difference, we must find the sum of 12 and 4 more than 9.
Write number sentences for each operation.
To find 4 more than 9, add 4 to 9:
4 + 9
Solve the number sentences and decide which answer is reasonable.
4 + 9 = 13
Write number sentences for each operation.
Now, find the sum of 12 and 13:
12 + 13
Solve the number sentences and decide which answer is reasonable.
12 + 13 = 25
Write number sentences for each operation.
Next, find the product of 8 and 14:
(8)(14)
Solve the number sentences and decide which answer is reasonable.
(8)(14) = 112
Write number sentences for each operation.
Finally, subtract from that product the sum of 12 and 4 more than 9:
112 – 25
Solve the number sentences and decide which answer is reasonable.
112 – 25 = 87
Check your work.
Retrace your steps. Add 25 to 87: 25 + 87 = 112. Divide 112 by 14: = 8, one of the given numbers. Subtract 12 from 25: 25 – 12 = 13. Thirteen is 4 more than 9, two of the other given numbers.
 Write seven as "7," a number as "x," and thirteen as "13." Product signals multiplication, so the product of x and 13 is 13x. More than signals addition, so 7 more than 13x is 13x + 7.
 Write half as "", a number as "x," and fifteen as "15." The square of x is x^{2}. Half the square of a number is x^{2}. Times signals multiplication, so 15 times x is 15x. Minus signals subtraction, so x^{2} minus 15x is x^{2} – 15x.
 Write three as "3," a number as "x," write eleven as "11," and write five as "5." Replace is equal to with the equals sign: the sum of 3 times x and 11 = 5. Sum signals addition and times signals multiplication: The sum of 3 times x and 11 = 5 becomes 3x + 11 = 5. Subtract 11 from both sides of the equation, and then divide both sides by 3: 3x + 11 = 5, 3x = –6, x = –2.
 Write one as "1," write nine as "9," write a number as "x," write five as "5," and write eight as "8." Replace is with the equals sign: 1 fewer than 9 times x = 5 more than 8 times x. Fewer than signals subtraction, times signals multiplication, and more than signals addition: 1 fewer than 9 times x = 5 more than 8 times x becomes 9x – 1 = 8x + 5. Subtract 8x from both sides of the equation and add 1 to both sides: 9x – 1 = 8x + 5, x = 6.
 Write negative six as "–6," write a number as "x," write four as "4," write twelve as "12," and write two as "2." Replace is less than with the less than symbol: –6 times x plus 4 < 12 fewer than 2 multiplied by x. Times signals multiplication, plus signals addition, fewer than signals subtraction, and multiplied by signals multiplication: –6 times x plus 4 < 12 fewer than 2 multiplied by x becomes –6x + 4 < 2x – 12. Subtract 2x and –4 from both sides of the inequality. –6x + 4 < 2x – 2, –8x < –16. Divide both sides of the inequality by –8 and reverse the direction of the inequality symbol: –8x < –16, x > 2.
 Read the entire word problem.
We are given the fraction of Marco's gas tank that was full and the fraction that he uses.
Identify the question being asked.
We are looking for the fraction of his gas tank that is full now.
Underline the keywords and words that indicate formulas.
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 Marco uses of the tank, he has less, which means that we must subtract from .
Write number sentences for each operation.
Before we can subtract, we must have common denominators. The least common multiple of 4 and 8 is 8, so convert to eighths: = .
–
Solve the number sentences and decide which answer is reasonable.
– =
of Marco's gas tank is full now.
Check your work.
Add the fraction of the tank that remains to the fraction of the tank Marco used over the weekend. This sum should equal the fraction of the tank that was originally full. + = + = .
 Read the entire word problem.
We are given the number of pounds of chocolate Gino buys, the fraction of pounds that is white chocolate, and the number of friends with whom Gino shares.
Identify the question being asked.
We are looking for the number of pounds of chocolate each friend receives.
Underline the keywords and words that indicate formulas.
The keyword shares signals division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
First, we must find what fraction of the chocolate is dark chocolate. Then, we must multiply that by 2, the total number of pounds of chocolate Gino bought. Finally, we will divide that quantity by 5, since Gino shared the dark chocolate with five friends.
Write number sentences for each operation.
If of the chocolate was white chocolate and the rest was dark chocolate, we must subtract from 1 to find the fraction of dark chocolate.
Write 1 as a fraction with a denominator of 10:
–
Solve the number sentences and decide which answer is reasonable.
– =
of the chocolate is dark chocolate.
Write number sentences for each operation.
of the 2 pounds of chocolate Gino bought is dark chocolate. Multiply by 2 to find the number of pounds of dark chocolate:
(2) ()
Solve the number sentences and decide which answer is reasonable.
(2) () =
Gino has pounds of dark chocolate.
Write number sentences for each operation.
Gino shares pounds of dark chocolate with five friends. Divide by 5:
Solve the number sentences and decide which answer is reasonable.
Each friend receives pounds of dark chocolate.
Check your work.
Multiply the number of pounds of chocolate each friend receives by the number of friends: () (5) = pounds. Since Gino bought 2 pounds of chocolate, and of it was dark chocolate, divided by should equal 2 pounds: = () () = 2 pounds.
 Read the entire word problem.
We are given the number of ounces of water the faucet loses per hour.
Identify the question being asked.
We are looking for the number of ounces of water it loses in a day.
Underline the keywords and words that indicate formulas.
The keyword per can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem. We are told how many ounces of water are lost each hour, and we are looking for how many ounces are lost in a day, so we must convert one day to 24 hours.
List the possible operations.
Multiply the number of ounces of water lost in one hour by 24 hours to find the number of ounces lost in a day.
Write number sentences for each operation.
(0.06)(24)
Solve the number sentences and decide which answer is reasonable.
(0.06)(24) = 1.44 ounces
The faucet loses 1.44 ounces in a day.
Check your work.
The number of ounces lost in a day divided by 24 should equal the number of ounces lost in an hour: = 0.06 ounces.
 Read the entire word problem.
We are given the size of Christian's hard drive, the amount of data he had saved, and the amount of data he deleted.
Identify the question being asked.
We are looking for the amount of free space on his hard drive.
Underline the keywords and words that indicate formulas.
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.
If Christian deletes data from his hard drive, then we must subtract that amount from the amount of data he had saved. Once we have the new amount of data he has saved, we can subtract that from the size of the hard drive to find how many gigabytes are free.
Write number sentences for each operation.
First, find how many gigabytes are used. Subtract the amount Christian deleted from the amount he had saved:
26.754 – 8.24
Solve the number sentences and decide which answer is reasonable.
26.754 – 8.24 = 18.514 gigabytes
Write number sentences for each operation.
Now, subtract the amount of space used from the size of the drive to find the number of gigabytes that are free:
50 – 18.514
Solve the number sentences and decide which answer is reasonable.
50 – 18.514= 31.486 gigabytes
Check your work.
Subtract the amount of free space from the total size of the drive. This will give you the amount of used space. Add to that the number of gigabytes Christian deleted, and this should give you the original number of gigabytes saved on the drive: 50 – 31.486 = 18.514, 18.514 + 8.24 = 26.754 gigabytes.
 Read the entire word problem.
We are given the number of appointments Dr. Wilcox can schedule in a week and the number of appointments he has scheduled.
Identify the question being asked.
We are looking for the percent of his schedule that is booked.
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.
We need to find what percent is 21 of 30. We can find what percent one number is of a second number by dividing the first number by the second number. Divide 21 by 30 and express the answer as a percent.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
= 0.70 = 70%
70% of Dr. Wilcox's schedule is booked.
Check your work.
Since we used division to find our answer, we can check our work with multiplication. Multiply the percent of Dr. Wilcox's schedule that is booked by the total number of appointments he could have. This should give us back the number of appointments he has booked: (0.70)(30) = 21 appointments.
 Read the entire word problem.
We are given the original price of a photo, the percent that it increased, and the percent that it then decreased.
Identify the question being asked.
We are looking for the final price of the photo.
Underline the keywords.
The keyword increased usually signals addition, but the problem does contain the percent symbol twice, so we will likely have to use at least one percent formula.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
First, we need to find how much the price of the photo increased. We will multiply the original price by 15%, or 0.15, and add that increase to the original price. Then, we will find 25% of that price, and subtract that amount to give us the final price.
Write number sentences for each operation.
Multiply the original price, $120, by 15%, or 0.15:
($120)(0.15)
Solve the number sentences and decide which answer is reasonable.
($120)(0.15) = $18
The price of the photo increased by $18.
Write number sentences for each operation.
Add $18 to the original price:
$120 + $18
Solve the number sentences and decide which answer is reasonable.
$120 + $18 = $138
The price of the photo became $138.
Write number sentences for each operation.
Next, find 25% of $138. Multiply $138 by 25%, or 0.25:
($138)(0.25)
Solve the number sentences and decide which answer is reasonable.
($138)(0.25) = $34.50
The price of the photo decreased by $34.50.
Write number sentences for each operation.
Find the final price by subtracting $34.50 from $138:
$138 – $34.50
Solve the number sentences and decide which answer is reasonable.
$138 – $34.50 = $103.50
The price of the photo now is $103.50.
Check your work.
There is no easy way to check a problem like this. Review your calculations to ensure that this answer is correct.
 Read the entire word problem.
We are given a temperature in degrees Fahrenheit.
Identify the question being asked.
We are looking for a temperature in degrees Celsius.
Underline the keywords and words that indicate formulas.
The words Celsius and Fahrenheit tell us that we need to use a temperature conversion 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 use the formula for converting temperature in degrees Fahrenheit to degrees Celsius: C = (F – 32).
Write number sentences for each operation.
Substitute the temperature in degrees Fahrenheit into the formula:
C = (77 – 32)
Solve the number sentences and decide which answer is reasonable.
C = (77 – 32) = (45) = 25° Celsius.
Check your work.
To check our answer, we can use the formula for converting Celsius to Fahrenheit to get back the temperature in degrees Fahrenheit. F = C + 32, F = (25) + 32 = 45 + 32 = 77° Fahrenheit.
 Read the entire word problem.
We are given the principal, rate, and time.
Identify the question being asked.
We are looking for the interest earned by that principal at that rate over that time.
Underline the keywords and words that indicate formulas.
The keyword per can signal multiplication or division. The words interest and rate tell us that we need to use the formula for interest.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
To find interest earned given the principal, rate, and time, we must multiply those three values. However, the interest rate is 4.2% per year, and our time is given in months. We must first convert three months to a number of years.
Write number sentences for each operation.
There are 12 months in a year, so we can convert the number of months to a number of years by dividing by 12:
Solve the number sentences and decide which answer is reasonable.
= 0.25
Janine had her money in the bank for 0.25 years. We are now ready to use the interest formula, I = prt, where I, interest, is equal to the product of p, principal, r, rate, and t, time.
Write number sentences for each operation.
Convert 4.2% to a decimal. Our number sentence is the interest formula, with 8,200 substituted for p, 0.042 substituted for r, and 0.25 substituted for t:
I = (8,200)(0.042)(0.25)
Solve the number sentences and decide which answer is reasonable.
I = (8,200)(0.042)(0.25) = $86.10
Check your work.
Since I = prt, we can check our interest answer by dividing the interest by the rate and time to see if that quotient equals the given principal, $8,200: = $8,200.
 Read the entire word problem.
We are given the rate at which the roller coaster travels and the time for which it travels.
Identify the question being asked.
We are looking for the distance it travels.
Underline the keywords and words that indicate formulas.
The keyword per can signal multiplication or division.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
To find the distance an object travels, we must multiply the rate by the time. However, the rate is given in miles per hour and the time is given in seconds. Both measurements must use the same unit of time. We will convert the number of seconds to hours. There are 60 seconds in a minute, and 60 minutes in an hour, which means that there are (60)(60) = 3,600 seconds in an hour. Divide the number of seconds by 3,600 to represent the time in hours.
Write number sentences for each operation.
=
The roller coaster traveled for of an hour. The keyword per signals multiplication, since the rate gives us the distance traveled in an hour, and we are looking for the distance traveled in of an hour.
Write number sentences for each operation.
Our number sentence is the distance formula, with 65 substituted for r and substituted for t: D = (65) ()
Solve the number sentences and decide which answer is reasonable.
(65) () = = 0.072 miles, to the nearest thousandth.
Check your work.
Since D = rt, we can check our distance answer by dividing the distance by the time to see if that quotient equals the given rate, 65 miles per hour:
= 64.8, or about 65 miles per hour.

Make a table with three rows and three columns. Since we are looking for how much salt will be added, our table will be about the percent and quantity of salt. The original solution is 5%, or 0.05, salt. It is 20 ounces in total, which means that 0.05 × 20 of it is salt. We are adding an unknown quantity of salt, 100% of which is salt. Our final solution will be 50% salt:
The total quantity of the final solution will be 20 + x, and 0.50 of it will be salt:
The salt quantity is the product of the percent concentration and the total quantity, 0.50(20 + x), and also the sum of the salt quantity in the original solution and the amount added, 0.05 × 20 and 1 × x. Set these two expressions equal to each other and solve for x:
0.50(20 + x) = (0.05 × 20) + (1 × x)
10 + 0.5x = 1 + x
9 = 0.5x
x = 18 ounces
Ida must add 18 ounces of salt to make the solution 50% salt.
 Make a table with three rows and three columns. Since we are looking for how much water will be evaporated, our table will be about the percent and quantity of water. The solution is 15% alcohol and 100% – 15% = 85% water. We are looking for a final solution that is only 100% – 75% = 25% water:
We are removing water, so the total quantity will be 45 – x, not 45 + x:
The final water quantity is equal to the percent concentration of the final solution times the total quantity of the solution: 0.25(45 – x). The final quantity of water is also equal to the original quantity of water minus the amount of water removed: (0.85 × 45) – (1 × x). Set these equations equal to each other and solve for x:
0.25(45 – x) = (0.85 × 45) – (1 × x)
11.25 – 0.25x = 38.25 – x
0.75x = 27
x = 36 liters
We must evaporate 36 liters of water to make the concentration of the solution 75% alcohol.
 Read the entire word problem.
We are given the number of necklaces and the number of earrings.
Identify the question being asked.
We are looking for the ratio of earrings to necklaces.
Underline the keywords and words that indicate formulas.
The word ratio usually means that we will be using a ratio to set up a proportion, but this problem simply asks us to find a ratio.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A ratio is a relationship between two numbers. There are no number sentences to write: the ratio of earrings to necklaces is the number of earrings, followed by a colon, and then the number of necklaces: 30:12. We can reduce this ratio by dividing both numbers by their greatest common factor. The greatest common factor of 30 and 12 is 6, so 30:12 reduces to 5:2.
 Read the entire word problem.
We are given the number of lefthanded players and righthanded players on a team.
Identify the question being asked.
We are looking for the ratio of lefthanded players to the total number of players.
Underline the keywords and words that indicate formulas.
The word ratio usually means that we will be using a ratio to set up a proportion, but this problem simply asks us to find a ratio. The keyword total signals addition.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A ratio is a relationship between two numbers. To find the ratio of lefthanded players to the total number of players, we must first the total number of players. Add the number of lefthanded players and the number of righthanded players.
Write number sentences for each operation.
14 + 6
Solve the number sentences and decide which answer is reasonable.
14 + 6 = 20 players
The ratio of lefthanded players to the total number of players is the number of lefthanded players, followed by a colon, and then the total number of players: 14:20. We can reduce this ratio by dividing both numbers by their greatest common factor. The greatest common factor of 14 and 20 is 2, so 14:20 reduces to 7:10.
Check your work.
We can check our addition by using subtraction. The total number of players minus the number of lefthanded players should equal the number of righthanded players: 20 – 14 = 6 players.
 Read the entire word problem.
We are given the ratio of cucumber slices to tomato slices and the number of cucumber slices altogether.
Identify the question being asked.
We are looking for the number of tomato slices.
Underline the keywords and words that indicate formulas.
The word ratio usually means that we will be using a ratio to set up a proportion.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
This problem contains two quantities, cucumber slices and tomato slices. We are given the value of one quantity, cucumber slices, and we are asked for the value of the other quantity, tomato slices. This is a parttopart problem. We must set up a proportion that compares the ratio of cucumber slices to tomato slices to the ratio of actual cucumber slices to actual tomato slices.
Write number sentences for each operation.
The ratio of cucumber slices to tomato slices is 4:3, or . Use x to represent the actual number of tomato slices, since that number is
unknown. The ratio of actual cucumber slices to actual tomato slices is 12:x, or . Set these fractions equal to each other: =
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x: 4x = (12)(3), 4x = 36, x = 9. If there are 12 cucumber slices in the salad, then there are nine tomato slices in the salad.
Check your work.
The ratio of cucumber slices to tomato slices is 4:3, so the ratio of actual cucumber slices to actual tomato slices should reduce to 4:3. The greatest common factor of 12 and 9 is 3, and 12:9 reduces to 4:3.
 Read the entire word problem.
We are given the ratio of firstclass seats to businessclass seats and the total number of seats altogether.
Identify the question being asked.
We are looking for the number of firstclass seats.
Underline the keywords and words that indicate formulas.
The word ratio means that we will likely be using a ratio to set up a proportion. The word total is a signal that this is a parttowhole ratio problem.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
This problem contains two quantities, firstclass seats and businessclass seats. We are given the total of the two quantities, and we are looking for the value of one quantity. This is a parttowhole problem. Since we are looking for the number of firstclass seats, we must set up a proportion that compares the ratio of firstclass seats to total seats to the ratio of actual firstclass seats to actual total seats.
Write number sentences for each operation.
The ratio of firstclass seats to businessclass seats is 2:25, which means that the ratio of firstclass seats to total seats is 2:25 + 2, which is 2:27, or . Use x to represent the number of firstclass seats, since that number is unknown. There are 324 total seats, so the ratio of actual first class seats to actual total seats is x:324, or . Set these fractions equal to each other:
=
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x:
27x = (2)(324), 27x = 648, x = 24. If there are 324 total seats, then 24 of them are firstclass seats.
Check your work.
The ratio of firstclass seats to total seats is 2:27, so the ratio of actual firstclass seats to total seats should reduce to 2:27. The greatest common factor of 24 and 324 is 12, and 24:324 reduces to 2:27.
 Read the entire word problem.
We are given the ratio of red tiles to blue tiles and the number of red tiles.
Identify the question being asked.
We are looking for the total number of tiles.
Underline the keywords and words that indicate formulas.
The word ratio means that we will likely be using a ratio to set up a proportion. The word total is a signal that this is a parttowhole ratio problem.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
This problem contains two quantities, red tiles and blue tiles. We are given the value of one quantity, and we are looking for the total of the two quantities. This is a parttowhole problem. Since we are looking for the total number of tiles, we must set up a proportion that compares the ratio of red tiles to the total number of tiles to the ratio of actual red tiles to the actual total number tiles.
Write number sentences for each operation.
The ratio of red tiles and blue tiles is 3:8, which means that the ratio of red tiles to the total number of tiles is 3:8 + 3, which is 3:11, or . Use x to represent the number of total tiles, since that number is unknown. There are 1,692 red tiles, so the ratio of actual red tiles to actual total tiles is 1,692:x, or . Set these fractions equal to each other:
=
Solve the number sentences and decide which answer is reasonable.
Cross multiply and solve for x:
3x = (11)(1,692), 3x = 18,612, x = 6,204. If there are 1,692 red tiles, then there are 6,204 total tiles.
Check your work.
The ratio of red tiles to the total number of tiles is 3:11, so the ratio of actual red tiles to the total number of actual tiles should reduce to 3:11. The greatest common factor of 1,692 and 6,204 is 564, and 1,692:6,204 reduces to 3:11.
 Read the entire word problem.
We are given the number of visitors to a library.
Identify the question being asked.
We are looking for the range of visitors to the library.
Underline the keywords and words that indicate formulas.
The word range actually appears in the problem, so we are told that we are looking for a range.
Cross out extra information and translate words into numbers.
The number of weeks, two, is not needed to solve this problem, so it can be crossed out.
List the possible operations.
The range is found by subtracting the smallest value from the largest value.
Write number sentences for each operation.
The smallest number in the set is 21 visitors and the largest number in the set is 77 visitors:
77 – 21
Solve the number sentences and decide which answer is reasonable.
77 – 21 = 56 visitors
Check your work.
Since we used subtraction to find our answer, we can use addition to check our answer. The smallest value plus the range should equal the largest value: 21 + 56 = 77, which is the largest value in the set.
 Read the entire word problem.
We are given the number of students in six math classes.
Identify the question being asked.
We are looking for the median number of students in a math class.
Underline the keywords and words that indicate formulas.
The word median actually appears in the problem, so we are told that we are looking for a median.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
To find the median value, put all of the class sizes in order from smallest to greatest and select the middle value. Since there is an even number of classes, the median value will be the average of the third and fourth scores.
Write number sentences for each operation.
First, we must place the numbers of students in order from smallest to greatest:
22, 23, 24, 26, 27, 27
The middle values are 24 and 26. The average of these numbers is equal to their sum divided by 2, so first we must find the sum:
24 + 26
Solve the number sentences and decide which answer is reasonable.
24 + 26 = 50
Write number sentences for each operation.
Now, divide that sum by 2:
Solve the number sentences and decide which answer is reasonable.
= 25
Check your work.
We can check that our median could be correct by comparing the number of values that are less than or equal to the median to the number of values that are greater than or equal to the median. These numbers should be equal. There are three values that are less than 25 (22, 23, 24) and three values that are greater than 25 (26, 27, 27), so 25 could be the median.
 Read the entire word problem.
We are told a car lot has 12 sports cars, eight sport utility vehicles, three convertibles, six pickup trucks, and six midsize cars.
Identify the question being asked.
We are looking for the likelihood that Tony bought either a sport utility vehicle or a pickup truck.
Underline the keywords and words that indicate formulas.
The word likelihood tells us that we are looking for a probability. The word or tells us that we are looking for two probabilities, which we will likely have to add.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The total number of possibilities is the total number of cars in the lot, so we must first find that total.
Write number sentences for each operation.
12 + 8 + 3 + 6 + 6
Write number sentences for each operation.
12 + 8 + 3 + 6 + 6 = 35
Since there are eight sport utility vehicles, the probability of Tony buying one of those is . There are six pickup trucks, so the probability of Tony buying one of those is . To find the probability that he bought either a sport utility vehicle or a pickup truck, add the two probabilities:
+
Solve the number sentences and decide which answer is reasonable.
+ = , or
The probability that Tony bought either a sport utility vehicle or a pickup truck is .
 Read the entire word problem.
We are told a display rack contains ten postcards of New York, 20 postcards of Paris, eight postcards of London, four postcards of Rome, and three postcards of Tokyo.
Identify the question being asked.
We are looking for the probability that Si bought a postcard of New York and a postcard of London.
Underline the keywords and words that indicate formulas.
We are told that we are looking for a probability, and the keyword and tells us that we are looking for two probabilities, which we will likely have to multiply. Since Si buys two postcards, the denominator of our second probability will not be the same as the denominator of our first probability.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The total number of possibilities is the total number of postcards, so we must first find that total.
Write number sentences for each operation.
10 + 20 + 8 + 4 + 3
Solve the number sentences and decide which answer is reasonable.
10 + 20 + 8 + 4 + 3 = 45
Write number sentences for each operation.
Since there are ten New York postcards, the probability of Si buying one of those is , or . After Si selects this postcard, there is one fewer postcard from which to choose. There are now 44 postcards left, eight of which are London postcards, so the probability of Si buying one of those is , or . To find the probability that he bought a New York postcard and a London postcard, multiply the two probabilities:
×
Solve the number sentences and decide which answer is reasonable.
× =
The probability that Si bought a postcard of New York and a postcard of London is .
 Read the entire word problem.
We are given the number of sides of a polygon.
Identify the question being asked.
We are looking for the sum of its interior angles.
Underline the keywords and words that indicate formulas.
The words polygon and interior angles indicate that we must use the formula for the sum of the interior angles of a polygon.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. We must subtract 2 from the number of sides of the polygon, and then multiply by 180.
Write number sentences for each operation.
8 – 2
Solve the number sentences and decide which answer is reasonable.
8 – 2 = 6
Write number sentences for each operation.
Now, multiply by 180:
(180)(6)
Solve the number sentences and decide which answer is reasonable.
(180)(6) = 1,080°
Check your work.
Since we used subtraction and multiplication to find our answer, we can check it by dividing and adding. Divide the total number of degrees by 180: = 6. Now, add 2 to 6, and this should give us the number of sides of the polygon: 6 + 2 = 8 sides.
 Read the entire word problem.
We are given the area and the height of a triangle.
Identify the question being asked.
We are looking for its base.
Underline the keywords and words that indicate formulas.
The keywords base, area, height, and triangle tell us that we need to use the formula for the area of a triangle.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We are given the area and the height. We can rewrite the formula A = bh by dividing both sides of
h. The base, b, is equal to .
Write number sentences for each operation.
First, multiply the area by 2:
2(1.98)
Solve the number sentences and decide which answer is reasonable.
2(1.98) = 3.96 yards^{2}
Write number sentences for each operation.
Now, divide that by the height:
Solve the number sentences and decide which answer is reasonable.
= 3.3 yards
The base of the triangle is 3.3 yards.
Check your work.
Since A = bh, check that half the product of the base and the height equals the area, 1.98 square yards: (3.3)(1.2) = (3.96) = 1.98 yards^{2}.
 Read the entire word problem.
We are given the measures of the hypotenuse and one leg of a triangle.
Identify the question being asked.
We are looking for the measure of the other leg.
Underline the keywords and words that indicate formulas.
We are told that we are working with a triangle, and the keyword hypotenuse tells us that we are working with a right triangle.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since we have a right triangle and the lengths of two sides, we can use the Pythagorean theorem to find the length of the missing side. Rewrite the formula a^{2} + b^{2} = c^{2} as b^{2} = c^{2} – a^{2}, since we are looking for the length of a leg.
Write number sentences for each operation.
First, find the difference between the square of the hypotenuse and the square of one leg:
(65)^{2} – (25)^{2}
Solve the number sentences and decide which answer is reasonable.
(65)^{2} – (25)^{2} = 4,225 – 625 = 3,600
Write number sentences for each operation.
The square of the leg is 3,600, so to find the length of the leg, we must take the square root of 3,600:
√3,600
Solve the number sentences and decide which answer is reasonable.
√3,600 = 60 inches
The length of the other leg is 60 inches.
Check your work.
The square of the hypotenuse should equal the sum of the squares of the legs: (25)^{2} + (60)^{2} = (65)^{2}, 625 + 3,600 = 4,225.
 Read the entire word problem.
We are given the area of a circle.
Identify the question being asked.
We are looking for a circumference.
Underline the keywords and words that indicate formulas.
The words area and circumference are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for area of a circle is A = πr^{2}, so we can find the radius by dividing the area by π, and then taking the square root of the quotient. Once we have the radius, we can multiply it by 2π to find the circumference.
Write number sentences for each operation.
Divide the area by π:
Solve the number sentences and decide which answer is reasonable.
= 169 square feet
Write number sentences for each operation.
Find the radius by taking the square root of 169:
√169
Solve the number sentences and decide which answer is reasonable.
√169 = 13 feet
The radius of the circle is 13 feet. To find the circumference, multiply the radius by 2π.
Write number sentences for each operation.
(13)(2π)
Solve the number sentences and decide which answer is reasonable.
(13)(2π) = 26π feet
Check your work
Divide the circumference by 2π to find the radius. Then, square the radius and multiply by π. This should give us back the area, 169π square feet. = 13, π(13)^{2} = 169π square feet.
 Read the entire word problem.
We are given the volume of a sphere.
Identify the question being asked.
We are looking for its surface area.
Underline the keywords and words that indicate formulas.
The words sphere, volume, and surface area are keywords. We will use the formula for the volume of a sphere to find the radius of the sphere, and then use the formula for the surface area of a sphere.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a sphere is V = πr^{3}, of the sphere and r is the radius. We can find the radius of the sphere by multiplying the volume by where V is the volume , dividing by π, and then taking the cube root. Since the formula for the surface area of a sphere is SA = 4πr^{2}, once we have the length of the radius, we can square it and multiply by 4π to find the surface area.
Write number sentences for each operation.
First, multiply the volume by :
() (288π)
Solve the number sentences and decide which answer is reasonable.
() (288π) = 216π
Write number sentences for each operation.
Next, divide by π:
Solve the number sentences and decide which answer is reasonable.
= 216
Write number sentences for each operation.
Now, take the cube root of 216 to find the radius:
3√216
Solve the number sentences and decide which answer is reasonable
3√216 = 6 centimeters
Write number sentences for each operation.
The surface area of a square is 4πr^{2}, so square the radius and multiply by 4π.
4π(6)^{2}
Solve the number sentences and decide which answer is reasonable.
4π(6)^{2} = 4π(36) = 144π centimeters^{2}
Check your work.
Divide the surface area by 4pi; and take the square root. This will give us the radius of the sphere. Cube the radius and multiply by π to get back the volume of the sphere: = 36, √36 = 6; π6^{3} = π(216) = 288π centimeters^{3}.