- Read the entire word problem.
We are given Joseph's ten scores.
Identify the question being asked.
We are looking for the number of points he scored most often.
Underline the keywords and words that indicate formulas.
The words most often mean that we are looking for the mode of the set.
Cross out extra information and translate words into numbers.
The number of times Joseph played the game is not needed to solve this problem, so the number ten can be crossed out.
List the possible operations.
The mode is found by counting the number of times each value occurs in a set.
Write number sentences for each operation.
We don't need a number sentence to solve this kind of problem. The scores 285, 290, 305, 310, and 330 occur once each. The score 300 occurs twice, and the score 320 occurs three times, which means that 320 is the mode.
Check your work.
The value 320 occurs three times in the set. No other value occurs more than twice, so 320 must be the mode.
- Read the entire word problem.
We are given Joseph's ten scores.
Identify the question being asked.
We are looking for the number of points he scored on average.
Underline the keywords and words that indicate formulas.
The word average means that we are looking for the mean of the set.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The mean is found by adding all of the scores and dividing by the number of scores, so we must use addition and division.
Write number sentences for each operation.
First, find the sum of all the scores. We will need the sum before we can divide:
320 + 285 + 300 + 290 + 320 + 310 + 300 + 305 + 330 + 320
Solve the number sentences and decide which answer is reasonable.
320 + 285 + 300 + 290 + 320 + 310 + 300 + 305 + 330 + 320 = 3,080
Write number sentences for each operation.
Now that we have the total of all the scores, divide by the number of scores, 10:
Solve the number sentences and decide which answer is reasonable.
Check your work.
The mean multiplied by the number of values should equal the sum of the values: 308 × 10 = 3,080 and 320 + 285 + 300 + 290 + 320 + 310 + 300 + 305 + 330 + 320 = 3,080, so 308 must be the mean.
- Read the entire word problem.
We are given Joseph's ten scores.
Identify the question being asked.
We are looking for the range of his scores.
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 times Joseph played the game is not needed to solve this problem, so that number 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 285 and the largest number in the set is 330:
330 – 285
Solve the number sentences and decide which answer is reasonable.
330 – 285 = 45
Check your work.
Since we used subtraction to find our answer, we can use addition to check our work. The smallest value plus the range should equal the largest value: 285 + 45 = 330, which is the largest value in the set.
- Read the entire word problem.
We are given Joseph's ten scores.
Identify the question being asked.
We are looking for the middle value of his scores.
Underline the keywords and words that indicate formulas.
The word middle means that we are looking for the median value of the set.
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 lengths in order from smallest to greatest and select the middle value. Since there is an even number of scores, the median value will be the average of the fifth and sixth scores.
Write number sentences for each operation.
First, we must place the scores in order from smallest to greatest:
285, 290, 300, 300, 305, 310, 320, 320, 320, 330
The middle values are 305 and 310. The average of these numbers is equal to their sum divided by 2, so first we must find the sum:
305 + 310
Solve the number sentences and decide which answer is reasonable.
305 + 310 = 615
Write number sentences for each operation.
Now, divide that sum by 2:
Solve the number sentences and decide which answer is reasonable.
Check your work.
We can check that our median is 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 five values that are less than 307.5 and five values that are greater than 307.5, so 307.5 is the median.
- Read the entire word problem.
We are told that Justin visits five parks, we are given the number of benches in four of those parks, and we are given the mean number of benches in the parks.
Identify the question being asked.
We are looking for the number of benches in Long Park.
Underline the keywords and words that indicate formulas.
The word mean usually indicates that we are looking for the mean value of the set, but we are given the mean, and we must use it to find a value in the set.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Normally, we find the mean by adding all of the values in a set and dividing by the number of values. Since we are given all but one of the values and the mean, we must work backward. First, multiply the mean by the number of values. Then, find the sum of the values and subtract it from that product to find the missing value.
Write number sentences for each operation.
Multiply the mean by the number of values:
10 × 5
Solve the number sentences and decide which answer is reasonable.
10 × 5 = 50
Write number sentences for each operation.
Next, find the total of the four given values:
8 + 13 + 10 + 8
Solve the number sentences and decide which answer is reasonable.
8 + 13 + 10 + 8 = 39
Write number sentences for each operation.
Finally, subtract the number of benches in four parks from the product of the mean and the number of parks:
50 – 39
Solve the number sentences and decide which answer is reasonable.
50 – 39 = 11
There are 11 benches in Long Park.
Check your work.
If there are 11 benches in Long Park, then the sum of the benches in the five parks divided by five should equal ten, since that is the mean number of benches: 8 + 13 + 10 + 8 + 11 = 50, = 10 benches.
- Read the entire word problemm.
We are given the number of pretzels Eric sells in each of seven hours.
Identify the question being asked.
We are looking for the difference between the greatest number of pretzels sold in an hour and the least number of pretzels sold in an hour.
Underline the keywords and words that indicate formulas.
The word span indicates that we are looking for the range of the set.
Cross out extra information and translate words into numbers.
The hours that Eric worked, 9 A.M. to 4 P.M., are not needed to solve the problem, so that information can be crossed out.
List the possible operations.
The range is found by subtracting the smallest value from the largest value. In this set, the largest value is 39 and the smallest value is 16.
Write number sentences for each operation.
39 – 16
Solve the number sentences and decide which answer is reasonable.
39 – 16 = 25
The difference from Eric's most successful hour to his least successful hour is 25 pretzels.
Check your work.
The smallest value plus the range should equal the largest value: 16 + 25 = 39, which is the largest value in the set.
- Read the entire word problem.
We are given nine test scores for Katie and told that she can drop the two lowest.
Identify the question being asked.
We are looking for her most common test score after dropping those two scores.
Underline the keywords and words that indicate formulas.
The words most common indicate that we are looking for the mode of the set.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We must put the set of scores in order:
85, 85, 85, 85, 90, 95, 95, 95, 100
and remove the two lowest scores:
85, 85, 90, 95, 95, 95, 100
Write number sentences for each operation.
There are no number sentences to write. The score 95 occurs three times, the score 85 occurs twice, and the scores 90 and 100 each occur once. Since 95 occurs the most often, it is the mode of the set and is Katie's most common test score.
Check your work.
After removing the two lowest scores, check that every other score occurs fewer than three times. Since 85 is the only score that occurs more than once, and it occurs only twice, 95 is the most common test score.
- Read the entire word problem.
We are given the amounts of six different vegetables that Annie buys and the median number of vegetables that she buys.
Identify the question being asked.
We are looking for the possible numbers of eggplant that she may have bought.
Underline the keywords and words that indicate formulas.
The word median usually indicates that we are looking for the middle value of a set, but we are given the median value in this problem. We must use the median to find the missing value in the set.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We must put the numbers of vegetables Annie bought in order:
3, 3, 4, 8, 8, 16
There are no operations to perform, but we know that the median number of vegetables Annie buys is 4. There are six values in the data set now, and if she buys four eggplants, 4 would be the middle number of the set:
3, 3, 4, 4, 8, 8, 16
However, if Annie buys more than four eggplants, then 4 would not be the median of the set, because there would be four values greater than 4 and only two values less than 4. If Annie buys fewer than four eggplants, 4 will also be the median of the set, since there will be three values less than 4 and three values greater than 4. If Annie buys 0, 1, 2, 3, or 4 eggplants, then the median number of vegetables she buys will be 4.
Check your work.
Insert each of those five answers into the set to see if the median value is 4 each time:
0, 3, 3, 4, 8, 8, 16
1, 3, 3, 4, 8, 8, 16
2, 3, 3, 4, 8, 8, 16
3, 3, 3, 4, 8, 8, 16
3, 3, 4, 4, 8, 8, 16
In each of these sets, 4 is the median value, so all of these answers are correct. If we insert any number other than these five, the median will not be 4.
- Read the entire word problem.
We are told that three sides of cube are red, one side is yellow, and two sides are blue and that the cube is rolled once.
Identify the question being asked.
We are looking for the chances that it will land on a red side.
Underline the keywords and words that indicate formulas.
The word chances tells us that we are looking for a probability.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A cube has six sides, so the total number of possibilities is six. There are three sides that are red.
Write number sentences for each operation.
A number sentence is not needed for this probability. The probability is equal to the number of outcomes that make the event true, 3, over the total number of possibilities, 6: , which reduces to .
- Read the entire word problem.
We are told that three sides of cube are red, one side is yellow, and two sides are blue and that the cube is rolled once.
Identify the question being asked.
We are looking for the probability that it will land on the yellow side.
Underline the keywords and words that indicate formulas.
The word probability tells us that we are looking for a probability.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A cube has six sides, so the total number of possibilities is six. There is one side that is yellow.
Write number sentences for each operation.
A number sentence is not needed for this probability. The probability is equal to the number of outcomes that make the event true, 1, over the total number of possibilities, 6: .
- Read the entire word problem.
We are told that three sides of the cube are red, one side is yellow, and two sides are blue and that the cube is rolled once.
Identify the question being asked.
We are looking for the odds that it will land on a blue side.
Underline the keywords and words that indicate formulas.
The word odds tells us that we are looking for a probability.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A cube has six sides, so the total number of possibilities is six. There are two sides that are blue.
Write number sentences for each operation.
A number sentence is not needed for this probability. The probability is equal to the number of outcomes that make the event true, 2, over the total number of possibilities, 6: , which reduces to .
- Read the entire word problem.
We are told that three sides of the cube are red, one side is yellow, and two sides are blue and that the cube is rolled once.
Identify the question being asked.
We are looking for the likelihood that it will not land on yellow.
Underline the keywords and words that indicate formulas.
The word likelihood tells us that we are looking for a probability.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
A cube has six sides, so the total number of possibilities is six. There are three sides that are red and two sides that are blue.
Write number sentences for each operation.
Add the number of red sides to the number of blue sides: 3 + 2
Solve the number sentences and decide which answer is reasonable.
3 + 2 = 5
The probability is equal to the number of outcomes that make the event true, 5, over the total number of possibilities, 6: .
- Read the entire word problem.
We are told that Al has eight white shirts, five blue shirts, two yellow shirts, and one red shirt in his closet.
Identify the question being asked.
We are looking for the chances that he will select a shirt that is blue or red.
Underline the keywords and words that indicate formulas.
The word chances 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 shirts in the closet, so we must first find that total.
Write number sentences for each operation.
8 + 5 + 2 + 1
Solve the number sentences and decide which answer is reasonable.
8 + 5 + 2 + 1 = 16
Write number sentences for each operation.
Since there are five blue shirts in the closet, the probability of selecting a blue shirt is . There is one red shirt in the closet, so the probability of selecting a red shirt is . To find the probability that a blue shirt or red shirt is selected, add the two probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability of Al selecting either a blue shirt or a red shirt is .
- Read the entire word problem.
We are told that Ingrid has 24 rock songs, 40 pop songs, ten classical songs, and six jazz songs.
Identify the question being asked.
We are looking for the probability that the player will select a rock song or a jazz song.
Underline the keywords and words that indicate formulas.
The word probability 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 songs on the player, so we must first find that total.
Write number sentences for each operation.
24 + 40 + 10 + 6
Solve the number sentences and decide which answer is reasonable.
24 + 40 + 10 + 6 = 80
Write number sentences for each operation.
Since there are 24 rock songs on the player, the probability of it selecting a rock song is . There are six jazz songs on the player, so the probability of it selecting a jazz song is . To find the probability that a rock song or a jazz song is selected, add the two robabilities:
Solve the number sentences and decide which answer is reasonable.
, or
The probability of the player selecting either a rock song or a jazz song is .
- Read the entire word problem.
We are told that a gumball machine contains 38 lemon gumballs, 12 grape gumballs, 19 strawberry gumballs, 16 orange gumballs, and 15 blueberry gumballs.
Identify the question being asked.
We are looking for the chances that Mohammed will select a lemon, strawberry, or orange gumball.
Underline the keywords and words that indicate formulas.
The word chances tells us that we are looking for a probability. The word or tells us that we are looking for more than one probability, 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 gumballs in the machine, so we must first find that total.
Write number sentences for each operation.
38 + 12 + 19 + 16 + 15
Solve the number sentences and decide which answer is reasonable.
38 + 12 + 19 + 16 + 15 = 100
Write number sentences for each operation.
Since there are 38 lemon gumballs, the probability of Mohammed selecting a lemon gumball is . There are 19 strawberry gumballs, so the probability of him selecting a strawberry gumball is , and since there are 16 orange gumballs, the probability of him selecting an orange gumball is . To find the probability that a lemon, strawberry, or orange gumball is selected, add the three probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability of Mohammed selecting a lemon, strawberry, or orange gumball is .
- Read the entire word problem.
We are told that there are 400 orchestra seats, 300 first mezzanine seats, 200 second mezzanine seats, 100 third mezzanine seats, and 100 fourth mezzanine seats.
Identify the question being asked.
We are looking for the odds that Kayla will not win an orchestra ticket.
Underline the keywords and words that indicate formulas.
The word odds tells us that we are looking for a 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 tickets in the hall, so we must first find that total.
Write number sentences for each operation.
400 + 300 + 200 + 100 + 100
Solve the number sentences and decide which answer is reasonable.
400 + 300 + 200 + 100 + 100 = 1,100
Write number sentences for each operation.
The probability of Kayla winning a ticket in the first mezzanine is , since there are 300 seats in that section out of a total of 1,100 seats. In the same way, the probability of her winning a ticket in the second mezzanine is , the probability of her winning a ticket in the third mezzanine is , and the probability of her winning a ticket in the fourth mezzanine is . To find the probability that she won a ticket in one of these sections, add the four probabilities:
Solve the number sentences and decide which answer is reasonable.
The odds of Kayla winning a ticket that is not in the orchestra is .
- Read the entire word problem.
We are given the number of students in Mr. Scott's class and told that only one perfect score was received on each of two exams.
Identify the question being asked.
We are looking for the odds that Stephanie has received both scores.
Underline the keywords and words that indicate formulas.
The word odds tells us 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.
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 students in the class, which we are told is 25. Stephanie is only one student, so the probability that she has received a perfect score on the math exam is . In the same way, the probability that she received a perfect score on the science exam is also .
Write number sentences for each operation.
The probability that Stephanie is the student to receive a perfect score on both exams is the product of each of those probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability that Stephanie was the student to receive a perfect score on both exams is .
- Read the entire word problem.
We are told that a spinner is divided into ten slices, numbered one through ten, respectively, and that the spinner is spun twice.
Identify the question being asked.
We are looking for the probability that it will land on five the first time and a number greater than five the second time.
Underline the keywords and words that indicate formulas.
The word probability tells us that we are looking for a probability, and the keyword and tells us that we are looking for two probabilitis, which we will likely have to multiply.
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 slices on the spinner, which we are told is ten. Since only one of the ten slices is numbered five, the probability of the spinner landing on five is . There are five slices that contain numbers that are greater than five (six, seven, eight, nine, ten), so the probability of the spinner landing on a number greater than five the second time is , or
Write number sentences for each operation.
The probability that the spinner lands on five and then on a number greater than five is the product of each of those probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability that the spinner lands on five the first time and a number greater than five the second time is .
- Read the entire word problem.
We are told that a number cube, which has six sides numbered one through six, respectively, is rolled three times.
Identify the question being asked.
We are looking for the chances that it will land on an even number the first time, an odd number the second time, and the number six the third time.
Underline the keywords and words that indicate formulas.
The word chances tells us that we are looking for a probability, and the keyword and tells us that we are looking for multiple probabilities, which we will likely have to multiply.
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 for each roll is the total number of sides of the cube, which is six. Since three sides contain even numbers (two, four, six), the probability of the cube landing on an even number is , or . , or . Only one side contains the number six, so the probability of the cube landing on the number six is .
Write number sentences for each operation.
The probability that the cube lands on an even number the first time, an odd number the second time, and the number six the third time is the product of each of those probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability that the cube lands on an even number the first time, an odd number the second time, and the number six the third time is .
- Read the entire word problem.
We are told that a package contains 12 chocolate cookies and ten vanilla cookies and that two cookies are taken from the package and not returned.
Identify the question being asked.
We are looking for the probability that both cookies were vanilla cookies.
Underline the keywords and words that indicate formulas.
The word probability tells us 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 the cookies were eaten, they were not returned to the package, which tells us that the denominator of our second probability will not be the same as the denominator of our first probability, and the numerator may change as well.
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 cookies in the package at any time. To start, there are 12 + 10 = 22 cookies in the package. Since ten of them are vanilla, the probability of Rita selecting a vanilla cookie is , or . Rita eats the cookie, which means that there is one less vanilla cookie (and one less total cookie), so the probability that she selects a vanilla cookie the second time is only , or .
Write number sentences for each operation.
The probability that Rita selects two vanilla cookies is the product of each of those probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability that Rita selects two vanilla cookies is .
- Read the entire word problem.
We are told that a program offers five science courses, eight math courses, four art courses, and three music courses, and that Mairead is placed in two courses that are not identical.
Identify the question being asked.
We are looking for the odds that she has been placed in an art course and a music course.
Underline the keywords and words that indicate formulas.
The word odds tells us 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 she cannot be placed in the same course twice, 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 courses at any time. To start, there are 5 + 8 + 4 + 3 = 20 courses. Since four of them are art courses, the probability of Mairead being placed in an art course is , or . After Mairead is placed in that art course, she cannot be placed in that course again, so there are now only 19 courses left. The probability that she is placed in a music course is , since there are three music courses offered out of the remaining 19 courses.
Write number sentences for each operation.
The probability that Mairead has been placed in an art course and a music course is the product of those probabilities:
Solve the number sentences and decide which answer is reasonable.
The probability that Mairead is placed in an art course and a music course is .
- Read the entire word problem.
We are told that a sock drawer contains eight pairs of brown socks, 18 pairs of white socks, and 20 pairs of black socks and that each of two girls selects a pair of socks.
Identify the question being asked.
We are looking for the likelihood that Molly will select a white pair and that Anna will select a black pair.
Underline the keywords and words that indicate formulas.
The word likelihood tells us 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 Molly and Anna cannot both wear the same pair of socks, 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 socks from which to choose at any time. To start, there are 8 + 18 + 20 = 46 pairs of socks. Since 18 of them are white, the probability of Molly selecting a white pair is , or .
Write number sentences for each operation.
The probability that Molly will select a white pair and that Anna will select a black pair is the product of those probabilities:
.
Solve the number sentences and decide which answer is reasonable.
.
The probability that Molly will select a white pair and that Anna will select a black pair is .