Roger is spending the afternoon house hunting. There are 4 properties on his list, and he has time to see 2 of them. How many sequences of 2 houses are possible?
sequences
There are 6 possible sequences of 2 houses for Roger to see.
To see why, we can use the formula for combinations:
nCr = n! / r!(n-r)!
In this case, n = 4 (the number of properties on his list) and r = 2 (the number of houses he will see).
4C2 = 4! / 2!(4-2)! = 6
So there are 6 possible sequences of 2 houses for Roger to see. They are:
- 1st and 2nd houses
- 1st and 3rd houses
- 1st and 4th houses
- 2nd and 3rd houses
- 2nd and 4th houses
- 3rd and 4th houses
At a dog show, a Pekingese, a Chihuahua, and a Great Dane won first place in their divisions. In how many different orders could these winners be announced?
orders
There are 6 different orders in which these winners could be announced.
To see why, we can list out all the possible orders:
1. Pekingese, Chihuahua, Great Dane
2. Pekingese, Great Dane, Chihuahua
3. Chihuahua, Pekingese, Great Dane
4. Chihuahua, Great Dane, Pekingese
5. Great Dane, Pekingese, Chihuahua
6. Great Dane, Chihuahua, Pekingese
So there are 6 different orders in which the winners could be announced.
There are 8 projects that made it to the final round of the science competition. In how many different orders can the judges rank these projects?
orders
There are 40,320 different orders in which the judges can rank these projects.
To see why, we can use the formula for permutations:
n! = n(n-1)(n-2)...(3)(2)(1)
In this case, n = 8 (the number of projects in the final round).
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
So there are 40,320 different orders in which the judges can rank these projects.
Rose has 10 books on her shelf, and she wants to read any 2 of them on vacation. How many sequences of 2 books are possible?
sequences
There are 45 sequences of 2 books that Rose can choose from her shelf.
To see why, we can use the formula for combinations:
nCr = n! / r!(n-r)!
In this case, n = 10 (the number of books on her shelf) and r = 2 (the number of books she will read on vacation).
10C2 = 10! / 2!(10-2)! = 45
So there are 45 sequences of 2 books that Rose can choose from her shelf.
Kevin is hanging framed photos above his fireplace. If he has 3 photos to choose from and 2 hooks on which to hang them, how many arrangements are possible?
ways
There are 6 different arrangements that are possible for Kevin to hang his 3 photos on the 2 hooks.
To see why, we can list out all the possible arrangements:
1. Photo 1 on hook 1, Photo 2 on hook 2, Photo 3 not hung
2. Photo 1 on hook 1, Photo 3 on hook 2, Photo 2 not hung
3. Photo 2 on hook 1, Photo 1 on hook 2, Photo 3 not hung
4. Photo 2 on hook 1, Photo 3 on hook 2, Photo 1 not hung
5. Photo 3 on hook 1, Photo 1 on hook 2, Photo 2 not hung
6. Photo 3 on hook 1, Photo 2 on hook 2, Photo 1 not hung
So there are 6 different arrangements that are possible for Kevin to hang his 3 photos on the 2 hooks.
At a spelling bee, there are 9 students competing. How many top-2 rankings are possible?
There are 36 different top-2 rankings possible.
To see why, we can use the formula for combinations:
nCr = n! / r!(n-r)!
In this case, n = 9 (the number of students competing) and r = 2 (the number of students who will be ranked in the top 2).
9C2 = 9! / 2!(9-2)! = 36
So there are 36 different top-2 rankings possible.
Is the dotted line a line of symmetry?
I'm sorry, I cannot provide an answer without more information or context. Please provide additional details or a visual reference for the dotted line.
Dana has a stamp collection she is very proud of. She buys a frame to display her favorite stamps. The frame is 6 inches wide, and a row of 7 stamps fits perfectly across it.
How wide is each stamp?
Write your answer as a proper fraction or mixed number.
inches
If a row of 7 stamps fits perfectly across a 6-inch wide frame, then each stamp is 6/7 inches wide.
We can write this as a fraction:
6/7 inches per stamp
So each stamp is 6/7 inches wide.
Mr. Maynard's class is taking a field trip to a science museum. They will be at the museum for 3 hours, and they want to divide their time evenly among the 10 exhibits.
How much time should they spend at each exhibit?
Write your answer as a proper fraction or mixed number.
hours
If the class wants to divide their time evenly among the 10 exhibits and they have 3 hours total, they should spend 3/10 of an hour, also known as 18 minutes, at each exhibit.
We can write this as a fraction:
3/10 hours per exhibit
So they should spend 3/10 of an hour, or 18 minutes, at each exhibit.
Omar baked 3 apple pies for Thanksgiving dinner. There were 16 people over at the house, including Omar. Each person ate the same amount of apple pie, and all of the pie was eaten.
How much pie did each person eat?
Write your answer as a proper fraction or mixed number.
of a pie
Since there were 16 people over at the house and 3 apple pies were eaten, we can start by finding out how many pieces of pie there are in total.
3 pies x 8 pieces per pie = 24 pieces of pie
So there are 24 pieces of pie in total. Since each person ate the same amount of pie, we can divide the total number of pieces by the number of people:
24 pieces / 16 people = 3/2 pieces per person
So each person ate 3/2, or 1 and 1/2, pieces of pie.
Trudy sells wedges of cheese at the local farmers' market. To make the wedges, she cuts a big 5-pound block of cheese into 16 pieces.
How much does each wedge of cheese weigh?
Write your answer as a proper fraction or mixed number.
pounds
To find out how much each wedge of cheese weighs, we need to divide the weight of the big 5-pound block of cheese by the number of wedges it is cut into.
5 pounds / 16 wedges = 5/16 pounds per wedge
So each wedge of cheese weighs 5/16 of a pound.
Mr. Ortega decided to make a healthy snack for the 20 students in his class. He gave each student a dish of yogurt, and divided 6 cups of strawberries equally among the dishes.
How many cups of strawberries did each student get in their yogurt?
Write your answer as a proper fraction or mixed number.
cups
If Mr. Ortega divided 6 cups of strawberries equally among the 20 students in his class, each student got 6/20 of a cup of strawberries.
We can simplify this fraction by dividing both the numerator and denominator by 2:
6/20 = 3/10
So each student got 3/10 of a cup of strawberries in their yogurt.
Aaliyah is packing up her room to move to a new city. Together, all the books on her bookshelf weigh 85 pounds. She wants to divide the books among 4 boxes, so that each box will weigh the same amount.
How many pounds of books will be in each box?
Write your answer as a proper fraction or mixed number.
pounds
If Aaliyah wants to divide her books among 4 boxes so that each box weighs the same amount, she needs to divide the total weight of her books (85 pounds) by the number of boxes (4):
85 pounds / 4 boxes = 21 and 1/4 pounds per box
So each box should weigh 21 and 1/4 pounds.
Harold sells hot apple cider at the Hendersonville Apple Festival each year. For a batch of cider that makes 25 servings, Harold uses 2 tablespoons of cinnamon.
How much cinnamon is in each serving of cider?
Write your answer as a proper fraction or mixed number.
tablespoons
If Harold uses 2 tablespoons of cinnamon in a batch of cider that makes 25 servings, we can find out how much cinnamon is in each serving by dividing the total amount of cinnamon by the number of servings:
2 tablespoons / 25 servings = 2/25 tablespoons per serving
So each serving of cider contains 2/25 tablespoons of cinnamon.
Ethan has 3 cups of frosting to use for 28 cupcakes. He is extra careful to make sure each cupcake gets the same amount of frosting.
How much frosting does each cupcake get?
Write your answer as a proper fraction or mixed number.
cups
To find out how much frosting each cupcake gets, we need to divide the total amount of frosting (3 cups) by the number of cupcakes (28):
3 cups / 28 cupcakes = 3/28 cups per cupcake
So each cupcake gets 3/28 of a cup of frosting.
Brendan, Nicole, Isabella, and Steven went to a baseball game on a bright sunny day, but they only brought 1 pair of sunglasses! The friends agreed that each person would wear the sunglasses for an equal amount of time during the 3-hour game.
How long did each person wear the sunglasses?
Write your answer as a proper fraction or mixed number.
hours
If each of the 4 friends wore the sunglasses for an equal amount of time during the 3-hour game, we can find out how long that would be by dividing the total time the sunglasses were worn (3 hours) by the number of friends (4):
3 hours / 4 friends = 3/4 hours per person
So each person wore the sunglasses for 3/4 of an hour, or 45 minutes.
Celine, Abby, Victoria, and Helen are on a team for a 2-mile relay race. Each person runs the same distance.
How far does each person run?
Write your answer as a proper fraction or mixed number.
miles
If the team is running a 2-mile relay race and there are 4 people on the team, each person would need to run 1/4 of the total distance:
2 miles / 4 people = 1/2 mile per person
So each person runs 1/2 mile.
Celine, Abby, Victoria, and Helen are on a team for a 2-mile relay race. Each person runs the same distance.
How far does each person run?
Write your answer as a proper fraction or mixed number.
miles
If the team is running a 2-mile relay race and there are 4 people on the team, each person would need to run 1/4 of the total distance:
2 miles / 4 people = 1/2 mile per person
So each person runs 1/2 mile.