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Let A = {1, 2} and B = {a, b}. Find (A x B) ∪ (A x {c}) and simplify the resulting Cartesian product. | tiger_lab_math | <think>
To find \((A \times B) \cup (A \times \{c\})\), I start by calculating each Cartesian product separately.
First, for \(A \times B\), I pair each element of set \(A\) with each element of set \(B\). This results in the ordered pairs \((1, a)\), \((1, b)\), \((2, a)\), and \((2, b)\).
Next, for \(A \times \{c\}... | 5 |
In Faye's baseball game, their team scored a total of sixty-eight points. The points were scored through singles (1 point), doubles (2 points), triples (3 points), and home runs (4 points). Faye scored a combination of 8 singles, 5 doubles, 2 triples, and 1 home run for a total of twenty-eight points. Each of her teamm... | tiger_math | <think>
First, I'll calculate Faye's total points by multiplying the number of each type of hit by its corresponding points and summing them up.
Next, I'll determine each teammate's score using the same method. For Teammate A, Teammate B, and Teammate C, I'll apply their respective hits to the point values and add th... | 6 |
In Faye's baseball game, their team scored a total of sixty-eight points. The points were scored through singles (1 point), doubles (2 points), triples (3 points), and home runs (4 points). Faye scored a combination of 8 singles, 5 doubles, 2 triples, and 1 home run for a total of twenty-eight points. Each of her teamm... | tiger_math | <think>
First, I'll determine Faye's total points by calculating the sum of her singles, doubles, triples, and home runs.
Next, I'll calculate each teammate's points using their respective combinations of singles, doubles, triples, and home runs.
After finding the total points scored by Faye and her three teammates... | 7 |
In Faye's baseball game, their team scored a total of sixty-eight points. The points were scored through singles (1 point), doubles (2 points), triples (3 points), and home runs (4 points). Faye scored a combination of 8 singles, 5 doubles, 2 triples, and 1 home run for a total of twenty-eight points. Each of her teamm... | tiger_math | <think>
First, I need to determine how many points each of Faye's teammates scored based on their respective combinations of singles, doubles, triples, and home runs.
For Teammate A, who scored 1 single, 3 doubles, and 1 home run:
- Singles contribute 1 point each: 1 × 1 = 1 point.
- Doubles contribute 2 points each: ... | 8 |
Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores, and he also plans to mow lawns in his neighborhood to earn extra money. If it takes Tom 4 hours to mow a lawn and he charge... | nvidia_math | <think>
First, I need to determine how much more money Tom needs to buy the bike. He currently has $120, and the bike costs $180. So, the remaining amount he needs is $180 - $120 = $60.
Tom's parents have agreed to give him an allowance of $5 per week. However, this allowance will only cover half of the remaining amou... | 9 |
Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores, and he also plans to mow lawns in his neighborhood to earn extra money. If it takes Tom 4 hours to mow a lawn and he charge... | nvidia_math | <think>
First, I need to calculate the total amount Tom needs to buy the bike, which is $180, and subtract the amount he already has in his piggy bank, which is $120. This leaves him needing an additional $60.
Next, Tom's parents have agreed to give him an allowance of $5 per week. They will cover half of the remainin... | 10 |
Let A = {1, 2} and B = {a, b}. Find (A x B) ∪ (A x {c}) and simplify the resulting Cartesian product. | tiger_lab_math | <think>
Alright, so I have this problem here where I'm given two sets, A and B. A is {1, 2} and B is {a, b}. I need to find the union of two Cartesian products: (A x B) and (A x {c}). Hmm, okay, let me think about how to approach this step by step.
First off, I remember that a Cartesian product of two sets, say A and ... | 4 |
Let A = {1, 2} and B = {a, b}. Find (A x B) ∪ (A x {c}) and simplify the resulting Cartesian product. | tiger_lab_math | <think>
Alright, so I have this problem where I need to find the union of two Cartesian products, specifically (A × B) ∪ (A × {c}), where A is {1, 2} and B is {a, b}. Then, I have to simplify the resulting Cartesian product. Hmm, let me think through this step by step.
First, I need to recall what a Cartesian product ... | 3 |
Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores, and he also plans to mow lawns in his neighborhood to earn extra money. If it takes Tom 4 hours to mow a lawn and he charge... | nvidia_math | <think>
First, I need to determine how much more money Tom needs to buy the bike. He currently has $120, and the bike costs $180.
Subtracting the amount he has from the bike's cost gives:
$180 - $120 = $60
So, Tom needs an additional $60.
Next, Tom's parents have agreed to give him an allowance of $5 per week for do... | 11 |
Eddy’s spider plant produces 2 baby plants x times a year. After 4 years, The mother plant will have produced 16 baby plants. What is the value of unknown variable x? | tiger_math | <think>
The problem states that Eddy's spider plant produces 2 baby plants x times each year. After 4 years, the total number of baby plants produced is 16.
To find the value of x, I can set up the equation 2 * x * 4 = 16.
Solving for x, I divide both sides by 8, which gives x = 2.
</think>
To determine the value of... | 18 |
Emily was planting vegetables in her garden. She started with some seeds and planted 29 of them in the big garden, and in each of her 3 small gardens, put 4 seeds each. How many seeds did Emily start with? | tiger_math | <think>
Emily started with an unknown number of seeds.
She planted 29 seeds in the big garden.
In each of her 3 small gardens, she planted 4 seeds per garden. So that's 3 times 4, which equals 12 seeds.
To find the total number of seeds she planted, we add the seeds in the big garden and the small gardens: 29 plus 1... | 27 |
Emily was planting vegetables in her garden. She started with some seeds and planted 29 of them in the big garden, and in each of her 3 small gardens, put 4 seeds each. How many seeds did Emily start with? | tiger_math | <think>
Emily started with an unknown number of seeds. She planted 29 seeds in the big garden. Additionally, she planted 4 seeds in each of her 3 small gardens, which totals 12 seeds. To find the total number of seeds she began with, we need to add the seeds planted in the big garden to those planted in the small garde... | 28 |
There are 46 rulers in the drawer. Tim put 25 more rulers into the drawer. There are _____ rulers in the drawer now . Please write a program to solve it | tiger_lab_math | <think>
To determine the total number of rulers in the drawer after Tim adds more, I start by identifying the initial number of rulers, which is 46.
Next, I note that Tim adds 25 rulers to the drawer.
To find the final total, I simply add the initial number of rulers to the number of rulers Tim added: 46 + 25.
This ... | 42 |
Emily was planting vegetables in her garden. She started with some seeds and planted 29 of them in the big garden, and in each of her 3 small gardens, put 4 seeds each. How many seeds did Emily start with? | tiger_math | <think>
First, I need to determine the total number of seeds Emily planted in the small gardens. She has 3 small gardens and planted 4 seeds in each, so that's 3 multiplied by 4, which equals 12 seeds.
Next, she planted 29 seeds in the big garden.
To find out how many seeds Emily started with, I add the seeds planted... | 29 |
At a book fair, 25% of the books sold were fiction books, and the average price of these books was $15. The remaining books sold were non-fiction books, with an average price of $20. If the total revenue from book sales was $14,000, what was the total number of books sold? | nvidia_math | <think>
First, I need to determine the total number of books sold at the book fair. Let's denote the total number of books as \( N \).
Out of these, 25% were fiction books. Therefore, the number of fiction books sold is \( 0.25N \), and the remaining 75% are non-fiction books, which is \( 0.75N \).
The average price ... | 33 |
There are 46 rulers in the drawer. Tim put 25 more rulers into the drawer. There are _____ rulers in the drawer now . Please write a program to solve it | tiger_lab_math | <think>
First, I need to identify the initial number of rulers in the drawer, which is 46.
Next, Tim adds 25 more rulers to the drawer.
To find the total number of rulers now, I should add the initial number of rulers to the number Tim added.
So, I'll perform the addition: 46 + 25 = 71.
Therefore, there are 71 rule... | 43 |
Eddy’s spider plant produces 2 baby plants x times a year. After 4 years, The mother plant will have produced 16 baby plants. What is the value of unknown variable x? | tiger_math | <think>
First, identify the information given. The spider plant produces 2 baby plants each time it reproduces, and this happens x times a year. Over 4 years, the total number of baby plants produced is 16.
Next, calculate the annual production of baby plants. Since it reproduces x times a year and each reproduction y... | 19 |
There are 46 rulers in the drawer. Tim put 25 more rulers into the drawer. There are _____ rulers in the drawer now . Please write a program to solve it | tiger_lab_math | <think>
I need to resolve the math problem first. There are originally 46 rulers in the drawer, and Tim adds 25 more.
To find the total number of rulers now, I should add 46 and 25 together.
After calculating, the total is 71 rulers.
Next, I'll create a simple Python program to solve this problem. The program will d... | 44 |
Eddy’s spider plant produces 2 baby plants x times a year. After 4 years, The mother plant will have produced 16 baby plants. What is the value of unknown variable x? | tiger_math | <think>
To determine the value of \( x \), I start by understanding the given information.
Each baby plant produces 2 new plants each time.
This reproduction occurs \( x \) times a year.
Over 4 years, the total number of baby plants produced is 16.
The equation based on this is:
\[ 2 \times 4x = 16 \]
Solving for ... | 20 |
Emily just purchased 2 pairs of curtains for $30.00 each and 9 wall prints at $x each. The store also offers an installation service. For $50.00 they will come to your house and professionally hang your curtains and prints. If Emily agrees to this service, Her entire order will cost 245. What is the value of unknown... | tiger_math | <think>
Emily bought 2 pairs of curtains and 9 wall prints. The total cost before installation was 2 multiplied by $30 plus 9 multiplied by $x, which is 2*30 + 9x.
Adding the $50 installation fee, her total cost becomes 2*30 + 9x + 50. According to the problem, this total equals $245.
Setting up the equation: 2*30 +... | 45 |
Emily just purchased 2 pairs of curtains for $30.00 each and 9 wall prints at $x each. The store also offers an installation service. For $50.00 they will come to your house and professionally hang your curtains and prints. If Emily agrees to this service, Her entire order will cost 245. What is the value of unknown... | tiger_math | <think>
First, determine the total cost of the curtains. Emily purchased 2 pairs at $30 each, so the cost is 2 multiplied by $30, which equals $60.
Next, calculate the total cost of the wall prints. She bought 9 prints at $x each, resulting in a cost of 9x.
The installation service costs $50.
The total cost of the o... | 47 |
Emily just purchased 2 pairs of curtains for $30.00 each and 9 wall prints at $x each. The store also offers an installation service. For $50.00 they will come to your house and professionally hang your curtains and prints. If Emily agrees to this service, Her entire order will cost 245. What is the value of unknown... | tiger_math | <think>
First, identify all the costs involved in Emily's purchase. She bought 2 pairs of curtains at $30 each and 9 wall prints at an unknown price of $x each. Additionally, she opted for the installation service which costs $50.
Next, calculate the total cost by summing up the cost of curtains, wall prints, and the ... | 46 |
Cherry put up a delivery service. She charges $2.50 for a 3-5 kilograms cargo and $4 for a 6-8 kilograms cargo. If she delivers four 5 kilograms cargo and two 8 kilograms cargo per day, how much money will she earn in a week? | tiger_math | <think>
First, I need to determine the total earnings Cherry makes from each type of cargo per day.
For the 5-kilogram cargo, which falls under the 3-5 kilograms category, there is a charge of $2.50 each. Delivering four of these would generate 4 multiplied by $2.50.
Similarly, for the 8-kilogram cargo, which falls u... | 25 |
Cherry put up a delivery service. She charges $2.50 for a 3-5 kilograms cargo and $4 for a 6-8 kilograms cargo. If she delivers four 5 kilograms cargo and two 8 kilograms cargo per day, how much money will she earn in a week? | tiger_math | <think>
First, I need to calculate the daily earnings by multiplying the number of each cargo type by its respective charge.
For a single 5-kilogram cargo, Cherry charges $2.50. Delivering four of these would earn her 4 * $2.50 = $10.00 per day.
For a single 8-kilogram cargo, she charges $4.00. Delivering two of thes... | 26 |
A train passes a man standing on a platform in 8 seconds and also crosses the platform which is 276 metres long in 20 seconds. The length of the train (in metres) is:
Answer Choices: (A) 184 (B) 176 (C) 175 (D) 96 (E) None of these | tiger_lab_math | <think>
To determine the length of the train, I'll start by defining the variables involved. Let's denote the length of the train as \( L \) and its speed as \( S \).
When the train passes the man standing on the platform, it covers its own length. The time taken for this is 8 seconds. Using the formula \( \text{Dista... | 38 |
At a book fair, 25% of the books sold were fiction books, and the average price of these books was $15. The remaining books sold were non-fiction books, with an average price of $20. If the total revenue from book sales was $14,000, what was the total number of books sold? | nvidia_math | <think>
First, I'll define the variables for the total number of books sold as \( N \). According to the problem, 25% of these books are fiction, so the number of fiction books sold is \( 0.25N \), and the number of non-fiction books sold is \( 0.75N \).
Next, I'll calculate the revenue generated from fiction books by... | 35 |
Let $a,$ $b,$ $c$ be distinct roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | tiger_math | <think>
Okay, so I need to compute the determinant of this 3x3 matrix where the rows are cyclic permutations of each other. The matrix is:
\[
\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}
\]
Alright, let me recall how to compute a determinant for a 3x3 matrix. The standard method is the rule of Sa... | 39 |
If $f^{-1}(g(x))=x^2+2$ and $g$ has an inverse, find $g^{-1}(f(4))$. | tiger_math | <think>
Okay, so I have this problem: If \( f^{-1}(g(x)) = x^2 + 2 \) and \( g \) has an inverse, find \( g^{-1}(f(4)) \). Hmm, alright, let's try to unpack this step by step.
First, let's recall what \( f^{-1} \) means. \( f^{-1}(y) \) is the value \( x \) such that \( f(x) = y \). So if \( f^{-1}(g(x)) = x^2 + 2 \),... | 1 |
Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AB$ is 8 cm. What is the area of triangle $ABC$, in square centimeters? | tiger_math | <think>
Okay, so I have this problem about an isosceles right triangle, ABC, where angle A is 90 degrees. The length of segment AB is 8 cm, and I need to find the area of triangle ABC. Hmm, let me try to visualize this.
First, an isosceles right triangle has two legs that are equal in length and the hypotenuse. Since... | 13 |
A train passes a man standing on a platform in 8 seconds and also crosses the platform which is 276 metres long in 20 seconds. The length of the train (in metres) is:
Answer Choices: (A) 184 (B) 176 (C) 175 (D) 96 (E) None of these | tiger_lab_math | <think>
Okay, so I have this problem here about a train passing a man standing on a platform. It takes 8 seconds to pass the man and 20 seconds to cross the platform, which is 276 meters long. I need to find the length of the train. The answer choices are A)184, B)176, C)175, D)96, E)None of these. Hmm, I think I need ... | 37 |
A train passes a man standing on a platform in 8 seconds and also crosses the platform which is 276 metres long in 20 seconds. The length of the train (in metres) is:
Answer Choices: (A) 184 (B) 176 (C) 175 (D) 96 (E) None of these | tiger_lab_math | <think>
Okay, so I have this problem here about a train passing a man on a platform. Let me see. It says the train passes the man in 8 seconds, and it also crosses the platform which is 276 meters long in 20 seconds. They're asking for the length of the train. The options are given, but I need to figure it out myself f... | 36 |
Let $a,$ $b,$ $c$ be distinct roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | tiger_math | <think>
Alright, so I've got this determinant to compute, and it's a 3x3 matrix where each row is a cyclic permutation of the roots a, b, c of the cubic equation x³ + px + q = 0. Hmm, interesting problem! Let me try to break it down.
First off, determinants can often be tricky, but maybe there's a pattern or a proper... | 40 |
Cherry put up a delivery service. She charges $2.50 for a 3-5 kilograms cargo and $4 for a 6-8 kilograms cargo. If she delivers four 5 kilograms cargo and two 8 kilograms cargo per day, how much money will she earn in a week? | tiger_math | <think>
First, I will calculate the daily earnings from the 5-kilogram cargoes. Cherry delivers 4 cargoes of 5 kilograms each, with each cargo costing $2.50. So, \( 4 \times \$2.50 = \$10 \) per day from these deliveries.
Next, I will calculate the daily earnings from the 8-kilogram cargoes. She delivers 2 cargoes of ... | 24 |
If $f^{-1}(g(x))=x^2+2$ and $g$ has an inverse, find $g^{-1}(f(4))$. | tiger_math | <think>
Okay, so I have this problem here: If \( f^{-1}(g(x)) = x^2 + 2 \) and \( g \) has an inverse, find \( g^{-1}(f(4)) \). Hmm, let's see. I need to find \( g^{-1}(f(4)) \). I know that \( f^{-1} \) is the inverse function of \( f \), which means that \( f^{-1}(y) = x \) is equivalent to \( f(x) = y \). Similarly,... | 2 |
Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AB$ is 8 cm. What is the area of triangle $ABC$, in square centimeters? | tiger_math | <think>
Okay, so, I need to figure out the area of triangle ABC, which is an isosceles right triangle with a right angle at A. They told me that the length of segment AB is 8 cm. Hmm, okay, let's break this down step by step.
First, let's visualize the triangle. Since it's an isosceles right triangle, that means two ... | 12 |
Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AB$ is 8 cm. What is the area of triangle $ABC$, in square centimeters? | tiger_math | <think>
Alright, so here I am trying to find the area of triangle ABC. It's given as an isosceles right triangle with angle A being 90 degrees. Hmm, okay, so let me break this down step by step.
First, what does it mean for a triangle to be an isosceles right triangle? Well, "isosceles" means that two sides are equal ... | 14 |
If $f^{-1}(g(x))=x^2+2$ and $g$ has an inverse, find $g^{-1}(f(4))$. | tiger_math | <think>
Okay, so I have this problem: If \( f^{-1}(g(x)) = x^2 + 2 \) and \( g \) has an inverse, find \( g^{-1}(f(4)) \). Hmm, let me try to figure this out step by step.
First, I know that \( f^{-1}(g(x)) = x^2 + 2 \). Since \( f^{-1} \) is the inverse function of \( f \), that means if I apply \( f \) to both sides... | 0 |
Let $a,$ $b,$ $c$ be distinct roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | tiger_math | <think>
Alright, so I've got this problem where I need to compute the determinant of a 3x3 matrix, and the elements of this matrix are the roots of a cubic equation. Specifically, the matrix is:
\[
\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}
\]
And the roots a, b, c all come from the equation \(... | 41 |
What is the Lebesgue measure of the set of all Liouville numbers, and how does it compare to the Lebesgue measure of the set of all real transcendental numbers? | automath | <think>
Alright, so I’m trying to figure out the Lebesgue measure of the set of Liouville numbers and how it compares to the Lebesgue measure of all real transcendental numbers. Let me start by recalling what Lebesgue measure is. From what I remember, Lebesgue measure is a way of assigning a "volume," "area," or "lengt... | 15 |
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