instruction stringlengths 10 13.6k | input stringclasses 1
value | output stringlengths 1 10.4k |
|---|---|---|
A box contains 5 different colored scarves. In how many ways can 3 scarves be chosen to make a set, considering that the set must include at least one scarf of a specific color (say, blue)? | We need to find the number of ways to choose 3 scarves out of 5, ensuring that at least one of them is blue. There are two scenarios to consider:
1. The set contains exactly one blue scarf.
In this case, we choose 1 blue scarf (which can be done in only 1 way since there's only 1 blue scarf) and 2 scarves from the ... | |
Ava and her family went on a vacation. On the first day, they drove 250 miles. On the second day, they covered 2/3 of the distance they traveled on the first day. On the third day, they drove 3/4 as many miles as the total distance traveled on the first two days. What is the total distance Ava and her family traveled o... | On the first day, Ava and her family drove 250 miles.
On the second day, they covered 2/3 of the distance they traveled on the first day which is (2/3)*250 = 500/3 = 166.67 miles.
So the total distance traveled in the first two days is 250 + 166.67 = 416.67 miles.
On the third day, they drove 3/4 as many miles as the t... | |
A three-digit number $ABC$ is reversed to form a second three-digit number $CBA$. If $A$, $B$, and $C$ are distinct digits, what is the greatest common divisor of the differences $ABC - CBA$ and $2 \times ABC + CBA$? | Let's consider the three-digit numbers $ABC$ and $CBA$.
The number $ABC$ can be expressed as:
\[ ABC = 100A + 10B + C \]
The reversed number $CBA$ can be expressed as:
\[ CBA = 100C + 10B + A \]
We are interested in the differences $ABC - CBA$ and $2 \times ABC + CBA$.
1. First, let's find $ABC - CBA$:
\[ ABC - CBA... | |
Emily aims to collect $800.00 over 8 months for her summer vacation. She already has $150.00 in her savings account to contribute towards this goal. How much money does Emily now need to save each month in order to reach her target in 8 months? | To solve this, let's calculate how much Emily still needs to save.
She aims to collect $800 and already has $150, so she still needs to save $800 - 150 = 650$.
She has 8 months to reach her target, so we divide the amount she still needs to save by the number of months she has to save it.
Thus, the amount she needs to ... | |
In a high school auditorium, there are a certain number of teachers and 2 more than 5 times that many students. If there are 72 people total, how many teachers are there? | Let $x$ be the number of teachers.
The number of students is $5*x + 2$.
The total number of people is the sum of the number of teachers and students, so we have $x + (5*x + 2) = 72$.
Solving for $x$, we have $6*x + 2 = 72$, which implies $6*x = 70$.
Thus $x = 70 / 6$, which gives $x = 11.6667$, but since $x$ must be an... | |
Find the sum of the positive divisors of 24. | First, let's find the positive divisors of 24. A divisor is a number that divides another number exactly without a remainder.
The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
Now, let's sum these divisors:
\[ 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 3 + 3 + 4 + 6 + 8 + 12 + 24 = 6 + 4 + 6 + 8 + 12 + 24 = 10 + 6 + 8 + 12 + ... | |
A bakery is having a sale on its famous sourdough bread. A regular loaf normally costs $2.50, but during the sale, it costs $2.25. A whole wheat loaf normally costs $3.25, but during the sale, it costs $2.90. If Sara buys 2 regular loaves and 3 whole wheat loaves during the sale, how much will she save compared to buyi... | To find out how much Sara will save, we need to calculate the total cost of the bread at the regular price and at the sale price, then find the difference.
Regular loaves:
- Regular price: 2 loaves $\times$ $2.50 = $5.00
- Sale price: 2 loaves $\times$ $2.25 = $4.50
Whole wheat loaves:
- Regular price: 3 loaves $\tim... | |
Given that $\cos A = \frac{3}{5}$ and $\cos B = \frac{2}{7}$, find the value of $\cos (A-B)$. | ## Step 1: Recall the formula for $\cos (A-B)$
The formula for $\cos (A-B)$ is given by $\cos (A-B) = \cos A \cos B + \sin A \sin B$.
## Step 2: Use the given values of $\cos A$ and $\cos B$
Given that $\cos A = \frac{3}{5}$ and $\cos B = \frac{2}{7}$, we can substitute these values into the formula.
## Step 3: Find ... | |
The value of the expression $\frac{\sqrt{2}+\sqrt{4}}{\sqrt{3}+\sqrt{3}}$ is closest to which whole number? | Evaluate the expression:
\[ \frac{\sqrt{2} + \sqrt{4}}{\sqrt{3} + \sqrt{3}} = \frac{\sqrt{2} + 2}{\sqrt{3} + \sqrt{3}} = \frac{\sqrt{2} + 2}{2\sqrt{3}} \]
Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$:
\[ \frac{\sqrt{2} + 2}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\... | |
Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If the area of $\triangle ABC$ is $150$ and $AC = 25,$ then what is $BD$? | Since the circle has diameter $BC$, we know that $\angle BDC$ is a right angle.
We can use the formula for the area of a triangle: $\text{Area} = \frac{1}{2}(\text{base})(\text{height})$. In $\triangle ABC$, $AC$ is the base and $BD$ is the height.
Given that the area of $\triangle ABC$ is $150$ and $AC = 25$, we can... | |
Three boxes contain different amounts of money. The total amount of money in all three boxes is $\$ 845$. Two of the boxes together contain $\$ 455$, and one of these two boxes contains $\$ 34$ more than the other. How much money is in the third box? | ## Step 1: Let's denote the amount of money in the three boxes as A, B, and C.
We know that the total amount of money in all three boxes is $845, so we can write the equation: A + B + C = 845.
## Step 2: Two of the boxes together contain $455.
We can assume that these two boxes are A and B, so we can write the equatio... | |
Let $a$ and $b$ be the roots of the equation $x^2-6x+13=0$. Find the value of $a^3+b^3$. | Use Vieta's formulas to find the sum and product of the roots:
\begin{align*}
a + b &= 6 \\
ab &= 13
\end{align*}
Then, use the formula for $a^3 + b^3$:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Substitute the known values:
\[ a^3 + b^3 = 6(a^2 - 13 + b^2) \]
To find $a^2 + b^2$, use the fact that:
\[ (a... | |
Mrs. Thompson purchased five times as many books for her daughter as she purchased for her son. If the son received 30 books, how many books did Mrs. Thompson buy? | If the son received 30 books, and Mrs. Thompson purchased five times as many books for her daughter, then the daughter received 30 * 5 = 150 books.
So, Mrs. Thompson purchased a total of 30 + 150 = 180 books.
The answer is \boxed{180}. | |
The area of the smallest circle that will contain a square of side length 6 is what? | The diagonal of the square will be the diameter of the smallest circle that contains the square.
Using the Pythagorean theorem to find the diagonal $d$ of the square:
\[ d^2 = 6^2 + 6^2 = 36 + 36 = 72 \Rightarrow d = \sqrt{72} \]
Since the diagonal of the square is the diameter of the circle, the radius $r$ of the ci... | |
Alex visited a bakery and bought 6 trays of cookies. Each tray had 15 cookies. He gave 2 trays to his friend and a dozen cookies to his brother. How many cookies did Alex have left? | Let's solve this problem.
First, we calculate the total number of cookies Alex bought.
He bought 6 trays of cookies, with 15 cookies per tray, for a total of 6 * 15 = 90 cookies.
Then, he gave 2 trays of cookies to his friend. Since each tray has 15 cookies, he gave 2 * 15 = 30 cookies to his friend.
Additionally, h... | |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | The prime numbers between 4 and 18 are: 5, 7, 11, 13, and 17.
We need to find the number that could be obtained by taking the product of two primes and subtracting their sum.
Let $x$ and $y$ be two different prime numbers.
We want to find:
\[ xy - (x + y) = xy - x - y \]
Consider each option:
Option A: $21$
Option... | |
It takes Emily 8 minutes to ride her bike a mile if she doesn't have to carry her backpack. If she has to carry her backpack, it takes 11 minutes to ride a mile. How many minutes longer would it take her to ride 25 miles if she had to carry her backpack? | Without her backpack, Emily rides 25 miles in $8*25 = 200$ minutes.
With her backpack, Emily rides 25 miles in $11*25 = 275$ minutes.
So it would take her $275-200 = 75$ minutes longer.
Thus, it would take Emily \boxed{75} minutes longer to ride 25 miles if she had to carry her backpack. | |
Max attended 40 college courses in 2 years. Sid attended four times as many college courses as Max in the same period. What's the total of college courses they both attended? | Max attended 40 courses, and Sid attended four times as many courses as Max.
Sid attended 40 * 4 = 160 courses.
To find the total number of courses they both attended, add the number of courses Max attended to the number of courses Sid attended.
40 + 160 = 200
So, Max and Sid attended \boxed{200} courses in total. | |
A quadrilateral has vertices at coordinates (0,0), (7,1), (4,3), and (1,5). What is the area of the quadrilateral? | To find the area of the quadrilateral, divide it into two triangles and find the area of each triangle.
Triangle 1: (0,0), (7,1), (4,3)
Triangle 2: (0,0), (4,3), (1,5)
For each triangle, use the formula:
\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]
For Triangle 1:
\[ \text{Area}... | |
Find the area in the plane contained by the graph of
\[|x + 2y| + |x - 2y| \le 8.\] | The given inequality is:
\[ |x + 2y| + |x - 2y| \le 8 \]
The graph of this inequality is a parallelogram with vertices at $(4, 2), (-4, 2), (-4, -2), (4, -2)$.
To find the area of the parallelogram, we need to find the length of the base and the height.
The length of the base is the distance between $(4, 2)$ and $(-... | |
The debate team at our school has 7 boys and 9 girls, and we need to choose a group of 5 students to participate in an upcoming tournament. How many ways can we select the group without any restrictions? | We need to select a group of 5 students from a total of 16 students (7 boys and 9 girls) without any restrictions.
This is a combination problem, because the order of selection does not matter.
The formula for combination is:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
where $n$ is the total number of items, and $k$ is the n... | |
Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that $f(f(x)) = g(g(x))$ for all real numbers $x$. Given that $a + b = 123$, find $c + d$. | We write out the expressions for $f(f(x))$ and $g(g(x))$:
\begin{align*}
f(f(x)) &= f(x^2 + ax + b)\\
&= (x^2 + ax + b)^2 + a(x^2 + ax + b) + b\\
&= x^4 + 2ax^3 + (a^2 + 2b)x^2 + (ab + a)x + b^2 + ab + b,
\end{align*}and
\begin{align*}
g(g(x)) &= g(x^2 + cx + d)\\
&= (x^2 + cx + d)^2 + c(x^2 + cx + d) + d\\
&= x^4 + 2c... | |
Solve for $n$: $5^{2n + 1} = \frac{1}{25}$. Express your answer as a common fraction. | To solve the equation for $n$, start by expressing $\frac{1}{25}$ as a power of 5:
\[ \frac{1}{25} = \frac{1}{5^2} = 5^{-2} \]
So, the equation becomes
\[ 5^{2n + 1} = 5^{-2} \]
Since the bases are the same, we can equate the exponents:
\[ 2n + 1 = -2 \]
Solve for $n$:
\[ 2n = -3 \Rightarrow n = \boxed{-\frac{3}{2}}... | |
Let $g(x)$ be a function such that $g(x) = g(x + k)$ for all real numbers $x$ and some constant $k > 0$. If $g(0) = 5$, what is the value of $g(k/2)$? | To find the value of $g(k/2)$, we can use the given periodicity condition.
Given that $g(x) = g(x + k)$ for all real numbers $x$, we can choose $x = -k/2$ to get
\[ g(-k/2) = g(-k/2 + k) = g(k/2) \]
Now, we can choose $x = -k/2 + k = k/2$ to get
\[ g(k/2) = g(k/2 + k) = g(3k/2) \]
We notice that we can continue addi... | |
What is the maximum number of consecutive even integers that can be added together before the sum exceeds 600? | To find the maximum number of consecutive even integers that can be added together before the sum exceeds 600, we can consider the sum of the first $n$ even integers.
The sum of the first $n$ even integers is given by the formula $n(n + 1)$.
We want to find the largest value of $n$ such that $n(n + 1) \leq 600$.
Let... | |
Let $a$, $b$, and $c$ be complex numbers such that $|a - b| = 3$, $|b - c| = 4$, and $|a - c| = 6$. Find the largest possible value of $|z - a| + |z - b| + |z - c|$, where $z$ is a complex number. | To approach this problem, consider the following steps:
1. Interpret the given conditions geometrically:
- $|a - b| = 3$ means $a$ and $b$ are 3 units apart.
- $|b - c| = 4$ means $b$ and $c$ are 4 units apart.
- $|a - c| = 6$ means $a$ and $c$ are 6 units apart.
2. Recognize that these conditions suggest a ... | |
Find the value of $\theta$ when $-2 + i \sqrt{2}$ is expressed in exponential form as $re^{i \theta}$. | We know that $e^{i \theta} = \cos \theta + i \sin \theta$. The complex number $-2 + i \sqrt{2}$ can be expressed in the form $re^{i \theta}$, where $r$ is the magnitude of the complex number and $\theta$ is the angle.
First, we find the magnitude $r$ of the complex number:
\[ r = \sqrt{(-2)^2 + (\sqrt{2})^2} = \sqrt{4... | |
What is the remainder when $17^9 \div 17^7$ is divided by 10? | We can simplify the expression $17^9 \div 17^7$ using the quotient of powers rule, which states that $a^m \div a^n = a^{m - n}$:
\[ 17^9 \div 17^7 = 17^{9 - 7} = 17^2 \]
Now, let's find the remainder when $17^2$ is divided by 10:
\[ 17^2 = 289 \]
\[ 289 \div 10 = 28 \quad \text{with a remainder of } 9 \]
So, the rema... | |
Emily and Ryan are competing in a chess tournament. Ryan is leading by 18 points when he makes a 12-point move. Emily responds with a 25-point counterattack. By how many points is Ryan now ahead? | Ryan had a lead of 18 points initially.
After his 12-point move, he is leading by 18 + 12 = 30 points.
However, after Emily's 25-point counterattack, his lead is reduced to 30 - 25 = 5 points.
So Ryan is now ahead by \boxed{5} points. | |
Emma has 12 marbles. 5 of them are blue, 4 of them are red, and 3 of them are purple. If she selects 2 marbles at random, what is the probability that both marbles she selects are purple? | ## Step 1: Calculate the total number of ways to choose 2 marbles out of 12.
There are $\binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2} = 66$ ways to choose 2 marbles out of 12.
## Step 2: Calculate the number of ways to choose 2 purple marbles out of 3.
There are $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \... | |
Tom and Jerry are two friends who have been saving money for a vacation. Tom has saved 30% more than Jerry, and together they have saved $2400. If Jerry has saved $100 less than the amount Tom has saved, how much has Tom saved? | Let's assume that Jerry has saved $x$. Since Tom has saved 30% more than Jerry, Tom has saved $x + 0.3x = 1.3x$.
We know that together they have saved $2400, so:
\[ x + 1.3x = 2400 \Rightarrow 2.3x = 2400 \Rightarrow x = 2400 : 2.3 \Rightarrow x = 1043.48 \]
We also know that Jerry has saved $100 less than Tom, so:
\... | |
A landscape designer is planning to cover a rectangular garden bed with mulch. The garden bed is 60 inches long and 30 inches wide. If a 50-pound bag of mulch can cover 120 square inches of the bed to a suitable depth, how many pounds of mulch are required to cover the entire garden bed? | To find the total amount of mulch required, we need to calculate the area of the garden bed and then divide it by the area covered by one bag of mulch.
The area of the garden bed is:
60 inches x 30 inches = 1800 square inches
One 50-pound bag of mulch covers 120 square inches, so:
1800 square inches / 120 square inch... | |
A map has a scale of 1 cm: x km, where x is an unknown value. The distance between two cities on the map is 120 cm. If the actual distance between the two cities is 1800 km, what is the value of x? | Since the map scale is 1 cm: x km, the distance between the two cities on the map (120 cm) represents 120x km in real life.
We are given that the actual distance between the two cities is 1800 km.
Set up the equation:
\[ 120x = 1800 \]
Solve for x:
\[ x = 1800 : 120 \Rightarrow x = 15 \]
The value of x is:
\[ \boxe... | |
What is the sum of the digits of the number $3^{10} + 2^{11} + 1^{12}$? | Calculate $3^{10}$, $2^{11}$ and $1^{12}$.
\[ 3^{10} = 59049 \]
\[ 2^{11} = 2048 \]
\[ 1^{12} = 1 \]
Now add these numbers:
\[ 3^{10} + 2^{11} + 1^{12} = 59049 + 2048 + 1 = 61097 + 1 = 61098 \]
Sum the digits of $61098$:
\[ 6 + 1 + 0 + 9 + 8 = 7 + 0 + 9 + 8 = 7 + 9 + 8 = 16 + 8 = 24 \]
So, the answer is:
\[ \boxed{2... | |
A quadrilateral $ABCD$ is inscribed in a circle with center $O$ and radius $r$. The diagonals $AC$ and $BD$ intersect at $E$, and the area of $\triangle ABE$ is twice the area of $\triangle CDE$. If $AB = r\sqrt{2}$ and $CD = r\sqrt{3}$, find the value of $OE + r$. | Since the quadrilateral is inscribed in a circle, we can use the properties of inscribed angles and triangles.
The diagonals $AC$ and $BD$ intersect at $E$, and the area of $\triangle ABE$ is twice the area of $\triangle CDE$.
Let's denote $AE = x$, $BE = y$, $CE = z$, and $DE = w$.
Since $AB = r\sqrt{2}$ and $CD = ... | |
A snail is at the bottom of a 20-foot well. Each day, it climbs up 4 feet, but at night, it slips back 3 feet. How many days will it take for the snail to reach the top of the well? | Let's break down the snail's progress day by day:
Day 1: The snail climbs 4 feet, so it is at 4 feet. At night, it slips back 3 feet, so it is at 1 foot.
Day 2: The snail climbs 4 feet, so it is at 5 feet. At night, it slips back 3 feet, so it is at 2 feet.
Day 3: The snail climbs 4 feet, so it is at 6 feet. At nigh... | |
A farmer is planning to sell a shipment of 120 boxes of apples at a market. Each box contains 8 apples, and the farmer wants to package them in bags of 4 apples each for easier sale. If the farmer can sell the apples in bags of 4 for $2 each, but the market charges a $1 commission for each bag sold, what is the minimum... | To find the minimum number of bags the farmer must sell, we first need to calculate the total cost and the selling price per bag.
Each box has 8 apples, and the farmer wants to package them in bags of 4 apples each. So, the number of bags from one box is $8 : 4 = 2$ bags.
The farmer has 120 boxes, so the total number... | |
Find the coefficient of x in the expansion of $(x^2+x+1)^4$. | To find the coefficient of $x$ in the expansion of $(x^2+x+1)^4$, we can use the Binomial Theorem.
Since the term we are interested in has $x$ raised to the power of 1, we must choose one $x$ from the four binomials and then choose the remaining terms such that the exponents add up to 1.
We can choose the $x$ term fr... | |
From Monday to Friday, Jackson exercises for 45 minutes a day. On Saturday and Sunday, he exercises for 1 hour and 30 minutes. He also takes an extra 30-minute walk on Wednesday and Friday. How many minutes does he exercise in 6 weeks? | Let's calculate the number of minutes Jackson exercises in a week.
From Monday to Friday, he exercises for 5 * 45 = 225 minutes.
On Saturday and Sunday, he exercises for 2 * 90 = 180 minutes.
Additionally, he walks for 30 minutes on Wednesday and Friday, so that's 2 * 30 = 60 minutes.
So, in total, he exercises for 225... | |
Find the sum of the reciprocals of the positive divisors of 24. | The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The sum of the reciprocals of these divisors is:
\[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{12} + \frac{1}{24} \]
Find a common denominator for the fractions:
\[ \frac{24}{24} + \frac{12}{24} + \frac{8}{... | |
Ashley borrowed money from her sister 8 months ago. She has been paying back $15 per month. Currently, she has paid off one-third of the amount she borrowed. How much will she still owe her sister 3 months from now? | Let's denote the amount Ashley borrowed as $x$.
Ashley has been paying back $15 per month for 8 months.
So, the total amount she paid back is $8 * 15 = 120$.
The problem states that this amount is one-third of what she borrowed.
This can be written as $120 = x / 3$ or $x = 120 * 3 = 360$.
So Ashley borrowed $360.
In 8 ... | |
A bag contains 12 red balls and 8 blue balls. If two balls are drawn randomly from the bag without replacement, what is the probability that the second ball drawn is blue, given that the first ball drawn is red? Express your answer as a common fraction. | The probability of the second ball being blue given that the first ball is red can be found by considering the total number of balls left after drawing the first red ball and the number of those that are blue.
Initially, there are 20 balls in the bag (12 red and 8 blue).
After drawing a red ball first, there are 19 b... | |
Maya, Jaxon, and Landon are participating in a school project where they need to purchase rabbits and study their behavior. Maya bought three times the number of rabbits that Jaxon bought, while Jaxon bought 20 fewer rabbits than Landon. If Maya bought 36 rabbits, what's the average number of rabbits the three students... | Let's break down the information given.
Maya bought 36 rabbits.
Maya bought three times the number of rabbits that Jaxon bought, so Jaxon bought 36 / 3 = 12 rabbits.
Jaxon bought 20 fewer rabbits than Landon, so Landon bought 12 + 20 = 32 rabbits.
Now we can calculate the average number of rabbits the three students b... | |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | The expression $(x+y+z)(xy+xz+yz)$ expands to
\[ (x+y+z)(xy+xz+yz) = x^2(y+z) + y^2(x+z) + z^2(x+y) + 3xyz \]
From the second equation, we know that:
\[ x^2(y+z) + y^2(x+z) + z^2(x+y) = 6 \]
Substitute this into the expanded expression:
\[ 6 + 3xyz = 18 \Rightarrow 3xyz = 12 \Rightarrow xyz = \boxed{4} \] | |
Simplify the expression: $\frac{1}{3}(2x^3 + \frac{1}{x^2}) + \frac{2}{3}(x^2 - 5x)$. | To simplify the expression, we need to apply the distributive property and combine like terms.
The expression is: $\frac{1}{3}(2x^3 + \frac{1}{x^2}) + \frac{2}{3}(x^2 - 5x)$
Distribute $\frac{1}{3}$ and $\frac{2}{3}$:
\[ \frac{2x^3}{3} + \frac{1}{3x^2} + \frac{2x^2}{3} - \frac{10x}{3} \]
This is the simplified expre... | |
Find the number of non-decreasing sequences of four numbers from the set {1, 2, 3, 4, 5, 6}, where each number can be used any number of times. | We can solve this problem by stars and bars method, where the stars represent the four numbers to be chosen and the bars represent the dividers between the numbers 1, 2, 3, 4, 5, and 6.
Let's represent the sequence as stars and bars. The stars will represent the numbers in the sequence and the bars will separate the d... | |
Consider a sequence of functions defined recursively as follows:
\begin{align*}
f_1(x) &= x^2 + 1 \\
f_{n+1}(x) &= f_n(f_n(x))
\end{align*}
Find the value of $f_3(1)$. | We need to find the value of $f_3(1)$ using the recursive definition of the sequence of functions.
Step 1: Calculate $f_1(1)$.
\[ f_1(1) = 1^2 + 1 = 2 \]
Step 2: Calculate $f_2(1)$ using the recursive definition.
\[ f_2(1) = f_1(f_1(1)) = f_1(2) = 2^2 + 1 = 5 \]
Step 3: Calculate $f_3(1)$ using the recursive definit... | |
The polynomial $P(x)$ is divisible by $(x - a)^3,$ and the polynomial $Q(x)$ is divisible by $(x - b)^4.$ Given that $P(x)Q(x)$ is divisible by $(x - ab)^7,$ what is the smallest possible degree of $P(x)Q(x)$? | ## Step 1: Understand the divisibility conditions for P(x) and Q(x)
P(x) is divisible by (x - a)^3, meaning it has a root 'a' with multiplicity of at least 3. Similarly, Q(x) is divisible by (x - b)^4, indicating it has a root 'b' with multiplicity of at least 4.
## Step 2: Understand the divisibility condition for P(... | |
Dave breaks 2 guitar strings per night when playing live. If he performs 6 shows a week for 12 weeks, how many guitar strings will he need to replace? | Dave performs 6 shows a week for 12 weeks, so he performs a total of 6 * 12 = 72 shows.
In each show, he breaks 2 guitar strings.
Thus, the total number of strings he needs to replace is 72 * 2 = 144.
So Dave will need to replace \boxed{144} guitar strings. | |
A water tank can be filled by two pipes, A and B. Pipe A fills the tank at a rate of 2 cubic meters per hour, while pipe B fills it at a rate of 3 cubic meters per hour. However, due to a leak, the tank is being drained at a rate of 0.5 cubic meters per hour. If the tank is filled in 4.5 hours when both pipes are open,... | Let's denote the capacity of the tank as $C$ cubic meters.
The combined rate at which both pipes fill the tank is $2 + 3 = 5$ cubic meters per hour.
However, due to the leak, the effective rate at which the tank is being filled is $5 - 0.5 = 4.5$ cubic meters per hour.
The time it takes to fill the tank is given as ... | |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| The line through the origin that cuts the parallelogram into two congruent polygons is the diagonal of the parallelogram.
To find the equation of this line, first we find the midpoint of the diagonal:
\[ \left( \frac{10 + 28}{2}, \frac{45 + 153}{2} \right) = \left( \frac{38}{2}, \frac{198}{2} \right) = (19, 99) \]
Th... | |
In right triangle $XYZ$ with $\angle YXZ = 90^\circ$, we have $XY = 24$ and $YZ = 25$. Find $\tan Y$. | Given the triangle $XYZ$ with $\angle YXZ = 90^\circ$, $XY = 24$, and $YZ = 25$, we need to find $\tan Y$. Recall that $\tan Y = \frac{\text{opposite side}}{\text{adjacent side}}$.
To find $\tan Y$, we need the length of side $XZ$. Use the Pythagorean theorem:
\[ XZ^2 = YZ^2 - XY^2 = 25^2 - 24^2 = 625 - 576 = 49 \Righ... | |
If Heidi can paint a wall in 45 minutes, what fractional part of the wall can she paint in 9 minutes? | Heidi paints a wall in 45 minutes. We want to know how much of the wall she can paint in 9 minutes.
Let's find the fraction of the wall she can paint in 1 minute:
\[ \frac{1}{45} \text{ of the wall per minute} \]
To find out how much of the wall she can paint in 9 minutes, multiply the fraction of the wall she can pa... | |
A certain organization holds a competition for its members in which each member can participate in one or more events out of ten. Every event has exactly four participants, and for each event, there is a unique winner who earns 3 points, while the other three participants receive 1 point each. After all the events are ... | Let $n$ be the number of participants.
Since each event has exactly four participants, the total number of participants in all events is $4 \times 10 = 40$.
Since the scores of all participants are distinct, we can list the scores in decreasing order:
\[ s_1 > s_2 > \cdots > s_n \]
Let $e_i$ be the number of events ... | |
Maria is planning a dessert of chocolate chip cookies for her book club meeting. Her recipe calls for 1/6 of a cup of sugar per batch of 8 cookies. Maria has 12 book club members, including herself. If she uses 3 cups of sugar to make cookies, and each member of the book club eats an equal number of cookies, how many c... | To find out how many cookies Maria will eat, we first need to determine the total number of cookies made and then divide that number by the total number of book club members.
Since the recipe calls for 1/6 of a cup of sugar per batch of 8 cookies, and Maria uses 3 cups of sugar, we first need to find out how many batc... | |
Five distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the second largest is $10$? | To find the probability that among the five distinct integers picked at random from $\{1,2,3,\ldots,12\}$, the second largest is $10$, we need to consider the total number of possible selections and the number of successful outcomes where $10$ is the second largest.
1. Total possible selections: The number of ways to ... | |
Determine the sum of all digits appearing in the decimal expansion of $\frac{1}{11}$ before the first repetition. | The decimal expansion of $\frac{1}{11}$ is $0.090909\ldots$.
Therefore, the sum of all digits appearing in the decimal expansion of $\frac{1}{11}$ before the first repetition is $0 + 9 = \boxed{9}$. | |
Let $f(x)$ be the function defined as follows:
$f(1) = 3$
$f(n) = \underbrace{f(f(\ldots f}_{n-1 \text{ times}}(1)\ldots)) + 2$
Let $N$ be the least value of $n$ for which $f(n)$ is divisible by $7$. Determine the value of $N$. | ## Step 1: Understand the function definition
The function $f(x)$ is defined recursively. The base case is $f(1) = 3$. For any other positive integer $n$, $f(n)$ is defined as applying the function $f$ to the result of $f(n-1)$ and then adding $2$.
## Step 2: Calculate the first few values of $f(n)$
To understand the ... | |
At the University of California, the ratio of students to professors in the Computer Science department is 30:2. How many new professors will the university need to hire to lower this ratio to 25:2, if currently there are 900 students? | ## Step 1: Calculate the current number of professors
To find the current number of professors, we can use the given ratio of students to professors, which is 30:2. Since there are 900 students, we can set up a proportion to find the number of professors. Let's denote the number of professors as P. The proportion will ... | |
When $\sqrt[3]{3^5\cdot5^2}$ is fully simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers. What is $c+d$? | To simplify the expression $\sqrt[3]{3^5\cdot5^2}$, we need to break it down into factors that can be expressed as cubes and those that cannot.
The expression can be rewritten as:
\[ \sqrt[3]{3^5\cdot5^2} = \sqrt[3]{3^3\cdot3^2\cdot5^2} \]
We can factor out the perfect cube $3^3$ from the expression:
\[ \sqrt[3]{3^3\... | |
What is $\sqrt[5]{243} \cdot \sqrt[4]{16} \cdot \sqrt[2]{25}$ expressed as a positive integer? | First, simplify each radical expression:
$\sqrt[5]{243} = \sqrt[5]{3^5} = 3$,
$\sqrt[4]{16} = \sqrt[4]{2^4} = 2$,
$\sqrt[2]{25} = \sqrt[2]{5^2} = 5$.
Now, multiply these values:
\[ 3 \cdot 2 \cdot 5 = \boxed{30} \] | |
How many integers between 1000 and 9999, inclusive, have at least one digit that is a power of 2 (i.e., 1, 2, 4, or 8)? | To find the number of integers between 1000 and 9999 that have at least one digit that is a power of 2, we first find the total number of integers between 1000 and 9999 and then subtract the number of integers that do not have any digit that is a power of 2.
The total number of integers between 1000 and 9999 is 9000.
... | |
Emily is baking cookies for a party. She needs to bake three batches, which takes 20 minutes per batch. Then she can decorate all the cookies, which takes 40 minutes. How long does her cookie preparation take in total? | To find the total time Emily needs for cookie preparation, we need to add up the time it takes for baking and decorating.
The baking time is 20 minutes per batch and she needs to bake 3 batches.
So the total baking time is $3 * 20 = 60$ minutes.
The decorating time is 40 minutes.
So the total preparation time is $60... | |
Ryan rode his skateboard at an average speed of 8 miles per hour for two and a half hours. If his friend, Alex, skates for 2 hours, at what average speed, in miles per hour, would he have to skate to travel the same distance that Ryan traveled? | Ryan's distance can be found by multiplying his speed by his time:
\[ \text{distance} = \text{speed} \times \text{time} = 8 \times 2.5 = 20 \text{ miles} \]
Alex needs to travel the same distance in 2 hours, so we set up the equation:
\[ \text{distance} = \text{speed} \times \text{time} \Rightarrow 20 = \text{speed} \... | |
The value of $\log_{5}{23}$ is between the consecutive integers $c$ and $d$. Find $c+d$. | To solve this problem, we need to find the two consecutive integers between which the value of $\log_{5}{23}$ lies.
Recall that the logarithmic function $\log_{a}{x}$ gives the exponent to which the base $a$ must be raised to produce the number $x$. In this case, we're looking for the power to which 5 must be raised t... | |
Let
$$g(x)=\left\{\begin{array}{lcl}\sqrt{x^2+1}&\text{if}&x<-1\\\frac{1}{x}&\text{if}&-1\le x\le 1\\x^2-c&\text{if}&x>1\end{array}\right.$$Find $c$ if the piecewise function $g$ is continuous. | The function $g(x)$ is continuous if and only if the three pieces of the function agree at the breakpoints $x=-1$ and $x=1$. So we need to find the value of $c$ that satisfies
\begin{align*}
\sqrt{(-1)^2+1}&=\frac{1}{-1}\\
\frac{1}{1}&=1^2-c
\end{align*}The first equation is satisfied, since $\sqrt{2}\neq -1$. The seco... | |
A function $f:\{1,2,3,4,5,6,7\}\rightarrow \{1,2,3,4,5,6,7\}$ has the property that for all $n\in \{1,2,3,4,5,6,7\}$,
$$f(f(n))+f(n+1)+1\equiv 0\pmod 7.$$
Determine the smallest possible value of
$$f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7).$$ | Let $S=f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7)$. Since $f(f(n))+f(n+1)\equiv -1\pmod 7$ for all $n$, we have
$$\sum_{n=1}^7 f(f(n))+\sum_{n=1}^7 f(n+1)\equiv -7\pmod 7$$
Notice that
$$\sum_{n=1}^7 f(f(n))=\sum_{n=1}^7 f(n)=S$$
since the two sums contain the same terms. Also, $\sum_{n=1}^7 f(n+1)=\sum_{n=2}^8 f(n)=S-f(1)+f(... | |
Ben rolls four fair 10-sided dice, each numbered with numbers from 1 to 10. What is the probability that exactly two of the dice show a prime number? | A prime number on a 10-sided die can be 2, 3, 5, or 7, so the probability of getting a prime number on one roll is $4 : 10 = 2 : 5$. The probability of not getting a prime number is $6 : 10 = 3 : 5$.
Since we want exactly two dice to show a prime number, we can choose which two of the four dice show primes in $\binom{... | |
On the island of Mumble, the Mumblian alphabet has only $5$ letters, and every word in the Mumblian language has no more than $3$ letters in it. How many words are possible? (A word can use a letter more than once, but $0$ letters does not count as a word.) | Count the number of possible words for each length separately and add them up.
\begin{itemize}
\item There are $5$ one-letter words, since there are $5$ possible letters.
\item There are $5\cdot 5=25$ two-letter words, since each letter can be any of the $5$ possible letters, and the choices are independent... | |
Find the value of $a$ such that the sum of the cubes of the roots of the equation $x^3 + ax^2 + x + a = 0$ is equal to $-1$. | Let the roots of the equation $x^3 + ax^2 + x + a = 0$ be $r$, $s$, and $t$.
The sum of the cubes of the roots is $r^3 + s^3 + t^3$.
We know that $r + s + t = -a$.
Also, $r^3 + s^3 + t^3 - 3rst = (r + s + t)(r^2 + s^2 + t^2 - rs - st - tr)$.
We can find $rst$ from the equation: $rst = -a$.
We can also find $r^2 + ... | |
How many different positive, five-digit integers can be formed using the digits 1, 1, 3, 7, and 7? | Since we have 5 digits with 2 repeated twice and 1 repeated twice, we can use the formula for permutations with repeated elements to calculate the number of different positive, five-digit integers that can be formed.
The formula is:
\[ \frac{n!}{n_1! \cdot n_2! \cdot... \cdot n_k!} \]
where:
- $n$ is the total numbe... | |
The digits of a three-digit number are randomly generated so that the hundreds digit is odd, the tens digit is prime, and the units digit is even. The prime digits are 2, 3, 5, and 7. How many such numbers can be generated, if the digits are not necessarily distinct, and each digit that meets the given condition is equ... | Let's break down the conditions for each digit:
- The hundreds digit is odd, so it can be 1, 3, 5, 7, or 9. That's 5 possibilities.
- The tens digit is prime, so it can be 2, 3, 5, or 7. That's 4 possibilities.
- The units digit is even, so it can be 0, 2, 4, 6, or 8. That's 5 possibilities.
To find the total number ... | |
Find the monic cubic polynomial, in $x,$ with real coefficients, which has $3 + i \sqrt{7}$ as a root. | If $3 + i \sqrt{7}$ is a root, then $3 - i \sqrt{7}$ must also be a root, since the polynomial has real coefficients.
Let's find the product of the corresponding factors:
\[ (x - (3 + i \sqrt{7}))(x - (3 - i \sqrt{7})) = (x - 3 - i \sqrt{7})(x - 3 + i \sqrt{7}) \]
\[ = (x - 3)^2 - (i \sqrt{7})^2 \]
\[ = x^2 - 6x + 9... | |
In a flower shop, there are 15 different types of flowers available. How many ways can Emma choose 5 flowers to create a bouquet if the order of selection does not matter? | Since the order of selection does not matter, this is a combination problem. The formula for combinations is given by:
\[ C(n, k) = \frac{n!}{k!(n - k)!} \]
where $n$ is the total number of items and $k$ is the number of items to choose.
In this case, Emma is choosing 5 flowers out of 15, so $n = 15$ and $k = 5$.
Sub... | |
Points $Q,$ $P,$ $R,$ and $T$ lie on a straight line. Point $S$ is positioned off the line such that $\angle QPS = 100^\circ$ and $\angle SRT = 60^\circ.$ If the reflex angle at $S$ is $y^\circ,$ what is the value of $y$? | Since $\angle QPS$ and $\angle SRT$ are angles formed by a line intersecting two other lines, we know that:
\[ \angle QPS + \angle SRT = 180^\circ \]
However, in this problem, we need to find the reflex angle at $S$, which is the total measure of the angle formed at $S$ minus the acute angle at $S$. To do this, we fir... | |
Consider the lines \begin{align*}
y&=2x-1 \\
3y&=4x+2 \\
2y&=6x-3 \\
3y&=x+1 \\
y&=x+2
\end{align*}Let's say that a pair of lines is $\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many pairs are good? | We have the lines:
\begin{align*}
y &= 2x - 1 \\
y &= \frac{4}{3}x + \frac{2}{3} \\
y &= 3x - \frac{3}{2} \\
y &= \frac{1}{3}x + \frac{1}{3} \\
y &= x + 2
\end{align*}
To determine whether a pair of lines is good, we need to find the slope of each line. Two lines are parallel if they have the same slope, and two lines... | |
Ashley has 15 dresses, 8 tops, and 10 jeans in her closet when everything is clean and put away. Currently, in her laundry basket, she has 60% of her dresses, 25% of her tops, and 80% of her jeans. How many items of clothing will Ashley need to wash? | To find out how many items of clothing Ashley needs to wash, we need to calculate 60% of her dresses, 25% of her tops, and 80% of her jeans.
For the dresses, 60% of 15 is $15 * 60 / 100 = 900 / 100 = 9$.
For the tops, 25% of 8 is $8 * 25 / 100 = 200 / 100 = 2$.
For the jeans, 80% of 10 is $10 * 80 / 100 = 800 / 100 = ... | |
Given $wz = 4$, determine the value of $(w + z)^2 - (w^2 + z^2)$. | The expression $(w + z)^2 - (w^2 + z^2)$ can be simplified as follows:
\begin{align*}
(w + z)^2 - (w^2 + z^2) &= (w^2 + 2wz + z^2) - (w^2 + z^2)\\
&= w^2 + 2wz + z^2 - w^2 - z^2\\
&= 2wz
\end{align*}
Given $wz = 4$, we substitute this value into the simplified expression:
\[ 2wz = 2(4) = 8 \]
So, the value of $(w + z... | |
Find the number of lattice points that satisfy the inequality $x^2 + y^2 \leq 100$ and lie on a circle with center at the origin and radius $\sqrt{2}$.
What is the number of lattice points on the graph of the equation $x^2 - 2y^2 = 3$? | For the inequality $x^2 + y^2 \leq 100$, the number of lattice points satisfying this condition is the number of points inside or on the circle with radius 10 centered at the origin.
The number of lattice points on the circle with radius $\sqrt{2}$ centered at the origin is 4: $(1, 1)$, $(1, -1)$, $(-1, 1)$, $(-1, -1)... | |
Alexandra is a freelance writer and receives her payment on a monthly basis. She always sets aside one-third of her income for taxes. In April, she earned $7200 and spent $1200 on a new laptop. What is her total amount of remaining money at the end of the month? | To find the total amount of remaining money at the end of the month, we need to subtract the amount Alexandra spent on the laptop and the amount she set aside for taxes from her total income.
She earned $7200 and spent $1200 on the laptop, so we first subtract $1200 from $7200 to get $7200 - $1200 = $6000.
Then, we n... | |
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other ar... | This problem can be approached by graph theory.
Consider each person as a vertex and each handshake as an edge in a graph. We are interested in counting the number of different possible graphs that satisfy the given conditions.
Since each person shakes hands with exactly two other people, each vertex must have a degr... | |
$\Delta ABC$ is isosceles with $AC = BC$. If $m\angle C = 40^{\circ}$, what is the number of degrees in $m\angle CBD$? [asy] pair A,B,C,D,E;
C = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);
draw(B--C--A--E); dot(D);
label("$A$",A,S); label("$B$",B,S); label("$D$",D,S); label("$C$",C,N);
[/asy] | The triangle $\Delta ABC$ is isosceles, meaning $AC = BC$, and $\angle C = 40^{\circ}$. Because the triangle is isosceles, $\angle A$ and $\angle B$ must be equal. Let $x = \angle A = \angle B$. The sum of angles in a triangle is $180^{\circ}$, so $x + x + 40^{\circ} = 180^{\circ} \Rightarrow 2x + 40^{\circ} = 180^{\ci... | |
Isabella is shopping at an electronics store. She has 12 gadgets worth $25 each in her basket. As she is about to checkout, she adds 15 smartwatches to her basket. If the $945 in Isabella's wallet is exactly enough to pay for all the gadgets and smartwatches, how much does each smartwatch cost? | Let's break this down step by step:
Isabella has 12 gadgets worth $25 each, so the total cost of the gadgets is:
12 x $25 = $300
She adds 15 smartwatches to her basket, and the total cost of the gadgets and smartwatches is $945.
To find the cost of the smartwatches, we need to subtract the cost of the gadgets from t... | |
A sociology student conducted a survey to find the average daily expenditure of college students on food. According to her survey, 3/5 of the students spend an average of $8 per day on food, while the rest spend an average of $5 per day. If she surveyed 75 students, what is the total amount of money those 75 students s... | The number of students spending an average of $8 per day is 3/5 * 75 = 0.6 * 75 = 45
The number of students spending an average of $5 per day is 2/5 * 75 = 0.4 * 75 = 30
The total amount of money those 75 students spend on food in a day is $8 * 45 + $5 * 30.
This equals to $360 + $150 = 510.
Thus, the total amount of m... | |
A farmer has 240 crates of apples. He was planning to sell them all for $60,000 when 90 crates were damaged during a storm. Due to the damage, his buyers became wary of the quality, forcing him to lower his price by $20 per crate. How much money would the farmer lose if he sold the remaining crates at the lowered price... | Let's break down the problem step by step:
1. The farmer had 240 crates of apples and wanted to sell them for $60,000. This means the price per crate was $60,000 / 240 = $250 per crate.
2. Due to the storm, 90 crates were damaged, leaving 240 - 90 = 150 crates.
3. The farmer had to lower the price by $20 per crate, ma... | |
A bakery sells a special type of bread that comes in 0.8 kg loaves. If a customer buys 3.5 loaves, how many 0.25 kg slices can they cut from the total amount of bread, assuming no crumbs or waste? | First, we need to find the total weight of bread the customer bought.
Since each loaf weighs 0.8 kg and the customer bought 3.5 loaves:
\[ \text{Total weight} = 3.5 \times 0.8 \, \text{kg} = 2.8 \, \text{kg} \]
Now, we need to find out how many 0.25 kg slices can be cut from 2.8 kg of bread:
\[ \text{Number of slices... | |
Suppose $x \not= 0$. We can write $(8x + 5 + 3x^2) + (4x - 2)$ in the form $px + q + rx^2$, where $p$, $q$, and $r$ are integers. Find $p + q + r$. | $(8x + 5 + 3x^2) + (4x - 2)$
$= 8x + 5 + 3x^2 + 4x - 2$
$= (8x + 4x) + (5 - 2) + 3x^2$
$= 12x + 3 + 3x^2$
$= 12x + 3 + 3x^2$
Therefore, $p = 12$, $q = 3$, and $r = 3$.
So, $p + q + r = 12 + 3 + 3 = 15 + 3 = 18$.
The answer is $\boxed{18}$. | |
Emily works for a company that offers a flexible leave policy: for every 12 days worked, she earns 1 day of paid leave. Last year, Emily worked 320 days and took 4 days off in June and three times as many in November. How many more paid leave days can Emily still take? | To determine how many more paid leave days Emily can still take, we first need to find out how many paid leave days she earned in total and then subtract the leave days she has already taken.
Emily earns 1 paid leave day for every 12 days worked. She worked 320 days, so she earned 320 / 12 = 26.67 paid leave days. Sin... | |
A right-angled triangle has side lengths which are consecutive integers and the length of its hypotenuse is also a perfect square. What is its smallest possible area? | Let the lengths of the triangle's legs be $x$ and $x+1$, and the hypotenuse be $x+2$.
By the Pythagorean theorem, we have:
\[ x^2 + (x+1)^2 = (x+2)^2 \Rightarrow x^2 + x^2 + 2x + 1 = x^2 + 4x + 4 \Rightarrow x^2 - 2x - 3 = 0 \]
Solving the quadratic equation gives:
\[ (x - 3)(x + 1) = 0 \Rightarrow x = 3 \text{ or } ... | |
There are 20 stickers on a page. If you have 12 pages of stickers but lose one of the pages, then how many stickers would you have? | Since there are 20 stickers on each page, having 12 pages of stickers amounts to having $12 * 20 = 240$ stickers.
But then one page of stickers was lost which amounts to losing 20 stickers.
Thus the total number of stickers after losing a page is $240 - 20 = 220$ stickers.
So you would have \boxed{220} stickers. | |
The infinite sequence $S=\{s_1,s_2,s_3,\ldots\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$? | Notice that the pattern in the remainders follows the pattern in the powers of 7, but it eventually gets stuck in an infinite loop. If we start calculating the sequence modulo 5 we get:
\[ s_1 = 7 \equiv 2 \pmod 5 \]
\[ s_2 = 7^{s_1} = 7^7 \equiv 2^7 = 128 \equiv 3 \pmod 5 \]
\[ s_3 = 7^{s_2} \equiv 7^3 \equiv... | |
Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. | To solve this problem, we can factor the first equation $8x^2 + 7x - 1 = 0$ as follows:
\[ (8x - 1)(x + 1) = 0 \]
This gives us two solutions for $x$:
\[ 8x - 1 = 0 \Rightarrow x = \frac{1}{8} \text{ and } x + 1 = 0 \Rightarrow x = -1 \]
Now we will use these two solutions in the second equation $24x^2 + 53x - 7 = 0$... | |
Julia has 5 times as many books as Emily and 3 fewer books than Olivia. If they have 201 books all together, how many books does Emily have? | Let's denote the number of books Emily has as $x$.
Since Julia has 5 times as many books as Emily, Julia has $5x$ books.
Since Julia has 3 fewer books than Olivia, Olivia has $5x + 3$ books.
Since they have 201 books all together, the total number of books they have can be written as: $x + 5x + (5x + 3) = 201$.
Combini... | |
The diagonals of a parallelogram intersect at the point $O$. Two sides of the parallelogram lie on the lines $y = 3x$ and $y = 5x$. What is the sum of the $x$- and $y$-coordinates of $O$? | Since the diagonals of a parallelogram bisect each other, $O$ is the midpoint of both diagonals.
The diagonals intersect at $O$. To find $O$, we need to find the intersection point of the lines $y = 3x$ and $y = 5x$.
Setting the equations equal to each other:
\[ 3x = 5x \Rightarrow 2x = 0 \Rightarrow x = 0 \]
Substi... | |
A museum charges $8 for an admission ticket. Three-fifths of the visitors will buy a guided tour for $10, one-fifth will purchase a souvenir for $6, and one-tenth will attend a workshop for $12. The museum sold tickets worth $1440. How much money did the museum make in total? | Let's break down the problem step by step:
1. The museum sold tickets worth $1440, and each ticket costs $8, so the number of visitors is:
1440 / 8 = 180
2. Three-fifths of the visitors will buy a guided tour, which is:
(3/5) x 180 = 108 visitors
The total amount spent on guided tours is:
108 x 10 = 1080
3. One-... | |
What is the maximum number of students that can be enrolled in the fall term of the `06-`07 academic year, given that the student headcount for the fall terms of the `02-`03, `03-`04, and `04-`05 academic years were 12,300, 12,800, and 13,100, respectively, and that the student headcount has been increasing by an avera... | To find the maximum number of students that can be enrolled in the fall term of the `06-`07 academic year, we first need to calculate the increase in the student headcount for each of the given years.
Year `02-`03 to `03-`04: $12,800 - 12,300 = 500$
Year `03-`04 to `04-`05: $13,100 - 12,800 = 300$
The average increa... | |
A box contains 15 coins, 10 of which are fair coins and 5 of which are biased coins with a probability of landing heads of $\frac{1}{3}$. A coin is randomly selected from the box and flipped 3 times. What is the probability that exactly two of the flips result in heads? Express your answer as a common fraction. | First, we need to find the probability of getting exactly two heads in three flips for a fair coin and a biased coin.
For a fair coin, the probability of getting exactly two heads in three flips is $\binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = \frac{3}{8}$.
For a biased coin, the probability o... | |
The logarithm of $2x$ is greater than $3$ and less than $4$. How many positive integer values of $x$ satisfy this condition? | The condition given is $3 < \log(2x) < 4$.
Recall that the logarithm is an increasing function, so we can exponentiate all parts of the inequality to get:
\[ 10^3 < 2x < 10^4 \Rightarrow 1000 < 2x < 10000 \]
Divide all parts of the inequality by 2:
\[ 500 < x < 5000 \]
We need to find how many positive integers are ... | |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some citie... | Consider the number of routes from $A$ to $B$ as $x$. Then, consider the number of routes from $B$ to $A$ as $y$.
We can see that for each route from $A$ to $B$, there is exactly one corresponding route from $B$ to $A$. Thus, the total number of routes is the same in both directions.
So $x = y$.
Let the number of ro... | |
Linda owns a store. She sells jeans at 11 dollars each and tees at 8 dollars each. If she sold 7 tees and 4 jeans in a day, how much money, in dollars, did she have at the end of the day? | To calculate how much money Linda had at the end of the day, we need to multiply the number of items sold by their respective prices and then add the two amounts together.
The total amount of money from selling jeans is $11 * 4 = 44$ dollars.
The total amount of money from selling tees is $8 * 7 = 56$ dollars.
So, Li... |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 12