problem
stringlengths 18
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|---|---|---|
Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$.
|
800
| 0.5 |
Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains.
|
60
| 0.333333 |
What is $\dfrac{2^2+4^2+6^2}{1^2+3^2+5^2} - \dfrac{1^2+3^2+5^2}{2^2+4^2+6^2}$
|
\dfrac{1911}{1960}
| 0.25 |
Given the numbers $5-2\sqrt{8}$, $2\sqrt{8}-5$, $12-3\sqrt{9}$, $27-5\sqrt{18}$, and $5\sqrt{18}-27$, find the smallest positive number.
|
2\sqrt{8}-5
| 0.75 |
If tree saplings are planted around a square garden, each sapling being 2 feet apart, calculate the number of saplings needed if one side of the garden measures 30 feet.
|
60
| 0.75 |
$R$ varies directly as $S$ and inversely as $T$. When $R = 2$ and $T = \frac {8}{5}$, $S = \frac{1}{2}$. Find $S$ when $R = 16$ and $T = \sqrt {50}$.
|
\frac{25 \sqrt{2}}{2}
| 0.75 |
Given Carlos took $60\%$ of a whole pie, and Maria took one fourth of the remainder, determine the portion of the whole pie that was left.
|
30\%
| 0.75 |
In a chess tournament, 64 players are wearing red hats, and 68 players are wearing green hats. A total of 132 players are paired into 66 teams of 2. If exactly 20 of these teams consist of players both wearing red hats, find the number of teams composed of players both wearing green hats.
|
22
| 0.666667 |
Points $A$ and $B$ are on a circle of radius $7$ and $AB = 8$. Point $C$ is the midpoint of the minor arc $AB$. Find the length of the line segment $AC$.
|
\sqrt{98 - 14\sqrt{33}}
| 0.083333 |
Triangle OAB has O=(0,0), B=(8,0), and A in the first quadrant. In addition, ∠ABO=90° and ∠AOB=45°. If OA is rotated 90° counterclockwise about O, find the coordinates of the image of A.
|
(-8, 8)
| 0.333333 |
Points A and B are 8 units apart. Determine the number of lines in a given plane containing A and B that are 3 units from A and 5 units from B.
|
3
| 0.166667 |
If each team consists of 18 players, and each player must be paid at least $20,000, and the total salary for the entire team cannot exceed $900,000, calculate the maximum possible salary that a single player on a team can receive.
|
560,000
| 0.666667 |
Given Shauna's scores on the first four tests are 79, 88, 94, and 91, and the maximum points for each test is 100, find the lowest score she could earn on one of the remaining two tests if she must score at least 75 on each test and maintain an average of 85 for all six tests.
|
75
| 0.75 |
Determine the coefficient of $x^6$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9$.
|
0
| 0.833333 |
A triangle $PQR$ is formed in the first quadrant such that vertex $P$ is at $(0,0)$ and points $Q$ and $R$ are on the lines $y=2x$ and $y=3x$, respectively, with both $Q$ and $R$ as lattice points. The area of triangle $PQR$ is $500,000$. Find the number of such triangles.
|
49
| 0.916667 |
$800$ students were surveyed about their pasta preferences. According to the survey results: $320$ students preferred spaghetti and $160$ students preferred manicotti. Calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti.
|
2
| 0.75 |
Given two poles that are $30''$ and $70''$ high standing $150''$ apart, find the height of the intersection of lines joining the top of each pole to the foot of the opposite pole.
|
21
| 0.583333 |
A bag contains five pieces of paper, each labeled with one of the digits $1$, $2$, $3$, $4$, or $5$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Determine the probability that the three-digit number is a multiple of $3$.
|
\frac{2}{5}
| 0.916667 |
Given that a and b are non-zero constants, evaluate the expression $(2a^{-1} + 3b^{-1})^{-1}$.
|
\frac{ab}{2b+3a}
| 0.666667 |
Josanna's test scores to date are $88, 92, 75, 81, 68, 70$. What is the minimum test score she would need to reach a new average of her test scores that is $5$ points higher than the current average?
|
114
| 0.75 |
Alice and Bob start at point 15 on a circle divided into 15 equally spaced points. Alice moves 7 points clockwise each turn, while Bob moves 11 points counterclockwise. Determine the number of turns it takes for them to meet at the same point.
|
5
| 0.916667 |
A rug is designed with three colors, creating three distinct rectangular regions. The areas of these regions form an arithmetic progression. The innermost rectangle has a width of 2 feet, and each of the two shaded regions surrounding it is 2 feet wide on all sides. Find the length in feet of the innermost rectangle.
|
4
| 0.166667 |
Given the expression $2-(-3)-4\times(-5)-6-(-7)-8\times(-9)+10$, evaluate this expression.
|
108
| 0.916667 |
Given a cell phone plan that costs $20$ dollars each month, plus $10$ cents per text message sent, plus $15$ cents for each minute used over 25 hours, determine Alex's total cost for the month of February given that he sent 150 text messages and talked for 32 hours.
|
\$98
| 0.916667 |
Consider the set of all fractions $\frac{x}{y}$ where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both the numerator and the denominator are increased by 1, the value of the fraction is increased by $20\%$?
|
0
| 0.25 |
Given the sum of the degree measures of the interior angles of a convex polygon is $3025^\circ$, calculate the degree measure of the missing angle.
|
35^\circ
| 0.833333 |
It takes Clea 70 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Her walking speed increases by 50% when the escalator is moving. Determine the time it would take Clea to ride the escalator down when she is not walking.
|
84
| 0.166667 |
Given a square with side length \( s \), find the ratio of the combined areas of the inscribed semicircle and quarter-circle to the area of the square.
|
\frac{3\pi}{16}
| 0.75 |
Marie starts working on four identical tasks at 9:00 AM. She finishes the first two tasks at 11:20 AM. Calculate the time when all four tasks will be completed.
|
1:40 \text{ PM}
| 0.916667 |
Given the volume expansion rate of $5$ cubic centimeters for every $4^\circ$ rise in temperature, and the initial volume of $30$ cubic centimeters at $28^\circ$, determine the volume of the gas in cubic centimeters when the temperature was $12^\circ$.
|
10
| 0.75 |
Let $x$ be the total number of marbles, then $\frac{2}{3}x$ of them are green and the rest, $\frac{1}{3}x$, are yellow. If the number of yellow marbles is tripled, find the fraction of marbles that will be yellow.
|
\frac{3}{5}
| 0.083333 |
The average of 4, 6.5, 8, x, and y is 18, so what is the average of x and y?
|
35.75
| 0.916667 |
Given the equation $y^2 + x^4 = 2x^2 y + 1$, find the absolute difference between the y-coordinates of the distinct points $(\sqrt{e}, a)$ and $(\sqrt{e}, b)$.
|
2
| 0.916667 |
Given the number $2025$, calculate the number of different four-digit numbers that can be formed by rearranging its digits.
|
9
| 0.75 |
A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape.
|
34
| 0.083333 |
Given Jason rolls three fair standard six-sided dice, determine the probability that he needs to reroll either exactly two or all three dice for the best chance of winning.
|
\frac{1}{2}
| 0.166667 |
Calculate the number of positive integers less than $1200$ that are divisible by neither $8$ nor $7$.
|
900
| 0.083333 |
At Fibonacci Middle School, the ratio of 10th-graders to 8th-graders is 7:4, and the ratio of 10th-graders to 9th-graders is 9:5. Find the smallest number of students that could be participating in a specific school activity.
|
134
| 0.833333 |
Given that Ms. Blue receives a 12% raise each year, calculate the total percent increase in her salary after five consecutive raises.
|
76.23\%
| 0.833333 |
Alex earned $x$ amount of money from a summer job. He used one-fourth of his money to buy one-half of the video games he wanted, where each video game costs the same amount. After that, he also spends one-sixth of his money on gaming accessories. Determine the fraction of his money he has left after buying all the video games and the accessories.
|
\frac{1}{3}
| 0.166667 |
Given m = \(\frac{c^2ab}{a - kb}\), express b in terms of m.
|
\frac{ma}{c^2a + mk}
| 0.416667 |
Two containers each contain the same number of marbles, and every marble is either red or yellow. In Container $A$ the ratio of red to yellow marbles is $7:3$, and the ratio of red to yellow marbles in Container $B$ is $4:1$. There are $120$ yellow marbles in total. Calculate the difference between the number of red marbles in Container $A$ and the number of red marbles in Container $B$.
|
24
| 0.916667 |
Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 7 times as old as Bonita. This year Ana's age is the cube of Bonita's age. What is n?
|
6
| 0.916667 |
Given that the calculator initially displays $\frac{1}{2}$, and the special key that replaces the number $x$ with $\frac{1}{1-x}$ is pressed 50 times, calculate the value displayed on the calculator after the 50th application of the key.
|
-1
| 0.833333 |
How many distinguishable rearrangements of the letters in the word "COMPLEX" have the vowels first?
|
240
| 0.833333 |
A list of $2030$ positive integers has a unique mode, which occurs exactly $11$ times, what is the least number of distinct values that can occur in the list?
|
203
| 0.166667 |
The vertex of the parabola $y = 2x^2 - 10x + c$ will be a point on the $x$-axis. Determine the value of $c$ for which this condition holds.
|
12.5
| 0.333333 |
Two lines with slopes $-1$ and $3$ intersect at $(1,3)$. Find the area of the triangle enclosed by these two lines and the line $x-y=2$.
|
8
| 0.916667 |
The graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$ intersect at points $(1,7)$ and $(9,1)$. Find the sum $a+c$.
|
10
| 0.5 |
Given that Lucy bakes a $24$-inch by $20$-inch pan of brownies and she wants to cut the brownies into rectangular pieces that measure $4$ inches by $3$ inches, calculate the number of brownie pieces she can cut from the pan.
|
40
| 0.166667 |
Given that 80 students were from Pinecrest Academy, 60 students were from Maple Grove School, 70 students attended from Maple Grove School, 30 of the boys were from Pinecrest Academy, and 90 students were girls, calculate the number of girls who were from Maple Grove School.
|
40
| 0.833333 |
Alberto bikes at a constant rate and covers 80 miles in 4 hours. Bjorn bikes at a rate of 20 miles per hour for the first 2 hours and then increases his speed to 25 miles per hour for the next 2 hours. Determine the difference in the distance Alberto has biked compared to Bjorn after four hours.
|
10
| 0.916667 |
Given that Liam's test score in base 7 has tens digit x and unit digit y, where 0 ≤ x, y < 7, determine the possible discrepancy between his incorrect sum and the correct one, after reversing the digits of his test score.
|
0, 6, 12, 18, 24, 30, 36
| 0.166667 |
Determine the least number of integers among $a, b, c, d$ that can be negative such that the equation $2^a + 2^b = 5^c + 5^d$ holds.
|
0
| 0.083333 |
Given the book's cover dimensions are 5 inches by 7 inches, where each reported dimension can be 0.5 inches more or less than stated, find the minimum possible area of the book cover.
|
29.25
| 0.916667 |
Evaluate \((x^x)^{(x^x)}\) at \(x = 3\).
|
27^{27}
| 0.833333 |
Given that each of a group of $60$ girls is either redhead or brunette, and either green-eyed or brown-eyed, 20 are green-eyed redheads, 35 are brunettes, and 25 are brown-eyed, determine the number of brown-eyed brunettes.
|
20
| 0.583333 |
Given two numbers whose sum is $10$ and the absolute value of whose difference is $12$, determine the equation that has these two numbers as its roots.
|
x^2-10x-11=0
| 0.25 |
For how many positive integers m is $\dfrac{2310}{m^2-2}$ a positive integer?
|
3
| 0.833333 |
The largest product one could obtain by multiplying two numbers in the set $\{ -8, -3, 0, 2, 4 \}$ is.
|
24
| 0.083333 |
Given the operation $a\textdollar b = (a - b)^2$, evaluate $(x - y + z)^2\textdollar(y - x - z)^2$.
|
0
| 0.916667 |
Estimate the population of the island of Atlantis in the year 2060, assuming the current population in 2000 is 400 and it doubles every 20 years, but reduces by 25% in 2040.
|
2400
| 0.5 |
Given two equiangular polygons \(P\) and \(Q\) with different numbers of sides; each angle of \(P\) is \(p\) degrees, and each angle of \(Q\) is \(q\) degrees, determine the number of possibilities for the pair \((p, q)\) where either \(p = 2q\) or \(q = 2p\).
|
2
| 0.333333 |
Given $100 \leq a \leq 300$ and $400 \leq b \leq 800$, and $a + b \leq 950$, find the smallest possible value of the quotient $\frac{a}{b}$.
|
\frac{1}{8}
| 0.666667 |
If $x$ cows produce $x+3$ cans of milk in $x+4$ days, calculate the number of days it will take $x+4$ cows to produce $x+6$ cans of milk.
|
\frac{x(x+6)}{x+3}
| 0.666667 |
Given that two points $(p, q)$ and $(r, s)$ lie on the line whose equation is $y=nx+m$, determine the distance between these two points in terms of $p, r, m,$ and $n$.
|
|r - p|\sqrt{1+n^2}
| 0.916667 |
Find the coefficient of \(x^9\) in the polynomial expansion of \((1+3x-2x^2)^5\).
|
240
| 0.916667 |
Let \( p \), \( q \), \( r \), and \( s \) be positive integers with \( p < 3q \), \( q < 4r \), and \( r < 5s \). If \( s < 90 \), find the largest possible value for \( p \).
|
5324
| 0.833333 |
On a different math exam, 15% of the students got 80 points, 20% got 85 points, 25% got 90 points, 30% got 95 points, and the rest got 100 points. Find the difference between the mean and the median score on this exam.
|
0
| 0.916667 |
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ so that the mean (average) of the remaining numbers is $7$?
|
3
| 0.916667 |
For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. Calculate $(2x + y)^2\textdollar(x - 2y)^2$.
|
\left(3x^2 + 8xy - 3y^2\right)^2
| 0.083333 |
Determine the number of distinct ordered pairs $(x,y)$ where both $x$ and $y$ have positive integral values, and they satisfy the equation $x^6y^6 - 19x^3y^3 + 18 = 0$.
|
1
| 0.916667 |
Given that Emma wrote down one integer four times and another integer three times, and the sum of these seven numbers is 140, and one of the numbers is 20, find the other number.
|
20
| 0.5 |
Let the base and exponent of $c^d$ be quadrupled, resulting in $s=(4c)^{4d}$. Also, let $s$ equal the square of the product of $c^d$ and $y^d$, resulting in $s=(c^d)(y^d)^2$. Express y in terms of c.
|
16c
| 0.083333 |
Given the expression \[4^{2+4+6} - (4^2 + 4^4 + 4^6)\], evaluate this expression.
|
16772848
| 0.666667 |
Given that runner A can beat runner B by 40 yards, and runner B can beat runner C by 30 yards, and runner A can beat runner C by 65 yards, determine the length of the race.
|
240
| 0.75 |
Given that Emily places 6 ounces of tea into a twelve-ounce cup and 6 ounces of honey into a second cup of the same size, and then adds 3 ounces of lemon juice to the second cup, then she pours half the tea from the first cup into the second, mixes thoroughly, and finally pours one third of the mixture in the second cup back into the first, calculate the fraction of the mixture in the first cup that is now lemon juice.
|
\frac{1}{7}
| 0.083333 |
Given points $E$ and $F$ lie on line segment $\overline{GH}$. The length of $\overline{GE}$ is $3$ times the length of $\overline{EH}$, and the length of $\overline{GF}$ is $7$ times the length of $\overline{FH}$. What fraction of the length of $\overline{GH}$ is the length of $\overline{EF}$?
|
\frac{1}{8}
| 0.583333 |
Let N be the second smallest positive integer that is divisible by every positive integer less than 8. Calculate the sum of the digits of N.
|
12
| 0.916667 |
Given that a 3-digit number has a digit-sum of $27$, how many of these numbers are even?
|
0
| 0.75 |
Given the average age of 8 people in Room A is 30, the average age of 5 people in Room B is 35, and the average age of 7 people in Room C is 40, find the average age of the people in the combined group of 20 individuals.
|
34.75
| 0.75 |
A grocer decides to create a larger display for a special event, stacking oranges in a pyramid-like structure whose new rectangular base measures $7$ oranges by $10$ oranges. Each layer above the first still follows the rule that each orange rests in a pocket formed by four below, reducing by one orange in each dimension per layer above. Moreover, this time the display features not one but three single oranges aligned vertically at the very top. Calculate the total number of oranges used in the entire display.
|
227
| 0.083333 |
Given that Alice has 30 apples, calculate the number of ways she can share them with Becky and Chris so that each of the three people has at least three apples.
|
253
| 0.833333 |
Given a group of students, consisting of boys and girls, such that $50\%$ are girls initially, and after two girls leave and two boys join, $40\%$ of the group are girls, determine the initial number of girls.
|
10
| 0.916667 |
Given that 63 students are wearing green shirts, 69 students are wearing red shirts, and there are 132 students in total, and the students are grouped into 66 pairs with exactly 27 pairs consisting of students wearing green shirts, determine how many pairs are composed of students both wearing red shirts.
|
30
| 0.416667 |
A triangle with side lengths $DB=EB=2$ is cut from an equilateral triangle ABC of side length $5$. Calculate the perimeter of the remaining quadrilateral.
|
13
| 0.25 |
Calculate the value of angle C in a triangle given that angle A = 45 degrees, angle B = 3x, and angle C = (1/2) angle B.
|
45^\circ
| 0.916667 |
Determine the number of different selections Jamie can make from 7 cookies, where he has the option to choose from 4 types of cookies.
|
120
| 0.5 |
Each of two boxes contains three chips numbered $1$, $2$, $4$. Calculate the probability that the sum of the numbers on the two chips drawn from the two boxes is even.
|
\frac{5}{9}
| 0.916667 |
Evaluate $(x^x)^{\sqrt{x^x}}$ at $x = 3$.
|
27^{3 \sqrt{3}}
| 0.333333 |
Given that the car ran exclusively on its battery for the first $60$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.03$ gallons per mile, and averaged $50$ miles per gallon on the whole trip, determine the total distance of the trip in miles.
|
180
| 0.833333 |
When simplified, $(-\frac{1}{216})^{-2/3}$, evaluate the expression.
|
36
| 0.916667 |
Let $S_n = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n$, where $n$ is a positive integer. Calculate $S_{21} + S_{34} + S_{45}$.
|
17
| 0.916667 |
If the product of two numbers is $4$ and their difference is $2$, calculate the sum of their fourth powers.
|
112
| 0.916667 |
A machine is scheduled to process jobs starting at $\text{9:45}\ {\small\text{AM}}$. It finished half of the jobs by $\text{1:00}\ {\small\text{PM}}$. At what time will all the jobs be completely processed?
|
4:15\text{ PM}
| 0.75 |
Given that the video score was $50$, and $75\%$ of the votes were likes, calculate how many votes had been cast on Sangho's video at that point.
|
100
| 0.583333 |
Given the sum of the interior angles of a convex polygon is $3239^\circ$, and one angle was incorrectly omitted, determine the measure of the missing angle.
|
1^\circ
| 0.666667 |
Given that Suzanna increases her distance by 1.5 miles every 7 minutes and takes a 5-minute break after 21 minutes, then continues at the same speed for another 14 minutes, calculate the total distance Suzanna rides.
|
7.5
| 0.916667 |
Two angles of an isosceles triangle measure $60^\circ$ and $x^\circ$. Find the sum of the three possible values of $x$.
|
180^\circ
| 0.833333 |
Given that segment $AB$ is both a diameter of a circle with radius $\sqrt{3}$ and a side of an equilateral triangle $ABC$, and the circle intersects side $AC$ at point $D$, find the length of $AE$, where $E$ is the midpoint of $AC$.
|
\sqrt{3}
| 0.666667 |
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