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stringlengths 18
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|---|---|---|
If $\frac{1}{4}$ of all ninth graders are paired with $\frac{1}{3}$ of all sixth graders, what fraction of the total number of sixth and ninth graders are paired?
|
\frac{2}{7}
| 0.166667 |
In a collection of red, blue, and yellow balls, all but 9 are red, all but 5 are blue, and all but 6 are yellow, calculate the total number of balls in the collection.
|
10
| 0.916667 |
A triangle and trapezoid share the same height. The triangle's base is expanded to 36 inches and their areas remain equal. What is the length of the median of the trapezoid?
|
18
| 0.916667 |
A square floor is covered with congruent square tiles, such that if the sum of the tiles on the two diagonals is equal to 25, determine the total number of tiles needed to cover the entire floor.
|
169
| 0.916667 |
Determine the number of solution-pairs in the non-negative integers of the equation $4x + 7y = 600$.
|
22
| 0.916667 |
A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock.
|
12.925
| 0.166667 |
Given that triangle ABC is right-angled at B, and AD=5, DC=2, determine the area of triangle ABC.
|
\frac{7\sqrt{10}}{2}
| 0.083333 |
In a physics test, 20% of the students scored 60 points, 40% scored 75 points, 25% scored 85 points, and the remaining students scored 95 points. Calculate the difference between the mean and median score of the students' scores on this test.
|
2.5
| 0.666667 |
The pressure of wind on a sail varies jointly as the area of the sail and the square of the velocity of the wind. When the velocity of the wind is 20 miles per hour, the pressure on a 2 square foot area is 4 pounds. Find the velocity of the wind when the pressure on 4 square feet is 64 pounds.
|
40 \sqrt{2} \text{ mph}
| 0.833333 |
Given that the average cost of a long-distance call in the USA in 1995 was $35$ cents per minute, and the average cost of a long-distance call in the USA in 2015 was $15$ cents per minute, find the approximate percent decrease in the cost per minute of a long-distance call.
|
57\%
| 0.166667 |
Given the expression $(1296^{\log_6 4096})^{\frac{1}{4}}$, calculate its value.
|
4096
| 0.583333 |
The sum of the digits of the greatest five-digit number M, whose digits have a product of 90.
|
18
| 0.75 |
Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride.
|
45
| 0.416667 |
Jenny has two identical 12-ounce cups. She pours six ounces of milk into the first cup and six ounces of juice into the second cup. She then transfers one-third of the milk from the first cup to the second cup. After mixing thoroughly, she transfers one-fourth of the mixture from the second cup back to the first cup. What fraction of the liquid in the first cup is now juice?
|
\frac{1}{4}
| 0.583333 |
A hamster is placed at a corner of a square garden with side length $12$ meters. It scurries $7.2$ meters along the diagonal towards the opposite corner, then takes a $90^{\circ}$ left turn and darts $3$ meters further. Calculate the average of the shortest distances from the hamster to each side of the square garden.
|
6
| 0.833333 |
Given that p, q, r, and s are integers in the set {0, 1, 2, 3, 4}, calculate the number of ordered quadruples (p, q, r, s) such that p·s + q·r is odd.
|
168
| 0.25 |
What is the ratio of the least common multiple of 280 and 476 to the greatest common factor of 280 and 476.
|
170
| 0.833333 |
Given the triangle with side lengths $12$, $30$, and $x$, determine how many integers $x$ satisfy the condition that the triangle has one right angle.
|
0
| 0.416667 |
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 30 miles per hour, he will be late by 5 minutes. If he drives at an average speed of 50 miles per hour, he will be early by 5 minutes. Calculate the speed in miles per hour that Mr. Bird needs to drive to get to work exactly on time.
|
37.5
| 0.833333 |
Susie buys $6$ muffins and $4$ bananas. Calvin spends thrice as much as Susie, paying for $3$ muffins and $24$ bananas. Calculate how many times as expensive as a banana is a muffin.
|
\frac{4}{5}
| 0.5 |
At Mountain Valley School, the ratio of $9^\text{th}$-graders to $7^\text{th}$-graders is $7:4$, and the ratio of $9^\text{th}$-graders to $8^\text{th}$-graders is $9:5$. Determine the smallest number of students that could be participating in this project.
|
134
| 0.75 |
A woman born in the first half of the twentieth century was $x$ years old in the year $x^2 + x$. Find the year in which she was born.
|
1936
| 0.25 |
Find the units digit of the decimal expansion of $\left(17 + \sqrt{210}\right)^{20} + \left(17 + \sqrt{210}\right)^{83}$.
|
8
| 0.083333 |
Given triangle XYZ has a total area of 180 square units, point M is the midpoint of XY, and point N is the midpoint of YZ. A line segment is drawn from X to N, dividing triangle XMN into two regions. Calculate the area of triangle XPN when point P is the midpoint of segment XM and line segment PN is drawn.
|
22.5
| 0.5 |
A professional soccer league consists of teams with 23 players each. Each player must be paid at least $20,000, and the total of all players' salaries for each team cannot exceed $800,000. Additionally, no single player can earn more than $450,000. Find the maximum possible salary, in dollars, for a single player.
|
360,000
| 0.75 |
A point Q(c,d) in the xy-plane is first rotated counterclockwise by 90° about the point (2,3) and then reflected about the line y = x. If the image of Q after these transformations is at (7, -4), calculate d - c.
|
3
| 0.5 |
Given the town's original population, it increases by $15\%$ and then decreases by $13\%$, and the resulting population is $50$ fewer people than the original population, find the original population.
|
100,000
| 0.333333 |
What is the probability that a randomly drawn positive factor of $90$ is less than $10$?
|
\frac{1}{2}
| 0.916667 |
What is the value of $\dfrac{13! - 12!}{10!}$?
|
1584
| 0.75 |
How many primes less than $150$ have $3$ as the ones digit?
|
9
| 0.75 |
How many whole numbers between $200$ and $500$ contain the digit $3$?
|
138
| 0.333333 |
Evaluate $(x^x)^{(x^x)}$ at $x = 3$.
|
27^{27}
| 0.916667 |
If $2x, 4x+4, 6x+6, \dots$ are in geometric progression, determine the fourth term.
|
-27
| 0.916667 |
Given that Abe holds 2 blue and 1 green jelly bean, Bob holds 1 blue, 2 green, and 1 yellow jelly bean, and Cara holds 3 blue, 2 green, and 1 red jelly bean, calculate the probability that all three pick jelly beans of the same color.
|
\frac{5}{36}
| 0.583333 |
An integer $N$ is selected at random in the range $1 \leq N \leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$.
|
\frac{6}{7}
| 0.083333 |
Elmer's new car provides 60% better fuel efficiency measured in kilometers per liter compared to his old car, but uses diesel fuel that is 25% more expensive per liter than the gasoline used by his old car. Calculate the percentage by which Elmer will save money when using his new car for a 1000 km trip.
|
21.875\%
| 0.916667 |
In $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. It is known that $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of segment $BC$.
|
125
| 0.5 |
In a similar mistake during multiplication, Tom reversed the digits of the two-digit number c and mistakenly multiplied the reversed number with d, resulting in the erroneous product of 143. Determine the correct product of c and d.
|
341
| 0.583333 |
The largest possible difference between the two primes which add up to the even number 138.
|
124
| 0.666667 |
Let $ABCD$ be a rectangle in the $xy$-plane where $AB=2$ and $BC=1$ with coordinates $A(0,0,0)$, $B(2,0,0)$, $C(2,1,0)$, and $D(0,1,0)$. Let $\overrightarrow{AA'}$, $\overrightarrow{BB'}$, $\overrightarrow{CC'}$, and $\overrightarrow{DD'}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA' = 12$, $BB' = 10$, $CC' = 16$, and $DD' = 20$, and $M$ and $N$ are the midpoints of $A'C'$ and $B'D'$ respectively, find the length of $MN$.
|
1
| 0.75 |
Given a regular pyramid with a square base area of $144 \mathrm{cm}^2$ and a height of $27 \mathrm{cm}$, determine the height in centimeters of the water in a cylindrical container with a base radius of $9 \mathrm{cm}$ after being filled with this water.
|
\frac{16}{\pi}
| 0.916667 |
Compute the difference between Mark's total and Mary's total bill for a coat originally priced at $120.00, which is being sold at a 25% discount, when the sales tax rate is 7%.
|
0
| 0.666667 |
Let $n$ be the smallest positive integer such that $n$ is divisible by 30, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?
|
9
| 0.333333 |
Given the containers A, B, and C, where A is $\tfrac{4}{5}$ full, B is initially empty and becomes $\tfrac{3}{5}$ full after pouring from A, and C is initially empty and becomes $\tfrac{3}{4}$ full after pouring from B, determine the ratio of the volume of A to the volume of C.
|
\frac{15}{16}
| 0.5 |
Given the expression $(64)^{-2^{-3}}$, calculate its value.
|
\frac{1}{\sqrt[4]{8}}
| 0.833333 |
Determine the largest number by which the expression n^4 - n^2 is divisible for all possible integral values of n.
|
12
| 0.333333 |
Triangle OCD has O = (0, 0) and D = (7, 0), with ∠CDO = 90° and ∠COD = 45°. Find the coordinates of the image of point C after OC is rotated 120° counterclockwise about O.
|
\left(-\frac{7(1+\sqrt{3})}{2}, \frac{7(\sqrt{3}-1)}{2}\right)
| 0.083333 |
Suppose in February a city recorded a total rainfall of 280 inches. Calculate the average rainfall in inches per hour during that month.
|
\frac{5}{12}
| 0.75 |
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$.
|
117
| 0.083333 |
Given that Emma's age is $E$ years and this is equal to the sum of the ages of her four children, and her age $M$ years ago was three times the sum of their ages then, calculate the value of $E/M$.
|
5.5
| 0.083333 |
What is the value of $1357 + 3571 + 5713 + 7135$?
|
17776
| 0.666667 |
Given rectangle ABCD where E is the midpoint of diagonal BD, point E is connected to point F on segment DA such that DF = 1/4 DA. Find the ratio of the area of triangle DFE to the area of quadrilateral ABEF.
|
\frac{1}{7}
| 0.25 |
A regular hexagon $ABCDEF$ with side length 2 is given. Two circles are positioned outside the hexagon. The first circle is tangent to line $\overline{AB}$ and the second circle is tangent to line $\overline{CD}$. Both circles are tangent to line $\overline{EF}$. Determine the ratio of the area of the second circle to the first circle.
|
1
| 0.25 |
Evaluate $3000 \cdot (3000^{1500} + 3000^{1500})$.
|
2 \cdot 3000^{1501}
| 0.833333 |
Given two concentric circles have radii of 2 and 4, find the probability that a chord joining two points chosen independently and uniformly at random on the outer circle intersects the inner circle.
|
\frac{1}{3}
| 0.25 |
Given a string of length 2, cut at a point chosen uniformly at random, calculate the probability that the longer piece is at least 3 times as large as the shorter piece, and the length of the smaller piece is exactly $\frac{1}{2}$.
|
0
| 0.75 |
The number of significant digits in the measurement of the side of a square whose computed area is $2.401$ square inches to the nearest thousandth of a square inch is $4$.
|
4
| 0.916667 |
Given hexadecimal (base-16) numbers only use digits $0$ through $9$ and letters $A$ through $F$ for values $10$ through $15$, determine the number of positive integers less than $500$ with a hexadecimal representation that contains only numeric digits.
|
199
| 0.083333 |
Given that a set $T$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, the line $y=x$, and the line $y=-x$, and that $(3,4)$ is in $T$, find the smallest number of points in $T$.
|
8
| 0.833333 |
How many 3-digit whole numbers, whose digits sum up to 27 and are even, and end specifically in 4?
|
0
| 0.75 |
If an examination consisting of 30 questions awards 4 points for each correct answer, -1 point for each incorrect answer, and 0 points for unanswered questions, and Amy's total score is 85, calculate the maximum number of questions she could have answered correctly.
|
23
| 0.916667 |
Given that a circle is divided into 15 sectors with all sector angles forming an arithmetic sequence and are all integers, determine the degree measure of the smallest possible sector angle.
|
3
| 0.833333 |
What is the number of terms with rational coefficients among the 726 terms in the expansion of \((x\sqrt[4]{3} + y\sqrt[5]{5})^{725}\)?
|
37
| 0.833333 |
If 105 customers purchased a total of 300 cans of soda, with every customer purchasing at least one can of soda, determine the maximum possible median number of cans of soda bought per customer that day.
|
4
| 0.083333 |
Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there?
|
64
| 0.166667 |
An iterative average of the numbers 1, 3, 5, 7, 9, and 11 is computed by arranging the six numbers in some order. First, find the mean of the first two numbers, then find the mean of that result with the third number, then continue this process with the fourth, fifth, and sixth numbers. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
|
6.125
| 0.083333 |
Given a set $\{8, -4, 3, 27, 10\}$, determine the smallest possible sum of three different numbers from this set such that their sum must be even.
|
14
| 0.833333 |
What is the smallest positive integer \( n \) such that \( \sqrt{n} - \sqrt{n-1} < 0.05 \)?
|
101
| 0.75 |
What is the value of $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$?
|
1457
| 0.666667 |
Given that $20\%$ of the students scored $60$ points, $40\%$ scored $75$ points, and $25\%$ scored $85$ points, with the remaining students scoring $95$ points, calculate the difference between the mean and median score of the students' scores on this test.
.
|
2.5
| 0.916667 |
In a plane, points $A$ and $B$ are $12$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $60$ units and the area of $\triangle ABC$ is $180$ square units?
|
0
| 0.666667 |
Given the graphs of $y = -|x-a| + b$ and $y = |x-c| -d$ intersect at points $(1,4)$ and $(7,2)$. Find the value of $a+c$.
|
8
| 0.666667 |
Given an archery contest with 120 shots, Chelsea is ahead by 70 points after 60 shots, with scoring as 10, 8, 5, 3, or 0 points, and Chelsea scoring at least 5 points per shot. Chelsea will achieve victory by scoring the next n shots as bullseyes (10 points) regardless of her opponent’s scores in the remaining shots. Find the smallest n needed to guarantee her win.
|
47
| 0.083333 |
Calculate the area of a polygon with vertices at points (1, -1), (4, 2), (6, 1), (3, 4), and (2, 0).
|
4.5
| 0.083333 |
Consider a scenario where $89$ out of the first $90$ balls are red. Afterwards, $8$ out of $9$ balls counted are also red. If at least $92\%$ of the total balls are red, find the maximum value of $n$, the total number of balls.
|
n = 90 + 9y = 90 + 9 \times 22 = 210 + 78 = 288
| 0.083333 |
Given circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle.
|
\frac{1}{12}
| 0.666667 |
When Neva was young, she could cycle 20 miles in 2 hours and 45 minutes. Now, as an older adult, she walks 8 miles in 3 hours. Calculate the difference in time it takes her to walk a mile now compared to when she was young.
|
14.25
| 0.75 |
Eight spheres, each with a radius of 2 and located in each octant, are tangent to the coordinate planes. Find the radius of the smallest sphere, centered at the origin, that contains these eight spheres.
|
2\sqrt{3} + 2
| 0.916667 |
Given x be a real number selected uniformly at random between 200 and 400, and \(\lfloor \sqrt{x} \rfloor = 18\), find the probability that \(\lfloor \sqrt{100x} \rfloor = 180\).
|
\frac{361}{3700}
| 0.75 |
Given $(n+i)^5$ is a real number, calculate the number of integer values of n.
|
0
| 0.333333 |
Given \( x \) is real and positive and grows beyond all bounds, find the limit as \( x \) approaches infinity of \( \log_4{(8x-7)}-\log_4{(3x+2)} \).
|
\log_4 \left(\frac{8}{3}\right)
| 0.083333 |
Evaluate the expression $\frac{2^{3002} \cdot 3^{3004}}{6^{3003}}$.
A) $\frac{1}{6}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{2}{3}$
E) $\frac{3}{2}$
|
\frac{3}{2}
| 0.75 |
A box contains triangular and pentagonal tiles totaling 30 tiles. There are 100 edges in total. Find the number of pentagonal tiles in the box.
|
5
| 0.416667 |
Points $P$ and $Q$ are on a circle of radius $10$ and $PQ = 12$. Point $R$ is the midpoint of the minor arc $PQ$. Determine the length of the line segment $PR$.
|
2\sqrt{10}
| 0.416667 |
An ordered pair (b, c) of integers is chosen such that b is non-negative and less than or equal to 5, and c is between -10 and 10, inclusive. Find the probability that the equation x^2 + bx + c = 0 will not have distinct positive real roots.
|
1
| 0.416667 |
The mean of three numbers is $20$ more than the least of the numbers and $10$ less than the greatest, and the median is $10$. What is their sum?
|
0
| 0.083333 |
Calculate the probability that all four of them flip their coins the same number of times.
|
\frac{1}{15}
| 0.083333 |
A rectangular box measuring $8$ cm by $4$ cm by $15$ cm is filled with liquid $Y$. This liquid forms a circular film of thickness $0.2$ cm when poured into a large body of water. What is the radius, in centimeters, of the resulting circular film?
|
\sqrt{\frac{2400}{\pi}}
| 0.083333 |
Given that a square-shaped area is covered entirely by congruent square tiles, and the total number of tiles that lie along the two diagonals of the square is 25, calculate the number of tiles used to cover the entire square area.
|
169
| 0.916667 |
Medians \( AF \) and \( BE \) of triangle \( ABC \) are perpendicular, \( AF = 10 \), and \( BE = 15 \). Find the area of triangle \( ABC \).
|
100
| 0.416667 |
Given a positive integer n not exceeding 120, the probability of choosing n when n ≤ 60 is q, and the probability of choosing n when n > 60 is 4q. Calculate the probability that an odd number is chosen.
|
\frac{1}{2}
| 0.75 |
Given $d$ is a digit, find the number of values of $d$ for which $2.0d05 > 2.015$.
|
8
| 0.833333 |
A painting measuring 20 inches by 30 inches is to be placed in a wooden frame, with the longer dimension horizontal this time. The wood at the top and bottom is three times as wide as the wood on the sides. If the area of the frame equals that of the painting itself, determine the ratio of the smaller to the larger dimension of the framed painting.
A) $1:3$
B) $1:2$
C) $2:3$
D) $3:4$
E) $1:1$
|
1:1 \text{ (E)}
| 0.166667 |
In a classroom test, $20\%$ of the students scored $65$ points, $25\%$ scored $75$ points, another $25\%$ scored $85$ points, and the rest scored $95$ points. Find the difference between the mean and median score of the students' scores in this test.
|
3.5
| 0.5 |
Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Halfway between the poles, there is an additional smaller pole of 10 feet high. The lines go from the top of each main pole to the foot of the opposite main pole, intersecting somewhere above the smaller pole. Determine the height of this intersection above the ground.
|
22.5
| 0.083333 |
Given that $(5z)^{10}=(20z)^5$, solve for $z$.
|
\frac{4}{5}
| 0.25 |
What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$?
|
-\frac{13}{9}
| 0.833333 |
Given that we have the digits 4, 4, 6, 6, 7, 7, 8, determine the number of 7-digit palindromes that can be formed.
|
6
| 0.083333 |
How many different integers can be expressed as the sum of three distinct members of the set $\{3, 5, 9, 13, 17, 21, 27\}$?
|
20
| 0.166667 |
Given that half of Marcy's marbles are red, three-eighths of them are blue, and seven of them are green, determine the smallest number of yellow marbles she could have.
|
0
| 0.916667 |
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