problem stringlengths 12 11.9k | answer stringlengths 1 2.55k | score float64 0.3 0.7 |
|---|---|---|
You have 1000 coins, exactly 999 genuine coins, each having the same equal value, but the 1000th coin is either heavier or light in value compared to the others. It is not know which one is either heavier or light. A friend who knows which one the fake is, offers to sell you 100 coins out each one is genuine, unless th... | 100 | 0.5 |
A cube's volume is 27 times the volume it had when each side was reduced to 50% the current side. If the current side is $3\text{cm}$, how much is the side when it is at the smaller size? | 1.5\text{cm | 0.5 |
You have two indistinguishable 3-inch metal rods. You drill smooth, round whole-number size whole-size diameter bored out on one end, such 1-inch, 2-inch, 3-inch, 4-inch, & 5-inch through each on one side. How many distinctly different from one another total size sizes could you create if the rods were completely glue... | 6 | 0.4 |
A certain species doubles in number every 20 days. If you start out observing 50 individuals on the first day, how many individuals will there be after 180 days?
What day should you start counting if you know there were only 10 individuals at day 0?
What day should you start counting if you know there were only 25 in... | 80 degrees Celsius | 0.3 |
In the first year, it rains 45 inches in Seattle. The subsequent year, the rainfall is 6 inches below the first year's total. In the fourth year, the rainfall is 3 inches above the second year's total. Calculate the average annual rainfall across these four years. | 31.5 | 0.4 |
In the Kingdom Islands, Jill lives on the 10th floor, Alex lives on the 20th floor, Beth lives on the 35th floor, Quinn lives on the 18th floor, Regan lives on the 23rd floor, Ashley lives on the 30th floor, Sally lives on the 14th floor, James lives on the 33rd floor, Evelyn lives on the 21 floor, April lives on the 3... | 27.10 | 0.4 |
In the planetarium, Dr. Zara is designing an interactive planetarium projector to showcase the cosmic distribution map. The map shows the locations, in light-years, from the origin point as coordinates in three dimensions: Earth's location is at \((0, 0, 0)\). Dr. Zara wants to display five new planets within the first... | 5 | 0.5 |
A circular garden is enclosed within an equilateral triangular fence. If the fence's side measures 4√3 meters, what is the radius, in meters, from the center to any point on the circle's boundary? | 1 | 0.7 |
A small town decides to implement solar panels on all municipal rooftoops. Initially, they install panels on 10% or 120 rooftoops. If the town council will continue to install solar panels at the same rate every year, how many rooftoops will have solar panels after 5 years? | 1800 | 0.4 |
A certain sequence is defined as $f(0) = 0$,$f(1) = \frac{3}{2}$,$f(2) = \frac{8}{3}$,$f(3) = \frac{15}{4}$,$f(4) = \frac{24}{5}$,$\cdots$,$f(36)=\frac{n}{n+1}$. Find the value if $n$. | n = 36^2 - 36 | 0.4 |
In the World War II Battle near St. Diem, there were three times as many civilian soldiers as allied soldiers. If the battle lasted three hours, how many civilian soldiers were there? | 9 | 0.3 |
Consider the following scenario: A group consists 16 girls who've each attended three science fairs, three art fairs, or both. If 5 girls attended only the science fairs, how many girls attended both science fairs as well as art fairs? | 11 | 0.3 |
There is an intricate relationship among the four basic forces in nature: gravitational, weak, strong, electric forces. An interaction in space-time often demands the special manifestation(s) from these forces, which act at the fundamental level. If the gravitational forces suddenly disappeared, how many forces out her... | 3 | 0.5 |
In the year 2023, if the average temperature in New York City was 58.5°F, what was the total temperature anomaly if the average temperature over the previous 30 years had been 52°F? | 195 | 0.5 |
A block is sliding on an inclined plane at an unknown velocity v0. When it collides elastically on the inclined plane, it gains another velocity v1. It is noted in this case 2v0 = v1. If v0 < 2, which one is correct among the following options:
A) v0 < v1
R) v0 > v1
T) 2v0 = v1
Y) v0 <= v1 | Y | 0.4 |
A train leaves Station A heading towards Station B at 60 mph. At the same time, another train leaves Station B heading towards Station A at 80 mph. The two stations, A & B, lie along the same straight railway track. If the total separation from Station A to Station B is 360 miles, determine after how many hours the two... | 2.57 | 0.7 |
In the set $\{1, 2, 3, \ldots, 2015\}$, find the minimum number \(n\) such hat any subset \(S\) oč the set contains at cast one pair \(\{k, m\}\) such as \(k + m = 2016\). | 1007 | 0.5 |
A certain type 1 student starts at the origin, $\mathbf{0}$, facing due north. At each step, they walk one unit, either east or west, in the direction they're facing. The student makes their choices at random. Thus at each step the student either walks one unit east or one unit west, each direction having probability $... | 1 | 0.3 |
A hypothetical alien species is capable only across pairs. When two alien pairs communicate, they combine their signals to send one signal, which is the sum, difference, product, or ratio. Consider four pairs: A, B, A, B. If after communicating, only one pair remains, which pair will it be?
Rules:
- The order in which... | A,B | 0.5 |
In an alien civilization, numbers follow their base-12 system. Given the equation \(2_{12} \times 5_{12} = \_ _{12}\). Fill the blank space to find the answer in base-12. | A | 0.6 |
A farmer is planning to plant three types: 3 potatoes, 2 carrots, 4 celery, 3 turnap, 5 corn, 4 peas, 6 pumpkinds, 4 mushrooms. Unfortunately, during the planting season, it rains severely on 2 days out 10. Find the difference in the number days when he does not plant any vegetables during the planting season? | 12 | 0.6 |
In the compound FeCl<sub>3</sub>, the oxidation state represented as FeXOYClZ, what is the value if X = 3, Y = 3, Z = 1, N = 1, M = 3, O = 3?
Note: X, Y, Z, N, M, O represent the coefficients in the oxidation state equation, 0 represents the oxidation state, 1 represents the element.
Answer the question: Which value re... | 3 | 0.3 |
A company is planning to manufacture two types, model A: hard work, model B: smart work. The manufacturing department estimates the resources requirement as given in the table:
|Resource |Model A|Model B|
|------------------ |-------|-------|
|Iron(ton) |3 |5 |
|Plastic(Kg) |2 ... | 7 | 0.4 |
In the townsville park, there is an arrangement in which 100 cactuses have been placed in circles. Every circle surrounds exactly five cactuses. How many circles surround all these cactuses? | 20 | 0.5 |
A right circular cone is inscribed in another right circular cone. The height from the apex to the base center is 12 units. The height from the apex to the base center is 18 units. If the larger cone’s base radius is 8 units, find the smaller cone's base radius. | 12 | 0.5 |
Consider the polynomial \( P(x) = x^4 - 10x^3 + 35x^2 - 50x + 24 \). Determine the sum \( p_1 + p_2 + p_3 \) after performing the following operations on it: For each coefficient \( p_i \) in the polynomial, replace \( p_i \) if \( p_i > 2 \) or \( p_i < -2 \) such it falls within the interval \([-2, 2)\) as close as p... | 4 | 0.7 |
In the town square, four friends—Alice, Bob, Charlie, David—each brought apples to sell at their local farmer's market. If David brought twice as many apples as Charlie, Charlie brought three fewer apples than Bob, Bob brought five fewer apples than twice the amount brought initially, initially, while Adam brought four... | 20 | 0.3 |
A physicist was calculating the factorial product \( 6! \times 12! \times 14! \times 16! \times 18! \times 20! \times 22! \times 24! \times 28! \) but accidentally added the sum-of-digits-of-the-product number. How many zeroes appear in the product, in the end? | 29 | 0.5 |
In the mystical townscape, from the vernal equinoxeve system, four temporal indicatenum ber eft nom-oi hePh margaints denoting different lunar days as depicted in the annex figure. The spotted pattern on the calendar leaves were definitely closely connected to the beginning day from lunar crontacjeslot scene in the sub... | 1 | 0.5 |
In the year 2000, the average human consumes 2,000 calories per day. If the world population in 2000 was 6 billion, how many calories were consumed globally each day in 2000?
[Show only the answer, not the solution. The question is incomplete as it asks you to put only the answer. So, I'll present the question, soluti... | 16761 billion calories | 0.3 |
In the year 2100, on Earth, people celebrate the 'Year Zero Festival.' The festival occurs on the exact day Earth completes 1 billion orbits around the sun since the year 2000. Given Earth's average orbital period is 365.25636 days, on which year will the first Year Zero Festival be celebrated? | 1,000,000,200 | 0.5 |
A right circular cylinder is inscribed in another right circular cylinder such as the bases lie in the same plane. If the height ratio is 2:3, what is the ratio, in simplest radical, R:r, if r is the radius R is the height? | 53.13 degrees | 0.4 |
What is the average atomic radius in picometres (\( \text{pm} \)), given the radiii (\( \text{pm} \) as well) below, to 2 decimal points precision?
\begin{itemize}
\item Hydrogen: \( 50 \) \( \text{pm} \).
\item Helium: \( 90 \) \( \text{pm} \).
\item Lithium: \( 150 \) \( \text{pm} \).
\item Beryl: \( 200 \) \( \text{... | 122.50 | 0.7 |
Consider an alien species on an uncharted planet. These creatures communicate using sound pulses. If they transmit sound in all three spatial dimensions, what happens to the intensity ($I$) at the receiver if the radius ($R$) is doubled, assuming sound intensity is inversely proportional to the square root? | I_{\text{new | 0.3 |
In ancient Egypt, the number 365 held great significance due to the number representing the days in their year. Suppose the ancient Egyptians used the notation AatN to represent the sum A + At + N, giving their calendar an additional 15 days. If N represents the days in the month as the ancient Egyptians conceptualized... | 380 | 0.7 |
A point source emits sound uniformly in all direction. The intensity at source is $I_0$. Two sound detectors $D_1$ & $D_2$ initially stand 1 m away from source in the same straight-line. If $D_1$ is moved to 2 m away, what will be the ratio $\frac{I_{D2}}{I_{D1}}$, if $I_{D1} = I_0$ ? | 2 | 0.6 |
Alice is looking at Bob, who is looking at Charlie. If Bob lies if he is told to look at Charlie, which is the truth: Does Bob look at Charlie? | Alice \text{ or \textbf{Does not | 0.7 |
A bus is traveling along the x-axis from left to right. At time \(0\,\text{seconds}\) the bus starts at \((0,0)\) but is accelerating toward the right at \(4\,\text{ft/s}^2\). Then, 2000 feet away from the bus, there's an oak stump at \((x,0)\) on the x-axis. If the stump keeps an eye on the bus the whole time, what is... | 16000 | 0.3 |
In an ecological study, researchers found three types: p, q, r, to be present in an ecosystem. The growth rate in this system is given as:
n2 = p^3 * q^2 * r
Here, p, q, r > 0.
Determine the net growth rate w.r.t. r. | p^3 * q^2 | 0.4 |
In the year 2050, the global population is projected to reach 9.7 billion. If the annual population growth rate is 1.0%, how many years will it take to reach this population level starting from the year 2000?
Given:
- Year 2050 population = 9.7 billion
- Annual growth rate = 1.0% = 0.01
- Year 2000 population = X bill... | 5.902 billion | 0.3 |
A 2-liter bottle is filled halfway. Three-quarters liter is added to the bottle. How many liters does the bottle contain?
Is this bottle now closer to full or to empty? | 1.75 liters; closer to full | 0.7 |
What is the probability, in simplest fraction, to get all five coins, each one have same front value as the one at the back, when you randomly choosing 5 coins from 10 coins having same value on both side from each 25c, 50c, $1, $2, $5, $10, $20, $50, $100, $200? | 1/45 | 0.4 |
In the field crop, what is the last letter in the technical term used to describe the individual sections or blades in the visual display device made from thin wafern? | s | 0.6 |
How many different three-digit numbers, ranging from 100 to 999, inclusive, have the property when the sum is 11? | 227 | 0.4 |
You need to transform the string `abracabra` to the string `abrrrara` using the minimum number DEX operations. In one DEX operation, you have to replace every ex- in the string, or ex. An ex-substring is an uninterrupted substring consisting only from exes, so this operation could also be defined as replacing all unbro... | 1 | 0.4 |
A certain species in the animal kingdom is studied, which after reaching adulthood, every individual produces an egg every year. However, this species experiences periodic population reduction due to an unknown factor. It is found out every five years, there is an epidemic causing 50% reduction in the population. If th... | 2,560,000 | 0.3 |
In the year 2000, the population in country R was \( P \) people. However, in 2002 it was discovered to be \( P^2 + 3 \). In 2005 it was discovered to be \( P^4 + 6P^2 + 8 \). In 2007 it was discovered to be \( P^6 + 9P^4 + 24P^2 + 16 \). In what year was it discovered to be \( P^8 + 12P^6 + 54P^4 + 108P^2 + 81 \)?
(A... | 2008 | 0.7 |
What is the value represented on the binary number $10011100111$. | 6 | 0.7 |
What is the largest integer \(n\) such \( \frac{12n}{10} \) is also an integer? | 5 | 0.5 |
An archaeologist discovers three ancient artifacts, labeled A, B, B. Two clues provide information on the artifacts’ age. Clue 1: The sum 10 times the age A + the age B + the age B = 88. Clue 2: The sum 12 times the age A + 11 times the age B + the age B = 125. How many years younger is the oldest discovered item than ... | 2.572683043 | 0.3 |
A car is traveling at 60 mph. After 3 hours, it slows to 40 mph. After another 2 hours, it stops. How long was the car traveling in total? | 5 | 0.7 |
In how many different number systems does the fraction 0.111... represent the repeating decimal equivalent to 1/7? | infinite | 0.7 |
In the early 17th century, which scientist disproved Aristotle's belief in the geocentrist model through his observation during his first set space journey? | Kepler | 0.6 |
An apple orchird 🍌 selected the number 44 to be 2016 year's total number since 2004 going back 5 years, what is the total number apples he got in 2020, if there is an increase rate from 2004 to 2020 which is 100%. | 88 | 0.4 |
A town's population starts at 5000 people. It is predicted to increase at an annual rate such if it doubles every 5 years. After 15 years, how many people will there be in the town?
After 25 years, the population triples. Calculate the total number people in the town.
After 35 years, 25% percent move to live in anoth... | 90000 | 0.7 |
What is the optimal strategy to maximize the probability in the following coin tossing scenario:
A fair coin is flipped 3 times. If all three outcomes match, you win the amount shown on the last coin. If they do not match, you get $0. Compute the expected value when using optimal strategy. | $0.125 | 0.4 |
In an ecosystem, 3 rabbits eat 1 carrot each. If each rabbit needs to eat 10 carrots to survive the winter, how many carrots does each rabbit need to eat every day?
Answer Choices:
A. 3
B. 10
B. 15
B. 20
B. 30 | A | 0.5 |
A cat is in the center maze. It needs to reach the cheese at the furthest corner in the fewest steps. However, there's only one correct sequence. How many steps does it need?
! | 6 | 0.5 |
A train travels from City A to City B in 5 hours at an average speed. If the train had traveled at 60 km/h, it could have made the journey in 4 hours. However, due to technical issues, the train’s speed reduced to 75% throughout the journey. Calculate the total time, in hours, the train took to complete the journey aft... | 5.33 | 0.5 |
At what temperature do water molecules transition from liquid to gas, measured in degrees Celsius? | 100 | 0.7 |
In the figure below, the blue square's side is twice the size, while the green square's side is thrise the size. Given the area ratio, how many times greater is the green square's area than the blue square? | 2.25 | 0.6 |
In 2023, the population density in New York City was 27,005 people per square mile, while the population density in Tokyo was 6,086 people per square mile. If both cities had the same area, how many times larger was the population in New York City compared to Tokyo? | 4.43 | 0.3 |
A scientist discovers two new elements: Element X, which is twice as reactive as hydrogen, exists in nature as an allotropous crystal, varying in crystalline forms. Element Y, discovered in the atmosphere, is non-toxic but reacts vigorously upon contamination. Within the first year, Element X is found to have 4 differe... | 20/121 | 0.3 |
What is the ratio 109 3 9 7 5 2 8? | 109 : 39 : 75 : 28 | 0.4 |
What is the sum, in degrees, when you add up the angles in all the vertices, if there is one right-angle corner in the figure? | 171 | 0.4 |
In an isolated island, there is only one species left: snakes. Due to the population's rapid decline, researchers decide to start an artificial breeding project to preserve the species. After their observations, they find out the optimal breeding conditions. Which should they apply if they'd prefer to have the best num... | A | 0.5 |
In the year 2023, an unknown number is defined as \( A \). Given the following rules:
- If \( A \) is divisible only 3 distinct prime numbers.
- \( A \) is greater than 500 but smaller than 1000.
- \( A \) must be an even number.
- The sum total when \( A \) is broken up as sum atomic in three prime numbers is the larg... | 36 | 0.4 |
What is the sum X to base 9 if the sum in base 10 is 570? | 703 | 0.7 |
You have three bags labeled A, B, anf P. Bag A contains 2 red balls, 5 blue balls, an d1 green balls. Bag B contains 3 red balls, 4 blue balls, an d2 green balls. Bag P contains 1 red balls, 6 blue balls, an d3 green balls. If you randomly choose one bag, take out one red
ball, put it back, take out one blue
ball, put ... | P | 0.3 |
A flock starts at 10 p.m. walking along the shoreline. On the following morning, another flock starts at the same time, going north. At noon the next day they meet. How much time did it take to meet if each flock walked at the same speed as the speed at which the two combined flocked? | 8 | 0.7 |
A fair 6-faced dice is rolled 5 times. If the probability distribution is followed, what is the probability, rounded to four decimal points, to get an even number exactly three times? | 0.2778 | 0.7 |
In an isoscale trpear = 5y + x = 10, 5y = 3x + 8. Find the value you two z, if z = x - 6y? | -10.9 | 0.5 |
In which year did the first successful ascent to the summit take place?
[Show step-by-step reasoning]
[Your reasoning here.] | 1953 | 0.6 |
In a secret code, the number 351 represents a hidden message. If the digits in the code are each incremented (increased) sequentially from right to the base 10, i.e., 1 to 2 to 3 and so on until the final digit in the code is incremented cyclically, (1 to 2 to 3 to 4 to 5 to 6 to 7 to 8 to 9 to 0 to 1 to 2 and so on), ... | C | 0.6 |
In a certain town, every person who loves cats has a cat. Among those who do not have cats, some still love cats. If the town has 100 people and there are exactly 30 people who do not love cats, how many people in the town do not have cats but still love cats? | 0 | 0.6 |
If a right circular cone has a base radius of 3 cm and a height of 4 cm, and if the cone is cut into two pieces horizontally, through the midpoint of its height, how many times larger is the volume of the top piece than the bottom piece? | 0.14 | 0.5 |
In a certain game, a rectangle with length 8 units and width 5 units is divided into squares of 1x1 units. If each square is colored either blue or yellow, how many distinct colorings are possible so that no two squares sharing an edge have the same color? | 4,294,967,296 | 0.7 |
If a square's area is \(256\) square units, and a rectangle has one side length that is half the side length of the square and another side length that is \(20\%\) longer than the side length of the square, then the area of the rectangle is \(\frac{1}{2}\) of the area of another square. What is the side length of the l... | 16 | 0.4 |
In the game of Settlers of Catan, you are on a quest to acquire the longest Road. To achieve this, you must construct roads following these specific guidelines:
1. You can only construct a new Road adjacent to an existing one you own, following the layout of the board. Roads must be straight, either horizontally or ve... | it's not possible to claim the longest Road in this scenario with only one more Road construct that follows the given guidelines. | 0.6 |
On a number line, the point representing the number \(x\) is 5 units to the right of the point representing the number \(y\). If \(y\) is 3, and the absolute value of \(z\) is 7, how much greater is the sum \(|x| - |z|\) than \(|x - z|\)? | 0 | 0.6 |
In a small village, the local baker decides to host a baking contest. There are 5 participants: Alice, Bob, Carol, Dave, and Eric. Each contestant is required to bake a different number of pies, determined sequentially from the sum of the numbers of pies they are immediately preceding. For example, if the first contest... | 110 | 0.7 |
In a specific sport, each team has to choose their uniform's top color from five options and their uniform's bottom color from another five options. Each team's uniform consists of one top color and one bottom color. There is an additional rule that no two players on the same team can wear uniforms of the same color co... | 10 | 0.5 |
Let \( S = \{1, 2, \ldots, 10\} \) be the usual 10-element sets. If \( a, \ldots, 10\) (missing comma in original) are assigned to these ten numbers so that \( 1 \leq i \leq 10 \) is assigned to 1 in this same order, how many distinct values are possible for the sum \( f(1)+f(2)+f(3)+\cdots+f(9)+f(10)\) (sum of numbers... | \text{Test students' ability to solve complex logical problems with a twist on simple mathematical concepts. | 0.6 |
On a standard 8x8 chessboard, a square is removed from one of the corners. How many ways can two rooks be place on the remaining squares such that they do not attack each? | 32 | 0.3 |
In the town of Arcadia, there are two types of insects, beetles and bugs. Each day, the number of beetles doubles, but the number of bugs triples. On day 0, there are 3 beetles and 5 bugs. On which day will the number of bugs be exactly twice the number of beetles? | Day would more directly calculate more solve optimally AS precise follow-way or recursive series scheme advanced mathematical computation finite close values (manual check day's that day-calculated ensuring directly solving log heuristics validation applying precipitously anticipated).
### Solution Method Proceeds Quic... | 0.4 |
In a hypothetical universe, there are two types of planets: Type-Alpha and Type-Bet. Type-Alpha planets are perfectly spherical with a radius of 10,000 kilometers. Type-Bet planets, on the same radius, have an eccentricity that makes their shape resemble a rugby-ball more than a sphere. If a spacecraft travels from the... | 0.9682 | 0.3 |
In the year 2023, the average lifespan of a house in a certain region is known to follow a geometric series with the first term representing the age of the youngest house (1 year old), and the common difference is such that every term increases the lifespan of the previous house in a way that the series converges. If t... | 1999 | 0.4 |
Consider the number 3729. If each digit of this number represents a different color in a four-stage color code system (where the first digit is the saturation level, the second the hue, the third the brightness, and the fourth the temperature), how many possible different color codes can be created if no color code can... | 8000 | 0.4 |
**
In a right-ang *le**d** triangu*lar feild with legs measuring 9 units and 40 units respectively, a gardener walks from the bottom-left corner to the top-right corner following the path along the two legs and then along the diagonal. Calculate the total walktime if his speed is 3 units/ minute along the legs and 4 un... | 13 minutes | 0.5 |
In a laboratory, a scientist is studying the behavior of sound waves in a resonator. They notice that a pure tone of frequency \( f \) Hz produces a standing sound pattern with a certain number of nodal lines. When the frequency is doubled to \( 2f \) Hz, the number of nodal lines observed in the same resonator increas... | 1 | 0.5 |
Who was the American astronaut who famously fell asleep on the job during the Apollo 11 mission? | Michael Collins | 0.4 |
In the Renaissance Period, Nicolaus Copernicus proposed a heliocentric model of the universe, which was revolutionary for its era. However, one of the common misconceptions is that he believed in the notion that the sun moved in an epicycle, and not the earth. Given this premise, which of the following could be the inc... | bi | 0.5 |
In a hypothetical universe, every element's atomic number doubles every year. If the element X had an atomic number of 5 in the year 2000, and you observe that element X's atomic number has now quadruped in the year 2022, how many years after the observation will it take for the atomic number of element X to surpass 10... | 6 | 0.7 |
Alice and Bob are playing a game with a standard six-sided dice. They take turns rolling the dice, with Alice going first. If a person rolls a 6, they win the game. However, if a person rolls a 1, the next person gets an additional roll. What is the probability that Alice rolls a 6 and Bob rolls a 1? | 1/36 | 0.3 |
Among the following historical figures, who is known for founding the State of Qin and ultimately unifying China, which was the first major state to establish an empire?
[Here are the options, labeled with Roman numerals]
I. Genghis Khan
II. Qin Shi Huang
III. Confucius | II | 0.7 |
Consider a right-anglar pyramid (aka square pyramid) with base length \( l = 4 \) units and lateral height \( h = 5 \) units. What is the volume of the pyramid? | \frac{80 | 0.5 |
Alice and Bob are participating in a tournament. Each match can either be a win or a draw. If Alice plays against Bob and she loses, she must play against Bob one more match. If she loses the second match, she must play against Bob one more match. This pattern continues until Alice has either a win or a draw against Bo... | \frac{756 | 0.7 |
In a hypothetical universe, particles can have one of two charges, denoted \(+1\) or \(-1\). When two particles interact, their charges combine to form a new charge, which is calculated using the following rule:
\(
\text{New Charge} =
\begin{cases}
\text{Particle 1 Charge} & \text{if Particle 1 Charge} = \text{Parti... | 23 | 0.3 |
If \( f(x) = \frac{1}{x} \) and \( g(x) = \sqrt{x-1} \), for how many real values of \(x\) does \( f(g(x)) = g(f(x)) \)? | = \frac{\sqrt{1 -x | 0.7 |
How many days will it take for a colony of insects to grow from 100 insects to 10,000 insects if it doubles in size every 2 days? | 14 | 0.7 |
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