Appearance
❓A side of beef lost 35 percent of its weight in processing. If the side of beef weighed a certain amount after processing, and it weighed 876.9230769230769 pounds before processing, how many pounds did it weigh after processing?
💡First, I need to determine the weight of the side of beef after processing. The beef lost 35% of its weight during processing, which means it retains 65% of its original weight.The original weight of the beef is 876.9230769230769 pounds. To find the weight after processing, I will calculate 65% of this original weight.I'll start by converting the percentage to a decimal by dividing by 100:65% = 0.65Next, I'll multiply the original weight by this decimal to find the weight after processing:Weight after processing = 0.65 * 876.9230769230769Performing the multiplication:0.65 * 876.9230769230769 = 569.9999999999999Since the result is very close to 570, I'll round it to the nearest whole number for simplicity.Therefore, the side of beef weighed approximately 570 pounds after processing.
❓If you divide a number by 10, the result is 6. Find the exact result of subtracting 15 from the number.
💡First, I need to find the original number. The problem states that when the number is divided by 10, the result is 6. I'll represent the unknown number as ( x ).So, I set up the equation:[frac{x}{10} = 6]To solve for ( x ), I'll multiply both sides of the equation by 10:[x = 6 times 10 = 60]Now that I know the original number is 60, the next step is to subtract 15 from this number:[60 - 15 = 45]Therefore, the exact result of subtracting 15 from the number is 45.
❓Chandra is going on a monster hunt. After a big storm, there have been lost monsters all around the castle grounds for the last 5 days. Chandra wants to rope all of them up and return them to the swamps where they will be the most comfortable. On the first day, she saw 2 monsters near the castle walls. Every day after that, the number of monsters she saw increased exponentially by a different factor. On the second day, the number tripled, on the third day it quadrupled, on the fourth day it quintupled, and on the fifth day, the number of monsters sextupled. After 5 days, how many monsters does Chandra need to lead back to the swamps?
💡First, identify the number of monsters Chandra saw each day.On the first day, she saw 2 monsters.On the second day, the number tripled: 2 * 3 = 6 monsters.On the third day, the number quadrupled: 6 * 4 = 24 monsters.On the fourth day, the number quintupled: 24 * 5 = 120 monsters.On the fifth day, the number sextupled: 120 * 6 = 720 monsters.Next, sum the number of monsters seen each day to find the total number of monsters Chandra needs to lead back.Total monsters = 2 + 6 + 24 + 120 + 720 = 872 monsters.
❓A computer virus specialized in occupying memory starts by taking up 2KB of memory at boot time, and then duplicates itself every 3 minutes. After duplication, the memory occupied is twice the original amount. Determine how many minutes after booting the computer will the virus occupy 64MB of memory (Note: 1MB = 2^{10}KB).
💡First, I need to understand the problem. The virus starts with 2KB of memory and duplicates every 3 minutes, doubling its memory each time. The goal is to determine how many minutes it will take for the virus to occupy 64MB of memory.Next, I'll convert 64MB to kilobytes since the initial memory is given in KB. Knowing that 1MB equals (2^{10})KB, I calculate:[64 times 2^{10} = 64 times 1024 = 65536 text{ KB}]Now, I'll establish a pattern for the virus's memory occupation over time. At boot time (0 minutes), it's 2KB. After each 3-minute interval, the memory doubles:- After 3 minutes: (2^2 = 4) KB- After 6 minutes: (2^3 = 8) KB- After 9 minutes: (2^4 = 16) KBFollowing this pattern, after (n) sets of 3 minutes, the virus occupies (2^{n+1}) KB of memory.To find when the virus reaches 64MB (65536 KB), I set up the equation:[2^{n+1} = 65536]Since (65536 = 2^{16}), the equation simplifies to:[n + 1 = 16][n = 15]Each (n) represents a 3-minute interval, so the total time is:[15 times 3 = 45 text{ minutes}]
❓There is a total of 250 marbles in a box, each of which is red, green, blue, white, or yellow. If one marble is drawn from the box at random, the probability that it will be white is 2 / 5, the probability that it will be green is 1 / 4, and the probability that it will be yellow is 1 / 10. What is the probability that the marble will be either red or blue?
💡First, I need to determine the probabilities of drawing a white, green, or yellow marble. These probabilities are given as 2/5, 1/4, and 1/10 respectively.Next, I'll add these probabilities together to find the combined probability of drawing a white, green, or yellow marble. To do this, I'll find a common denominator for the fractions. The least common multiple of 5, 4, and 10 is 20.Converting each probability to have a denominator of 20:- Probability of white = 2/5 = 8/20- Probability of green = 1/4 = 5/20- Probability of yellow = 1/10 = 2/20Adding these together:8/20 + 5/20 + 2/20 = 15/20Now, to find the probability of drawing either a red or blue marble, I'll subtract the combined probability of white, green, and yellow marbles from 1:1 - 15/20 = 5/20Finally, I'll simplify the fraction:5/20 = 1/4Therefore, the probability of drawing either a red or blue marble is 1/4.
❓Some identical machines, working alone and at their constant rates, take 6 hours to complete a job lot. It takes 2 such machines 24 hours to perform the same job. How many machines were working together in the first scenario?
💡First, I need to determine the work rate of a single machine. I know that 2 machines take 24 hours to complete the job. So, the combined work rate of 2 machines is ( frac{1}{24} ) jobs per hour. Therefore, the work rate of one machine is ( frac{1}{2 times 24} = frac{1}{48} ) jobs per hour.Next, I want to find out how many machines (( n )) are needed to complete the job in 6 hours. The combined work rate of ( n ) machines should be ( frac{1}{6} ) jobs per hour.Setting up the equation:[n times frac{1}{48} = frac{1}{6}]Solving for ( n ):[n = frac{1}{6} times 48 = 8]Therefore, 8 machines are required to complete the job in 6 hours.