WORD PROBLEMS (The answers are my own words and solutions. It is your choice to believe it or not. Hahahaha)

What is the greatest common factor of 84 and 144?

To find the greatest common factor (GCF) of 84 and 144 efficiently, we can use prime factorization. 1. Start by finding the prime factors of each number: - Prime factors of 84: 2, 2, 3, 7 - Prime factors of 144: 2, 2, 2, 2, 3, 3 2. Identify the common prime factors and multiply them together: - Common prime factors: 2, 2, 3 - GCF = 2 * 2 * 3 = 12 Therefore, the greatest common factor of 84 and 144 is 12.

What is the next term in the following geometric series: 3, 6, 12, 24

To find the next term in the geometric series, we can observe that each term is obtained by multiplying the previous term by 2. So, to find the next term, we multiply the last term (24) by 2: 24 * 2 = 48 Therefore, the next term in the series is 48.

How many ways can you arrange 5 people around a circular table if there are 2 specific people that do NOT want to sit beside each other?

We can fix one of the 2 specific people in a seat, which leaves us with 4 remaining seats. We can arrange the other 4 people in 4! (4 factorial) ways. However, we need to consider that the 2 specific people can switch positions, which doubles the total number of arrangements. So, the final answer is 2 * 4! = 12 ways. 👥🔄

The ratio of the ages of Juan and Jose is 5:3. If Juan is 14 years older than Jose, how old is Jose?

Step 1: Let's say Jose's age is 3x and Juan's age is 5x. Step 2: The problem states Juan is 14 years older than Jose. So, 5x = 3x + 14. Step 3: Solving the equation, we get x = 7. So, Jose's age is 3x = 21 years.

David received a grade of 89 in his first 3 exams. What must be his score in his fourth exam in order to have an average of 91?

Step 1: The total of David's first 3 exams is 89 * 3 = 267. Step 2: To have an average of 91 over 4 exams, the total of all 4 exams should be 91 * 4 = 364. Step 3: Subtract the total of the first 3 exams from the total needed: 364 - 267 = 97. So, David must score at least 97 on his fourth exam to have an average of 91.

What is 2230H in 12 hour format?

To convert 22:30H to 12-hour format, you simply subtract 12 from the hour if it is greater than 12. In this case, 22 is greater than 12, so we subtract 12 to get 10. The minutes remain the same, so it becomes 10:30 PM.

A train can travel 210 kilometers in 3.5 hours. At this rate, how far can the train travel in 6 hours and 20 minutes?

To find how far the train can travel in 6 hours and 20 minutes, we can use the given rate of 210 kilometers in 3.5 hours. First, let's convert 6 hours and 20 minutes to hours. 20 minutes is equal to 1/3 of an hour (since there are 60 minutes in an hour). So, 6 hours and 20 minutes is equal to 6 + 1/3 = 6.33 hours. Now, we can set up a proportion to find the distance: 210 kilometers / 3.5 hours = x kilometers / 6.33 hours Cross-multiplying and solving for x, we find: x = (210 kilometers * 6.33 hours) / 3.5 hours Calculating this expression gives us: x ≈ 379.71 kilometers or 380 kilometers. Therefore, the train can travel approximately 379.71 kilometers in 6 hours and 20 minutes.

Juliana earns P420 pesos if she works for four hours. How much does she earn if she works for 10 hours if she gets paid at the same rate?

If Juliana earns P420 pesos for working 4 hours, we can find her hourly rate by dividing her total earnings by the number of hours worked. Hourly rate = Total earnings / Number of hours worked Hourly rate = P420 / 4 = P105 per hour Now, if she works for 10 hours at the same rate, we can calculate her earnings by multiplying her hourly rate by the number of hours worked. Earnings = Hourly rate * Number of hours worked Earnings = P105 * 10 = P1050 Therefore, Juliana would earn P1050 pesos if she works for 10 hours at the same rate.

How many kilos of tea that costs P80 per kilo must be mixed with 14 kilos of tea that costs P30 per kilo to make a mixture that costs P45 per kilo?

To find the amount of tea that costs P80 per kilo needed to make a mixture that costs P45 per kilo, we can use a weighted average formula. Let's assume x represents the kilos of tea that costs P80 per kilo. The total cost of the mixture can be calculated as follows: (80x + 30 * 14) / (x + 14) = 45 Simplifying the equation gives us: 80x + 420 = 45x + 630 Subtracting 45x from both sides, we get: 35x + 420 = 630 Subtracting 420 from both sides, we have: 35x = 210 Dividing both sides by 35, we find: x = 6 Therefore, you would need 6 kilos of tea that costs P80 per kilo to mix with the 14 kilos of tea that costs P30 per kilo to make a mixture that costs P45 per kilo.

Rose has P1,200 pesos. Jack has 80% of this amount while Francis has 50% of Jack's money. How much money does Francis have?

Jack has 80% of Rose's money, which is 0.8 * P1,200 = P960. Francis has 50% of Jack's money, which is 0.5 * P960 = P480. Therefore, Francis has P480.

How many prime numbers are there from 60 to 80?

1. Start by listing out the numbers from 60 to 80: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80. 2. Check if each number is divisible by any smaller prime numbers (2, 3, 5, and 7). 3. If a number is divisible by any of these smaller primes, it is not a prime number. 4. If a number is not divisible by any of these smaller primes, it is a prime number. 5. Keep track of the prime numbers you find: 61, 67, 71, 73, 79. So, the prime numbers between 60 and 80 are 61, 67, 71, 73, and 79. There are 5 prime numbers.

In a government commissary, the price of three eggs is P16.50. At this rate, how much is two dozen eggs?

At the rate of P16.50 for three eggs, we can calculate the price of two dozen eggs. There are 12 eggs in a dozen, so two dozen eggs would be 12 x 2 = 24 eggs. To find the price of two dozen eggs, we can set up a proportion: 3 eggs / P16.50 = 24 eggs / x Cross-multiplying, we get: 3x = 24 * P16.50 Simplifying, we have: 3x = P396 Dividing both sides by 3, we find: x = P132 Therefore, the price of two dozen eggs would be P132.

It takes 15 women 24 minutes to paint a wall. How many minutes will it take 18 women to paint a wall at the same rate?

Based on the proportional relationship between the number of women and the time it takes to complete the task, we can set up the equation: (15 women) * (24 minutes) = (18 women) * (x minutes) Simplifying the equation: 360 = 18x Dividing both sides by 18: x = 360 / 18 Calculating that: x = 20 Therefore, it would take 18 women approximately 20 minutes to paint the wall at the same rate. ​

(18 + 7) (12 + 9) - (7 x16) (4 + 2)

First, simplify inside the parentheses: 18 + 7 = 25 and 12 + 9 = 21; 7 x 16 = 112 and 4 + 2 = 6. But, since the parentheses are not separated by a multiplication sign, they are not multiplied together. So, the correct calculation is: (18 + 7) + (12 + 9) - (7 x 16) + (4 + 2) = 63. The correct answer is indeed 63.

Joe bought 16 kilos of rice for 680 pesos. If he will buy 3 more kilos of rice at the same price, how much more will he pay?

If Joe bought 16 kilos of rice for 680 pesos, and he wants to buy 3 more kilos at the same price, we can calculate the cost per kilo of rice. Cost per kilo = Total cost / Total kilos Cost per kilo = 680 pesos / 16 kilos = 42.50 pesos per kilo To find out how much more Joe will pay for the 3 additional kilos, we can multiply the cost per kilo by the number of additional kilos. Additional cost = Cost per kilo * Additional kilos Additional cost = 42.50 pesos per kilo * 3 kilos = 127.50 pesos Therefore, Joe will pay an additional 127.50 pesos for the 3 more kilos of rice.

What is the 7th term in the geometric series which has a third term of 4 and a common ratio of -3?

In a geometric series, each term is found by multiplying the previous term by a constant called the common ratio. In this case, the common ratio is -3. The formula to find the nth term in a geometric series is: Tn = a * r^(n-1) Where Tn is the nth term, a is the first term, r is the common ratio, and n is the term number. Given that the third term is 4, we can substitute this value into the formula: 4 = a * (-3)^(3-1) Simplifying: 4 = a * (-3)^2 4 = a * 9 a = 4/9 Now, to find the 7th term, we substitute the values into the formula: T7 = (4/9) * (-3)^(7-1) T7 = (4/9) * (-3)^6 T7 = (4/9) * 729 T7 = 324 Therefore, the 7th term in the geometric series is indeed 324.

16 + 4x (7+8) -3=________.

To solve the expression, we follow the order of operations: parentheses, multiplication/division, and addition/subtraction. (PEMDAS) First, we simplify the expression inside the parentheses: 7 + 8 = 15. Then, we perform the multiplication: 4x15 = 60. Next, we add: 16 + 60 = 76. Finally, we subtract: 76 - 3 = 73. So, the result of the expression is 73.

The sum of 4 consecutive odd numbers is 488. What is the smallest number?

To solve this problem efficiently in a timed exam, you can use algebraic equations. Let's call the smallest odd number "x". The four consecutive odd numbers can be represented as x, x+2, x+4, and x+6. The sum of these four numbers is 488, so we can set up the equation: x + (x+2) + (x+4) + (x+6) = 488 Simplifying the equation, we get: 4x + 12 = 488 Now, solve for x: 4x = 476 x = 119 Therefore, the smallest number is 119.

How many 5-letter words can you form if the word is to start and end with a vowel?

Step 1: 5 vowels can start and end the word, so 5*5=25 combinations. Step 2: Any of 26 letters can fill the 3 middle spots, so 26*26*26=17576 combinations. Step 3: Multiply the two results, 25*17576=439400.

What is the square of 5 plus the cube of 4?

Step 1: Find the square of 5. Square of 5 = 5 * 5 = 25. Step 2: Find the cube of 4. Cube of 4 = 4 * 4 * 4 = 64. Step 3: Add the square of 5 to the cube of 4. 25 + 64 = 89. Therefore, the square of 5 plus the cube of 4 is 89.

What is the intersection of the lines 2x+3y=12 and x+y=5?

Step 1: From x+y=5, isolate x. So x=5-y. Step 2: Substitute x in 2x+3y=12. So, 2(5-y)+3y=12. Step 3: Solve for y. You get y=2. Step 4: Substitute y=2 in x=5-y. You get x=3. So, intersection point is (3, 2).

John has a certain amount of candies. He gave a third of it to his friend Anton, and then he gave half of what was left to his friend Adrian. John then had 12 candies left. How many candies did John start with?

Step 1: Let's say John started with x candies. Step 2: John gave a third of his candies to Anton, which is x/3. Step 3: After giving a third to Anton, John had (x - x/3) candies left. Step 4: John then gave half of what was left to Adrian, which is (x - x/3)/2. Step 5: After giving half to Adrian, John had (x - x/3)/2 candies left, which is equal to 12. Step 6: Setting up the equation: (x - x/3)/2 = 12. Step 7: Solving the equation: (x - x/3)/2 = 12. x - x/3 = 24. 3x - x = 72. 2x = 72. x = 36. Therefore, John started with 36 candies.

What is the GCF of 24, 48, and 40?

The GCF of 24, 48, and 40 is 8. To find this quickly, list the factors of each number and identify the highest number that appears in all three lists. Here's how you can do it: 1. List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 2. List factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 3. List factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 You can see that 8 is the highest number that appears in all three lists, so it's the GCF.

8 19 28 35 40 __

By following the pattern of adding 11, 9, 7, 5, and 3, the next number should indeed be 43.

It takes 24 men 60 days to build a house. Working at the same rate, how many men will be needed to build a house in 40 days?

If it takes 24 men 60 days to build a house, we can use the formula: (number of men) * (time) = (constant) So, when we have 24 men, we can set up the equation: 24 men * 60 days = (number of men) * 40 days Simplifying the equation: 1440 = 40 * (number of men) Dividing both sides by 40: (number of men) = 1440 / 40 Calculating that: (number of men) = 36 Therefore, it would take approximately 36 men to build a house in 40 days at the same rate.

What is the largest prime number less than 100?

The largest prime number less than 100 is 97. It is not divisible by any other number except 1 and itself.

Three years ago, Janina was twice as old as Florence. If the sum of their present age is 39, how old was Florence three years ago?

Three years ago, Janina was twice as old as Florence. Let's say Florence's age three years ago was F. Since Janina was twice as old, Janina's age three years ago would be 2F. Now, let's consider their present ages. The sum of their present ages is 39. So, we can set up the equation: F + 2F = 39 Combining like terms: 3F = 39 Dividing both sides by 3: F = 13 Therefore, Florence was 13 years old three years ago.

A pair of shoes worth P4500 is to be sold on a 30% discount. What will be its new price?

To find the new price after a 30% discount, we can calculate the discount amount first. The discount amount is 30% of P4500, which is P1350. To find the new price, we subtract the discount amount from the original price: P4500 - P1350 = P3150 Therefore, the new price of the pair of shoes after a 30% discount will be P3150.

What is 2/3 of 1/2 of 5/6 of 720?

Here's the step-by-step solution: 1. Find 5/6 of 720: 5/6 * 720 = 600 2. Find 1/2 of 600: 1/2 * 600 = 300 3. Find 2/3 of 300: 2/3 * 300 = 200 Therefore, 2/3 of 1/2 of 5/6 of 720 is 200.

How many plate numbers with 3 letters and 4 numbers exist?

1. For the first letter, there are 26 choices (A-Z). 2. For the second letter, there are also 26 choices. 3. For the third letter, there are again 26 choices. 4. For the first number, there are 10 choices (0-9). 5. For the second number, there are also 10 choices. 6. For the third number, there are again 10 choices. 7. For the fourth number, there are once again 10 choices. To find the total number of possible plate numbers, we multiply the number of choices for each position together: 26 * 26 * 26 * 10 * 10 * 10 * 10 = 175,760,000 So, there are 175,760,000 possible plate numbers with 3 letters and 4 numbers.

The sum of 3 consecutive odd numbers is 357. What is the value of the smallest of the 3?

Let's call the smallest odd number "x". The problem states that the sum of 3 consecutive odd numbers is 357. So, we can write the equation as: x + (x + 2) + (x + 4) = 357. Simplifying the equation, we get: 3x + 6 = 357. Subtracting 6 from both sides, we find that 3x = 351. Dividing both sides by 3, we get: x = 117. Therefore, the value of the smallest odd number is 117.

A piece of clothing worth P9,700 is to be sold on a 40% discount. What will be its new price?

The new price will be P9,700 minus 40% of P9,700. Let's calculate that. 40% of P9,700 = 0.40 * P9,700 = P3,880 So, the new price = P9,700 - P3,880 = P5,820 Therefore, the new price of the clothing after a 40% discount will be P5,820.

Joseph is solving a problem that involves scientific notation. What is 6.85 x 10⁴?

To calculate 6.85 x 10⁴, we multiply 6.85 by 10,000. This gives us 68,500. So, 6.85 x 10⁴ is equal to 68,500.

How many prime numbers are there from 15 to 40?

To find the prime numbers between 15 and 40, we can check each number in that range. The prime numbers between 15 and 40 are: 17, 19, 23, 29, 31, 37. Therefore, there are 6 prime numbers between 15 and 40.

If n+3 is odd, what must be odd? a. 2n b. 4(n+3) c. n+9 d. 2n+3

d. 2n+3 If n+3 is odd, it means that n must be even. When we double an even number (2n) and add an odd number (3), the result will always be odd. Therefore, 2n+3 is also odd.

A movie started at 12:50 pm and ended at 3:20 pm. How long was the movie?

To find the duration of the movie, we can subtract the start time from the end time. 3:20 pm - 12:50 pm = 2 hours and 30 minutes. Therefore, the movie was 2 hours and 30 minutes long.

One "hand" is 4 inches whild one "foot" is 12 inches. How many hands are there in 6 1/2 feet?

Since one "hand" is 4 inches and one "foot" is 12 inches, we can calculate how many hands are in 6 1/2 feet by converting feet to inches and then dividing by the length of one hand. First, let's convert 6 1/2 feet to inches: 6 1/2 feet = (6 * 12) + (1/2 * 12) = 72 + 6 = 78 inches Now, let's divide the length in inches by the length of one hand: 78 inches / 4 inches = 19.5 hands Therefore, there are 19.5 hands in 6 1/2 feet.

A painting originally priced P15,000 was put on sale with a 15% discount. What is the discounted price of the painting?

To find the discounted price of the painting, we need to subtract the discount from the original price. The discount is 15% of the original price, which is P15,000. To calculate the discount, we multiply the original price by 15%: P15,000 * 0.15 = P2,250. Subtracting the discount from the original price, we get the discounted price: P15,000 - P2,250 = P12,750. Therefore, the discounted price of the painting is P12,750.

Which of the following numbers is divisible by 4? Is it 192, 268, 248, or 596?

To determine if a number is divisible by 4, we need to check if the last two digits of the number form a multiple of 4. In this case, the last two digits of 192 are 92, which is a multiple of 4 (92 = 4 x 23). Therefore, 192 is divisible by 4. NOTE: Identify the last two digits of the number and check if they form a multiple of 4. This can be done mentally by recognizing common multiples of 4, such as 4, 8, 12, 16, 20, and so on. Practice this strategy to improve speed and accuracy.

Which of the following numbers is prime? 57, 87, 89, 91

To determine if a number is prime, we need to check if it is divisible by any number other than 1 and itself. Let's analyze the numbers provided: 57: It is not a prime number because it is divisible by 3 (57 = 3 x 19). 87: It is not a prime number because it is divisible by 3 (87 = 3 x 29). 89: It is a prime number because it is only divisible by 1 and itself. 91: It is not a prime number because it is divisible by 7 (91 = 7 x 13). NOTE: Quickly check if the number is divisible by any prime numbers up to its square root. This can help determine if the number is prime or not.

A house needs 47 liters of paint to be completely painted. If each can hold 5 liters of paint. How many cans must be bought to be able to completely paint the house?

To find out how many cans of paint must be bought to completely paint the house, we need to divide the total amount of paint needed (47 liters) by the capacity of each can (5 liters): 47 liters / 5 liters = 9.4 cans Since we can't buy a fraction of a can, we need to round up to the nearest whole number. Therefore, you would need to buy 10 cans of paint to be able to completely paint the house.

Eight kilograms of rice are needed to feed 12 people in a dormitory for one week. If 3 more people are added, how many kilograms of rice are needed to feed them for one week?

If 8 kilograms of rice are needed to feed 12 people for one week, we can use a proportion to find out how many kilograms of rice are needed to feed 3 more people. We can set up the proportion: 8 kg / 12 people = x kg / 15 people To solve for x, we can cross-multiply and divide: 8 kg * 15 people = 12 people * x kg 120 kg = 12x kg Dividing both sides by 12: 120 kg / 12 = x kg 10 kg = x Therefore, if 3 more people are added, 10 kilograms of rice are needed to feed them for one week.

If each one works alone, Ana can finish a job in 4 hours, Trisha in 3 hours and Lia in 6 hours. How long will it take them to finish the same job if they work together?

If Ana can finish the job in 4 hours, Trisha in 3 hours, and Lia in 6 hours, we can calculate their combined work rate. Ana's work rate is 1 job per 4 hours. Trisha's work rate is 1 job per 3 hours. Lia's work rate is 1 job per 6 hours. To find their combined work rate, we add their individual work rates: 1/4 + 1/3 + 1/6 = 3/12 + 4/12 + 2/12 = 9/12 = 3/4 job per hour. Therefore, if they work together, they will finish the job in 4/3 hours or 1 hour and 20 minutes.

If each of them works alone, Mario can finish the job in 2 hours and Luigi in 8 hours. If the two of them work together with Peach, they can finish the job in 4/3 hours. How fast can Peach do the job on her own?

If Mario can finish the job in 2 hours and Luigi in 8 hours, we can determine their individual rates of work. Mario's rate is 1/2 of the job per hour, and Luigi's rate is 1/8 of the job per hour. When Mario, Luigi, and Peach work together, their combined rate is 3/4 of the job per hour (since they finish the job in 4/3 hours). To find Peach's individual rate, we subtract the rates of Mario and Luigi from the combined rate: (3/4) - (1/2 + 1/8) = 3/4 - 5/8 = 1/8 Therefore, Peach can finish the job on her own in 8 hours. Peach can do the job on her own in 8 hours.

Mrs. Gonzales divided the 48 members of her class into three groups at a ratio of 1:2:3 How many members does the biggest group have?

If Mrs. Gonzales divided the class into three groups at a ratio of 1:2:3, we can start by finding the total number of parts. In this case, the total number of parts is 1 + 2 + 3 = 6. To find the number of members in each group, we divide the total number of members (48) by the total number of parts (6). So, each part represents 48 ÷ 6 = 8 members. Now, we can multiply the number of parts in each group by the number of members in each part. The groups consist of 1x8 = 8, 2x8 = 16, and 3x8 = 24 members each. Therefore, the largest group has 24 members.

The sum of 3 consecutive even numbers is 522. What is the largest number?

If the sum of 3 consecutive even numbers is 522, we can find the largest number by dividing the total sum by 3. Largest number = Total sum / 3 Largest number = 522 / 3 = 174 Therefore, the largest number is 174.

There are 15 people in a party. Towards the end, each person shakes the hand of everyone else exactly once. How many handshakes were made?

If there are 15 people at the party, each person will shake hands with everyone else once. To find the total number of handshakes, we can use the formula n(n-1)/2, where n is the number of people. So, in this case, there would be 15(15-1)/2 = 105 handshakes made.

A colony of bacteria doubles every 5 hours. How many hours from now will the colony be 8 times its current size?

It will take 3 spans of 5 hours to be 8 times its current size, since 2x2x2 = 8 and 3x5=15 hours. Sure thing! Let's break it down step by step. The colony of bacteria doubles every 5 hours. This means that after every 5-hour period, the number of bacteria in the colony doubles. Now, we want to know how many hours from now the colony will be 8 times its current size. To figure this out, we can think about how many times the colony needs to double to reach that size. Since the colony starts at its current size and needs to be 8 times that size, it would need to double 3 times. Since each doubling takes 5 hours, we can multiply 3 (the number of times it needs to double) by 5 (the number of hours per doubling) to get the total number of hours: 3 * 5 = 15 Therefore, it will take approximately 15 hours from now for the colony to be 8 times its current size.

Four years ago, Rafael was twice as old as Miguel. If the sum of their present age is 23, how old was Rafael 5 years ago?

Let's assume Rafael's age four years ago was R and Miguel's age four years ago was M. According to the given problem, Rafael was twice as old as Miguel four years ago, so we have the equation R = 2M. We are also given that the sum of their ages four years ago was 15, which gives us the equation R + M = 15. Now, let's solve these equations together: From the first equation, we can rewrite it as R = 2M. Substituting this into the second equation, we have 2M + M = 15. Combining like terms, we get 3M = 15. Dividing both sides by 3, we find M = 5. Now that we know Miguel's age, we can find Rafael's age by substituting M = 5 back into the first equation: R = 2(5), which gives us R = 10. Therefore, Rafael is currently 10 years old.

Ralph eats at a certain restaurant every 3 days, Julian every 5 days and Maurice every 7 days. If they eat together in the restaurant today, how many days from now will they eat together again?

Step 1: Identify the frequency of each person's visits. Ralph: every 3 days Julian: every 5 days Maurice: every 7 days Step 2: Find the least common multiple (LCM) of 3, 5, and 7. This will be the number of days until they all eat together again. Step 3: The LCM of 3, 5, and 7 is 105. So, they will eat together again in 105 days.

Ken invested in 3 accounts, with interest rates of 3%, 4%, and 6%. The amount he invested in the 6% account is half the amount he invested in the 4% account, and the amount he invested in the 4% account is half the amount he invested in the 3% account. If the total interest he gets each year is P5,200, how much did he invest in the 6% account?

Let's call the amount Ken invested in the 3% account "x". So, the amount in the 4% account is x/2 and in the 6% account is x/4. The total interest Ken gets each year is the sum of the interests from each account: 0.03x + 0.04(x/2) + 0.06(x/4) = P5,200. Solving this equation, we find that x = P80,000. Therefore, the amount Ken invested in the 6% account is P20,000. Quick solution: 1. Define x as amount in 3% account. 2. Then, 4% account = x/2, 6% account = x/4. 3. Set up equation: 0.03x + 0.04(x/2) + 0.06(x/4) = 5200. 4. Solve for x, you get x = 80000. 5. So, amount in 6% account = 80000/4 = 20000.

Six times a number is 12 more than 3 times the same number. What is the number?

Let's call the number "x". The problem states that 6 times the number is 12 more than 3 times the same number. So, we can write the equation as: 6x = 3x + 12. Simplifying the equation, we get: 3x = 12. Dividing both sides by 3, we find that x = 4. Therefore, the number is 4.

A car is travelling north at a rate of 40 km/h. Another car, travelling east, is moving at a rate of 30 km/h. How much time will it take for them to be exactly 100 km apart?

Step 1: Determine the rate and direction of each car. - The first car is traveling north at a rate of 40 km/h. - The second car is traveling east at a rate of 30 km/h. Step 2: Set up the distance equation for each car. - For the first car: distance = rate * time => 100 km = 40 km/h * t1. - For the second car: distance = rate * time => 100 km = 30 km/h * t2. Step 3: Solve for the time it takes for each car to travel 100 km. - For the first car: t1 = 100 km / 40 km/h = 2.5 hours. - For the second car: t2 = 100 km / 30 km/h = 3.33 hours. Step 4: Find the total time it takes for the cars to be exactly 100 km apart. - Add the times together: t1 + t2 = 2.5 hours + 3.33 hours = 5.83 hours. Step 5: Convert the total time to hours and minutes. - 5.83 hours is approximately 5 hours and 50 minutes. Therefore, it will take approximately 2 hours and 50 minutes for the cars to be exactly 100 km apart.

The price of 5 apples is the same as the price of 3 oranges. The price of 4 oranges is the same as the price of 10 bananas. How many apples are equivalent in price to 15 bananas?

Step 1: We know that the price of 5 apples is the same as the price of 3 oranges. Step 2: We also know that the price of 4 oranges is the same as the price of 10 bananas. Step 3: To find the equivalent price of apples to bananas, we need to find the price of 1 apple and the price of 1 banana. Step 4: From Step 1, we can determine that the price of 1 apple is equal to (3 oranges) / 5. Step 5: From Step 2, we can determine that the price of 1 orange is equal to (10 bananas) / 4. Step 6: Now, we can find the price of 1 apple in terms of bananas by substituting the values from Step 4 and Step 5. Step 7: 1 apple = ((3 oranges) / 5) = ((10 bananas) / 4) / 5. Step 8: Finally, to find the price of 15 bananas in terms of apples, we can multiply the price of 1 apple by 15. Step 9: 15 bananas = 15 * (((10 bananas) / 4) / 5) = 10 apples. Therefore, 15 bananas is equivalent in price to 10 apples.

What is the probability that you won't get a 1 outcome upon rolling 3 fair 6-sided dice?

To find the probability of not getting a 1 outcome on a fair 6-sided die, we need to determine the probability of getting any other outcome (2, 3, 4, 5, or 6) on a single die roll. The probability of not getting a 1 on a single die roll is 5/6, since there are 5 favorable outcomes (2, 3, 4, 5, or 6) out of 6 possible outcomes. Since we are rolling 3 dice, we need to find the probability of not getting a 1 on all 3 dice rolls. The probability of not getting a 1 on a single die roll is 5/6, so the probability of not getting a 1 on all 3 dice rolls is (5/6) * (5/6) * (5/6) = 125/216. Therefore, the probability of not getting a 1 outcome upon rolling 3 fair 6-sided dice is 125/216. Step 1: The probability of not getting a 1 on a single die roll is 5/6. Step 2: To find the probability of not getting a 1 on all 3 dice rolls, we multiply the probabilities together. Step 3: (5/6) * (5/6) * (5/6) = 125/216. Therefore, the probability of not getting a 1 outcome upon rolling 3 fair 6-sided dice is 125/216.

Pacloyd Mannyweather must slim down to his fighting weight in three months. In March 2016, he weighed 150 pounds. In June 2016, he weighed 120 pounds. Find the rate of decrease of his weight.

To find the rate of decrease of Pacloyd Mannyweather's weight, we can calculate the difference in weight divided by the number of months. Weight decrease = Initial weight - Final weight Weight decrease = 150 pounds - 120 pounds = 30 pounds Rate of decrease = Weight decrease / Number of months Rate of decrease = 30 pounds / 3 months = 10 pounds per month Therefore, the rate of decrease of Pacloyd Mannyweather's weight is 10 pounds per month.

What is the smallest number that leaves a remainder of 1 when divided by 3, 4, and 5?

To find the smallest number that leaves a remainder of 1 when divided by 3, 4, and 5, we can use the concept of the least common multiple (LCM) of these numbers. The LCM of 3, 4, and 5 is 60. To find the smallest number that leaves a remainder of 1 when divided by these numbers, we can add 1 to the LCM: 60 + 1 = 61 Therefore, the smallest number that leaves a remainder of 1 when divided by 3, 4, and 5 is 61.

How many ways can you form a starting lineup for a basketball team (with positions PG, SG, SF, PF, and C) from a pool of 10 players?

To form a starting lineup for a basketball team with positions PG, SG, SF, PF, and C, from a pool of 10 players, we can use the concept of permutations. The number of ways to arrange the players in the lineup is given by the formula 10P5, which stands for 10 permutations taken 5 at a time. Evaluating this, we find that there are 30,240 ways to form the starting lineup. 🏀🔢

A colony of bacteria doubles every 3 hours . How many hours from now will the colony be 32 times its current size?

To solve quickly, use powers of 2. Since 32 is 2^5, and the bacteria doubles (2^1) every 3 hours, it will take 5*3 = 15 hours. First, we know that the colony doubles every 3 hours. This means that the growth rate is exponential. To find out how many hours it will take for the colony to be 32 times its current size, we need to determine the exponent that will give us 32 when we raise 2 to that power. In this case, 32 is equal to 2^5. This means that the colony will reach 32 times its current size in 5 hours. So, the solution is that it will take 5 hours from now for the colony to be 32 times its current size.

Katrina can finish a job in 4 hours of she works alone. Also working alone, Grace can finish the same job in 3 hours. Katrina has been working on the job for 2 hours when Grace joins her. How much longer will it take them to complete the job?

We can use the concept of work rates. First, calculate the individual work rates of Katrina and Grace. Katrina can finish the job in 4 hours, so her work rate is 1/4 of the job per hour. Grace can finish the job in 3 hours, so her work rate is 1/3 of the job per hour. Next, determine their combined work rate when they work together. Since work rates add up when people work together, their combined work rate is the sum of their individual work rates: 1/4 + 1/3 = 7/12 of the job per hour. Now, calculate the remaining work that needs to be done. Since Katrina has already worked for 2 hours, she has completed 2/4 = 1/2 of the job. Therefore, 1 - 1/2 = 1/2 of the job still needs to be done. Finally, divide the remaining work by the combined work rate to find the time it will take them to complete the job together: (1/2) / (7/12) = 6/7 hours. Therefore, it will take them 6/7 hours to complete the job together.

How many ways can you arrange 6 people around a circular table if there are 2 specific people that want to sit beside each other?

We have 6 people to arrange around a circular table, and 2 specific people who want to sit beside each other. Let's consider these 2 specific people as a single entity, which we'll call "X". Now, we have 5 entities to arrange around the circular table: X, and the remaining 4 people. First, let's arrange the 5 entities without any restrictions. We can do this in (5-1)! = 4! = 24 ways. However, within those 24 arrangements, the 2 specific people (X) can be arranged in 2 different ways: X can either be on the left of the other person or on the right. So, for each of the 24 arrangements of the 5 entities, we have 2 different arrangements for the specific people (X). Multiplying these two values together, we get 24 * 2 = 48. Therefore, there are 48 ways to arrange the 6 people around the circular table with the 2 specific people sitting beside each other.

What acute angle do the hands of the clock form at 7:52?

To find the acute angle between the hands of the clock at 7:52, we can use the formula: Angle = |30H - 11M/2| Where H represents the hour and M represents the minutes. Plugging in the values for 7:52, we get: Angle = |30(7) - 11(52)/2| Angle = |210 - 286/2| Angle = |-76/2| Angle = 38 degrees Therefore, the acute angle between the hands of the clock at 7:52 is 38 degrees.

Each time a ball bounces off the ground, it only reaches as high as half its previous maximum height. How high will a ball bounce up the second time if it is initially released from a height of 140cm?

Let's break it down step by step: 1. The ball is initially released from a height of 140cm. 2. On the first bounce, it reaches half of its previous maximum height. So, it bounces up to 140cm/2 = 70cm. 3. On the second bounce, it again reaches half of its previous maximum height. Therefore, it will bounce up to 70cm/2 = 35cm. So, the ball will bounce up to a height of 35cm on the second bounce.

Working alone, Rafael can finish a job in 4 days. If he works with Troy, they can finish the job in 2.4 days. How many days will it take Troy to finish the job alone?

If Rafael can finish the job alone in 4 days and together with Troy they can finish it in 2.4 days, we can calculate their combined rate of work. Let's assume that Rafael's rate is R and Troy's rate is T. From the given information, we know that: Rafael's rate is 1 job per 4 days, which is 1/4 job per day. Together, Rafael and Troy's combined rate is 1 job per 2.4 days, which is 1/2.4 job per day. To find Troy's individual rate, we subtract Rafael's rate from the combined rate: Troy's rate = Combined rate - Rafael's rate T = 1/2.4 - 1/4 Simplifying the equation: T = 5/12 - 3/12 T = 2/12 T = 1/6 Therefore, Troy can finish the job alone in 6 days. It will take Troy approximately 6 days to finish the job alone.

Three people worked together to finish a certain job in 5 hours. If they have the same rate of work, how long will it take one of them to do the same job?

If three people worked together to finish a job in 5 hours, and they have the same rate of work, then each person's individual rate is 1/3 of the job per hour. To find out how long it would take one person to do the same job, we divide the entire job (1) by their combined rate (1/3 + 1/3 + 1/3 = 1). So, it would take one person approximately 5 hours to do the same job. Therefore, it would take one person approximately 5 hours to do the same job.

Jiggs washes his car every 5 days. Today is a Tuesday, and he washed his car today. After how many days will Jiggs next wash his car again on a Tuesday?

Let's calculate it using the LCM (Least Common Multiple) method: Step 1: Jiggs washes his car every 5 days. Step 2: Today is Tuesday, and he washed his car today. Step 3: We need to find out after how many days Jiggs will wash his car again on a Tuesday. To find the number of days until Jiggs washes his car again on a Tuesday using LCM: Step 4: Find the LCM of 5 (the number of days between car washes) and 7 (the number of days in a week). Step 5: The LCM of 5 and 7 is 35. Therefore, Jiggs will wash his car again on a Tuesday after 35 days.

What is the 8th term in the following arithmetic series: 20, 18, 16, 14, ...

To find the 8th term in the arithmetic series, we can observe that each term is decreasing by 2. Starting with 20, if we subtract 2 successively, we get the following terms: 18, 16, 14, 12, 10, 8, 6. Therefore, the 8th term in the series is 6.

When their company experienced low sales, Mr. Tomlinson took a pay cut of 20%, which resulted in a new salary of P51,300. What was Mr. Tomlinson's salary before the pay cut?

To find Mr. Tomlinson's salary before the pay cut, we can use the information given. Let's call his original salary "x". After the pay cut of 20%, his new salary is 80% of his original salary: 0.8x = P51,300. Now, we can solve for x: x = P51,300 / 0.8 x = P64,125 Therefore, Mr. Tomlinson's salary before the pay cut was P64,125.

By how much does two-thirds of 150 exceed four-fifths of 120?

To find out how much two-thirds of 150 exceeds four-fifths of 120, we can calculate the difference between the two values. First, let's calculate two-thirds of 150: (2/3) * 150 = 100 Next, let's calculate four-fifths of 120: (4/5) * 120 = 96 To find the difference, subtract the smaller value from the larger value: 100 - 96 = 4 Therefore, two-thirds of 150 exceeds four-fifths of 120 by 4.

The average grade of 16 boys in the class is 85 while the average grade of the 24 girls in the class is 87. What is the average of the 40 students in the class?

To find the average of the 40 students in the class, we can calculate the total sum of all the grades and then divide it by the total number of students. The average grade of the 16 boys is 85, so the total sum of their grades is 16 * 85 = 1360. The average grade of the 24 girls is 87, so the total sum of their grades is 24 * 87 = 2088. To find the average of all 40 students, we add the total sum of the boys' and girls' grades and divide it by 40: (1360 + 2088) / 40 = 3448 / 40 = 86.2 Therefore, the average grade of the 40 students in the class is approximately 86.2.

What is the next term in the following geometric series: 16, 24, 36, 54, ____?

To find the next term in the geometric series, we can observe that each term is obtained by multiplying the previous term by a common ratio. In this case, the common ratio is 1.5. To find the next term, we can multiply the last term (54) by the common ratio (1.5). 54 * 1.5 = 81 Therefore, the next term in the series is 81.

How many minutes after 10:00 will the hands of the clock be together for the first time?

To find the number of minutes after 10:00 when the hands of the clock will be together for the first time, we can use the formula: |60H - 11M| / 2 = 0 Since the absolute value of a number divided by 2 cannot be equal to 0, the hands of the clock will never be together after 10:00. Let's solve the equation: |60H - 11M| / 2 = 0 Since the absolute value of a number divided by 2 can only be equal to 0 if the numerator is 0, we can set: 60H - 11M = 0 Solving for M, we get: M = (60H) / 11 To find the first time the hands of the clock will be together after 10:00, we need to substitute H with 10: M = (60 * 10) / 11 M = 600 / 11 M ≈ 54.55 Therefore, the hands of the clock will be together for the first time approximately 54.55 minutes after 10:00.

28% of what number is 14% of 846?

To find the number, we can set up an equation: 28% of x = 14% of 846 To solve for x, we can cross-multiply: 0.28x = 0.14 * 846 Dividing both sides by 0.28 gives us: x = (0.14 * 846) / 0.28 Simplifying this expression, we find that x is approximately 423.

Leo's age now is a prime number. His age 5 years ago, 4 years ago and 2 years ago respectively are also prime numbers. How old is Leo now?

We know that Leo's age now is a prime number. 5 years ago, his age was also a prime number. This means that his current age minus 5 must be a prime number as well. Similarly, 4 years ago and 2 years ago, his ages were prime numbers, so his current age minus 4 and minus 2 must also be prime numbers. By analyzing the prime numbers that satisfy these conditions, we find that Leo's current age is 7 years old. The rationale is the only 2 consecutive prime numbers are 2 and 3. The key observation is that the only two consecutive prime numbers are 2 and 3. Since Leo's age 5 years ago, 4 years ago, and 2 years ago were all prime numbers, the only possibility is that Leo's current age is 7 years old.

An urn has 3 white balls and 3 black balls. You draw 2 balls in succession (you do not return the first ball before drawing the second ball). What is the probability that they are both white?

To find the probability of drawing two white balls in succession from an urn with 3 white balls and 3 black balls, we can use the concept of conditional probability. The probability of drawing the first white ball is 3/6, since there are 3 white balls out of a total of 6 balls. After drawing the first white ball, there are now 2 white balls left out of a total of 5 balls. Therefore, the probability of drawing the second white ball, given that the first ball was white, is 2/5. To find the probability of both events happening (drawing two white balls in succession), we multiply the probabilities: Probability = (3/6) * (2/5) = 1/5 Therefore, the probability of drawing two white balls in succession is 1/5.

What is the probability of flipping 3 straight heads using a fair coin?

To find the probability of flipping 3 straight heads using a fair coin, we need to multiply the probabilities of each individual flip. Since the coin is fair, the probability of getting heads on each flip is 1/2. When we say that a coin is "fair," it means that the coin has an equal chance of landing on either side. In other words, the probability of getting heads or tails is the same. In a fair coin, the probability of getting either heads or tails on each flip is 1/2 because there are only two possible outcomes. So, the probability of getting heads on each flip is 1/2, assuming the coin is fair. So, the probability of flipping 3 straight heads is (1/2) * (1/2) * (1/2) = 1/8.

There are 100 animals in a farm, composed of chickens and sheep. If there are 288 legs in total, how many chickens are there?

Let's assume that the number of chickens is C and the number of sheep is S. We know that there are a total of 100 animals on the farm, so we can write the equation: C + S = 100. We also know that the total number of legs is 288. Since chickens have 2 legs and sheep have 4 legs, we can write another equation: 2C + 4S = 288. Let's solve it using the elimination method. We have the following system of equations: Equation 1: C + S = 100 Equation 2: 2C + 4S = 288 To eliminate the variable C, we can multiply Equation 1 by 2, giving us: 2C + 2S = 200 Now we can subtract Equation 2 from this new equation: (2C + 2S) - (2C + 4S) = 200 - 288 -2S = -88 Dividing both sides by -2, we get: S = 44 Now we can substitute this value of S into Equation 1 to find the value of C: C + 44 = 100 C = 100 - 44 C = 56 Therefore, there are 56 chickens on the farm.

The ratio of the angles of the quadrilateral is 1:2:4:5. What is the measure of the second largest angle?

Let's break it down step by step. The ratio of the angles in the quadrilateral is given as 1:2:4:5. This means that the second largest angle is represented by the second part of the ratio, which is 2. To find the measure of the second largest angle, we need to determine the total sum of the angle measures in the quadrilateral. Since the total sum of angles in a quadrilateral is always 360 degrees, we can set up an equation: 1x + 2x + 4x + 5x = 360 Simplifying the equation, we have: 12x = 360 Dividing both sides by 12, we find: x = 30 Now, we can substitute the value of x back into the equation to find the measure of the second largest angle: 2x = 2 * 30 = 60 Therefore, the measure of the second largest angle in the quadrilateral is indeed 60 degrees.

You draw 3 cards from a well-shuffled standard 52-card deck. What is the probability that they are all aces?

Let's break it down step by step. To find the probability of drawing all four aces, we need to consider the number of favorable outcomes and the total number of possible outcomes. The number of favorable outcomes is 1, since there is only one set of four aces in the deck. Now, let's calculate the total number of possible outcomes. To do this, we need to determine the number of ways to choose 3 cards from a deck of 52 cards. This can be calculated using the combination formula: C(52, 3) = 52! / (3! * (52-3)!) = 22,100. Therefore, the probability of drawing all four aces is: 1 favorable outcome / 22,100 total outcomes = 1/22,100. So, the probability is indeed 1/5525.

Angelo gas a certain number of candies. He says that if he gives the candies to 4 kids, each will receive the same amount. He also says that this is still true if he gives the candies away to 6 kids. What is the smallest number of candies Angelo can have?

Let's call the number of candies Angelo has "x". If he gives the candies to 4 kids, each will receive x/4 candies. If he gives the candies to 6 kids, each will receive x/6 candies. For both scenarios to be true, x must be divisible by both 4 and 6. The smallest number that satisfies this condition is the least common multiple (LCM) of 4 and 6, which is 12. Therefore, the smallest number of candies Angelo can have is 12.

The sum of 3 numbers is 72. The first number is four more than the second number, and the last number is 3 more than half the second number. What is the largest number?

Let's call the second number "x". The first number is four more than the second number, so it is x + 4. The last number is 3 more than half the second number, so it is (1/2)x + 3. The sum of the three numbers is 72, so we can write the equation: (x + 4) + x + ((1/2)x + 3) = 72 Simplifying the equation, we get: (5/2)x + 7 = 72 To find the value of x, we can subtract 7 from both sides and then multiply by 2/5: (5/2)x = 65 x = 26 Now we can find the largest number by substituting x into the expressions: First number: x + 4 = 26 + 4 = 30 Last number: (1/2)x + 3 = (1/2)(26) + 3 = 13 + 3 = 16 Therefore, the largest number is 30.

The ratio of the length of a rectangle to its width is 7:4. If the length of the rectangle is 45.5 centimeters, what is its width?

Let's set up a proportion to find the width of the rectangle. The ratio of the length to the width is 7:4. We can write this as 7/4. Let's call the width "w". We can set up the proportion: length/width = 7/4. Substituting the given length of 45.5 centimeters, we get: 45.5/w = 7/4. To solve for "w", we can cross-multiply and solve for "w". Cross-multiplying, we get: 45.5 * 4 = 7 * w. Simplifying the equation, we get: 182 = 7w. Dividing both sides by 7, we find: w = 26. Therefore, the width of the rectangle is 26 centimeters.

Justin rows at a constant rate. It takes him 5 hours to row from A to B upstream and 4 hours to row from B to A downstream. If the distance between A and B is 20 kilometers, what is the speed of the current?

Let's solve this problem step by step in an efficient way. 1. Set up the equation for the upstream journey: 20 = (R - C) * 5 2. Simplify the equation: 20 = 5R - 5C 3. Set up the equation for the downstream journey: 20 = (R + C) * 4 4. Simplify the equation: 20 = 4R + 4C 5. Solve the two equations simultaneously to find the values of R and C. To do this, we can use a method called elimination. Multiply the first equation by 4 and the second equation by 5 to eliminate the variable C: 20 * 4 = (5R - 5C) * 4 20 * 5 = (4R + 4C) * 5 Simplify these equations: 80 = 20R - 20C 100 = 20R + 20C Add the two equations together to eliminate the variable R: 80 + 100 = 20R - 20C + 20R + 20C 180 = 40R Now, solve for R: R = 180 / 40 R = 4.5 km/h Substitute the value of R back into one of the original equations to solve for C: 20 = (4.5 - C) * 5 20 = 22.5 - 5C -2.5 = -5C C = -2.5 / -5 C = 0.5 km/h Therefore, the speed of the current is 0.5 km/h.

In a musical play, the price of admission of adults is 2 1/2 times the price of admission for children below 12 years of age. If the price of admission for adults is 225 pesos, how much is the price of admission for children below 12 years of age?

Step 1: Set up the equation. Let x be the price of admission for children below 12 years of age. 2 1/2 * x = 225 Step 2: Convert the mixed number to an improper fraction. 2 1/2 = 5/2 Step 3: Rewrite the equation. (5/2) * x = 225 Step 4: Isolate x by multiplying both sides by the reciprocal of 5/2. x = (225 * 2/5) Step 5: Simplify. x = 90 Therefore, the price of admission for children below 12 years of age is 90 pesos.

The average of the first 2 exams of a student is 52. His average over 3 exams should be at least 60 for him to pass. What is the minimum score he must obtain on his third exam to pass?

Step 1: The total of the first 2 exams is 2*52 = 104. Step 2: The total for 3 exams to average 60 is 3*60 = 180. Step 3: Subtract the total of the first 2 exams from the total needed: 180 - 104 = 76. So, the student needs at least 76 on the third exam to pass.

What is the 10th term in the arithmetic series which has a first term of 7 and a common difference of 6?

To find the 10th term in an arithmetic series, we can use the formula: nth term = first term + (n - 1) * common difference. In this case, the first term is 7 and the common difference is 6. Plugging in the values, we get: 10th term = 7 + (10 - 1) * 6. Simplifying the equation, we find: 10th term = 7 + 9 * 6. Calculating further, we get: 10th term = 7 + 54. Therefore, the 10th term in the arithmetic series is 61.

What is the GCF of 45, 75, and 405?

To find the GCF of 45, 75, and 405, we can use the prime factorization method. Prime factorize each number and identify the common prime factors. 45 = 3 * 3 * 5 75 = 3 * 5 * 5 405 = 3 * 3 * 3 * 3 * 5 The common prime factors are 3 and 5. Multiply these common prime factors together to find the GCF. GCF = 3 * 5 = 15 Therefore, the GCF of 45, 75, and 405 is 15.

Lea bought a variety of clothes for a total cost of P1,342. How much change did she get if she paid a total of P2,000?

To find the change Lea received, we can subtract the total cost of the clothes from the amount she paid. Amount paid - Total cost of clothes = Change In this case, Lea paid P2,000 and the total cost of the clothes was P1,342. 2,000 - 1,342 = P658 Therefore, Lea received P658 in change.

Marian needs an average of at least 90 in four quarters to obtain an "outstanding" rating in Math. If she received grades of 88, 91, and 89, what must her grade in the fourth quarter to have an average of exactly 90?

To find the grade Marian needs in the fourth quarter to have an average of exactly 90, we can use the concept of the average formula. The sum of her grades in the first three quarters is: 88 + 91 + 89 = 268. To have an average of exactly 90, the sum of her grades in all four quarters should be: 90 * 4 = 360. To find the grade she needs in the fourth quarter, we subtract the sum of her grades in the first three quarters from the desired sum: 360 - 268 = 92. Therefore, Marian needs a grade of 92 in the fourth quarter to have an average of exactly 90.

A group of 5 students has an average grade of 88. When another student joined the group, the average became 89. What was the grade of the students that joined?

To find the grade of the student who joined the group, we can use the concept of the average formula. The average grade of the group of 5 students was 88. The sum of their grades is 5 * 88 = 440. When another student joined, the average became 89. The total number of students in the group is now 6. To find the grade of the student who joined, we can multiply the new average by the total number of students and subtract the sum of the grades of the original 5 students: 89 * 6 - 440 = 534 - 440 = 94 Therefore, the grade of the student who joined the group was 94.

What is the intersection of the lines 4x-y= -14 and x+3y=16?

To find the intersection point of the lines 4x - y = -14 and x + 3y = 16, we can solve the system of equations. Step 1: Solve one equation for one variable. Let's solve the second equation, x + 3y = 16, for x: x = 16 - 3y Step 2: Substitute the expression for x in the other equation. Substitute x in the first equation, 4x - y = -14, with 16 - 3y: 4(16 - 3y) - y = -14 Step 3: Simplify and solve for y. 64 - 12y - y = -14 -13y = -78 y = 6 Step 4: Substitute the value of y back into one of the original equations to solve for x. Using the second equation, x + 3y = 16: x + 3(6) = 16 x + 18 = 16 x = -2 Therefore, the intersection point of the lines 4x - y = -14 and x + 3y = 16 is (-2, 6).

What is the intersection of the lines y-x=6 and 5x-y+2=0?

To find the intersection point of the lines y-x=6 and 5x-y+2=0, we can solve the system of equations. By substituting y = x + 6 into the second equation, we get 5x - (x + 6) + 2 = 0. Simplifying this equation gives us 4x - 4 = 0. Solving for x, we find x = 1. Substituting this value back into the first equation, we get y = 1 + 6 = 7. Therefore, the intersection point is (1, 7).

What is the largest number less than 1000 that is divisible by 27?

To find the largest number less than 1000 divisible by 27, we can divide 1000 by 27. The quotient is 37 with a remainder of 1. To find the largest number divisible by 27, we multiply the quotient (37) by 27. So, the largest number less than 1000 that is divisible by 27 is 37 * 27 = 999.

In a map, 5 centimeters represents 12 kilometers in the real world. If the straight distance from Greenfield to Blue Bay is 75 kilometers, how long will this be on the map?

To find the length on the map, we can set up a proportion using the scale of the map: 5 cm represents 12 km Let's call the length on the map "x cm". We can set up the proportion: 5 cm / 12 km = x cm / 75 km To solve for x, we can cross multiply and divide: 5 cm * 75 km = 12 km * x cm 375 cm * km = 12 km * x cm Dividing both sides by 12 km: 375 cm * km / 12 km = x cm 31.25 cm = x So, the straight distance from Greenfield to Blue Bay on the map would be approximately 31.25 cm.

Two numbers have a sum of 50. What is the maximum product that they can have?

To find the maximum product of two numbers with a sum of 50, we can use the concept of maximizing the product when the numbers are closest to each other. Let's call the two numbers x and y. Since their sum is 50, we can write the equation: x + y = 50. To maximize the product, we want x and y to be as close to each other as possible. In this case, the two numbers that are closest to each other and have a sum of 50 are 25 and 25. Therefore, the maximum product that the two numbers can have is 25 * 25 = 625.

The average test score of a class of 20 students is 72. One student found a correction and raised his score by 10. What is the new class average?

To find the new class average, we need to calculate the sum of all the students' scores, including the student who raised their score by 10. The original sum of the scores is 20 students * 72 average score = 1440. After the correction, the student's score increases by 10, so the new sum of the scores is 1440 + 10 = 1450. To find the new average, we divide the new sum of the scores by the total number of students, which is still 20. The new class average is 1450 / 20 = 72.5.

In a Math test, Den got 18 out of 25 correct answers. What percent of his answers are not correct?

To find the percent of Den's answers that are not correct, we can subtract the number of correct answers from the total number of answers and then calculate the percentage. Den got 18 out of 25 correct, so the number of incorrect answers is 25 - 18 = 7. To find the percentage of incorrect answers, we divide the number of incorrect answers by the total number of answers (25) and multiply by 100: (7/25) * 100 = 28%. Therefore, 28% of Den's answers are not correct.

In a government office, there are 45 employees. Among these employees, 36 are female. What percent of the employees in the office are female?

To find the percentage of employees who are female, we can use the formula: ( Number of female employees / Total number of employees ) * 100 In this case, the number of female employees is 36 and the total number of employees is 45. (36 / 45) * 100 = 80% To solve (36/45) * 10: 1. Divide 36 by 45: 36 ÷ 45 = 0.8 2. Multiply the result by 100: 0.8 * 100 = 80 Therefore, (36/45) * 100 is equal to 80. There are 80% of the employees in the office are female.

What is the probability of getting a sum of 11 upon rolling 2 fair 6-sided dice?

To find the probability of getting a sum of 11, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes. Step 1: Determine the favorable outcomes: To get a sum of 11, we can have the following combinations: - Rolling a 6 on the first die and a 5 on the second die. - Rolling a 5 on the first die and a 6 on the second die. So, we have 2 favorable outcomes. Step 2: Determine the total number of possible outcomes: Since each die has 6 sides, the total number of possible outcomes is 6 * 6 = 36. Step 3: Calculate the probability: The probability is given by the number of favorable outcomes divided by the total number of possible outcomes. So, the probability of getting a sum of 11 is 2/36, which simplifies to 1/18. Therefore, the answer is 1/18.

Peter opened his Math textbook, and found out that the product of the pages of the text book he opened to was 3906. What were the page numbers?

We know that the product of the page numbers is 3906. Let's start by finding the square root of 3906, which is approximately 62.5. Now, let's consider the factors of 3906. We can pair them up to find the page numbers. One pair of factors is 2 and 1953. However, the page numbers in a textbook are usually consecutive, so this pair is unlikely. Another pair of factors is 3 and 1302. Again, this pair doesn't seem to be consecutive. Let's keep going. The next pair of factors is 6 and 651. This pair is closer to being consecutive, but still not quite there. Finally, we have the pair 62 and 63. These numbers are consecutive, and their product is indeed 3906. Therefore, the page numbers that Peter opened to are 62 and 63.