Answer:
(x + 4) x 2
Step-by-step explanation:
x represents number of cookies Alice ate.
x + 4 represents the number of cookies Tim ate.
As Bob had eaten twice the number of cookies that Alice ate the expression should be
(x + 4) x 2
Twice The sum of a number and three times the second number is four. The difference of 10 times the second number And five times the first is 90 find the numbers
Answer: The first number is [tex]-10[/tex] and the second number is [tex]4[/tex]
Step-by-step explanation:
Let be "x" the first number and "y" the second number.
The word "Twice" indicates multiplication.
By definition, the sum is the result of an addition.
"is" indicates an equal sign.
Therefore, "Twice the sum of a number and three times the second number is four" can be expressed as:
[tex]2(x+3y)=4[/tex] [Equation 1]
A difference is the ressult of a subtraction, then " The difference of 10 times the second number and five times the first is 90" can be expressed as:
[tex]10y-5x=90[/tex] [Equation 2]
To find the numbers:
1. Solve for "x" from the Equation 1:
[tex]2(x+3y)=4\\\\2x+6y=4\\\\x=\frac{4-6y}{2}\\\\x=2-3y[/tex]
2. Substitute this equation into the Equation 2 and solve for "y":
[tex]10y-5(2-3y)=90\\\\10y-10+15y=90\\\\25y=100\\\\y=\frac{100}{25}\\\\y=4[/tex]
3. Substitute the value of "y" into the equation [tex]x=2-3y[/tex] and evaluate:
[tex]x=2-3(4)\\\\x=-10[/tex]
Find the difference.
(5x2 + 2x + 11) - (7 + 4x - 2x2)
A.
9 - 2x - 2x2
B.
3x2 - 2x + 4
C.
3x2 + 6x + 4
D.
7x2 - 2x + 4
[tex]\text{Solve:}\\\\(5x^2 + 2x + 11) - (7 + 4x - 2x^2)\\\\5x^2+2x+11-7-4x+2x^2\\\\7x^2+2x+11-7-4x\\\\7x^2-2x+11-7\\\\\boxed{7x^2-2x+4}[/tex]
Answer:
D. 7x2 - 2x + 4
Step-by-step explanation:
5x2 + 2x + 11 - 7 - 4x +2x2
7x2 + 2x + 11 - 7 - 4x
7x2 - 2x + 11 - 7
7x2 - 2x + 4
-6y-3=3-6y(if there is no solution,type in ''no solution'')y= Answer
Answer:
No solution
Step-by-step explanation:
-6y - 3 = 3 - 6y , add 6y on both sides to cancel them out
-3 = 3 , you're left with -3=3, which is impossible because they're
different numbers, therefore not solution.
-3 ≠ 3
In triangle $ABC$, $BC = 20 \sqrt{3}$ and $\angle C = 30^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.
Answer:
DE=10
Step-by-step explanation:
Given that in triangle ABC, $BC = 20 \sqrt{3}$ and $\angle C = 30^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$,
To find length of DE
Please refer to the attachment for solution
Since perpendicular bisector we have DC = 1/2 BC = 10 sqrt 3
Using right triangle CDE, we get DE = 10
Lynn rents a luxury car at the Edmonton Internacional Airport. She wants a nice
vehicle for her 3 day trip. She figures she will put on about 400 km during the three
days.
a. What would be the cost of the Standard Daily Rate plus Mileage plan?
b. What would be the cost of the Unlimited Mileage plan?
c. Which is the better plan?
Answer:
Step-by-step explanation:
Looking at the table,
a) the standard daily rate for a luxury car is $70. Since Lynn wants to rent the luxury car for 3 days, it means that the standard rate for 3 days would be
3 × 70 = $210
She figures she will put on about 400 km during the three
days. The cost of renting the luxury car per kilometer(mileage) is 30 cents. Therefore, the cost for 400 kilometers would be
400 × 30 = 12000 cents.
Converting 12000 cents to dollars, it becomes
12000/100 = $120
Total cost of renting the luxury car would be
120 + 210 = $330
b) the daily cost of the unlimited mileage daily for the luxury car is $105
Total cost of renting the luxury car for 3 days would be
105 × 3 = $315
c) the standard daily rate is better because it is cheaper
Kyle sold 28 boxes of fruit for a fundraiser.Omar sold 2 times as many boxes of fruit ad Kyle sold.What is the total number of boxes that Kyle and Omar sold?
Answer: 84
Step-by-step explanation:
K=28
O=2K=2*28=56
O+K=56+28=84
The total number of boxes that Kyle and Omar sold is 84.
What is the arithmetic operator?Arithmetic operators are four basic mathematical operations in which summation, subtraction, division, and multiplication involve.,
Division = divide any two numbers or variables called division.
Multiplication = to multiply any two or more numbers or variables called multiplication.
As per the given,
Kyle sold 28 boxes of fruit.
Omar sold 2 times,
28 x 2 = 56 boxes
Total boxes = 56 + 28 = 84 boxes
Hence "The total number of boxes that Kyle and Omar sold is 84".
To learn more about the arithmetic operators,
brainly.com/question/25834626
#SPJ5
The population of locusts in a certain swarm doubles every two hours. If 4 hours ago there were 1,000 locusts in the swarm, in approximately how many hours will the swarm population exceed 250,000 locusts?A. 6B. 8C. 10D. 12E. 14
4 hours ago = 1,000
2 hours ago = 2,000
Now = 4,000
In 2 hours = 8,000
In 4 hours = 16,000
In 6 hours = 32,000
In 8 hours = 64,000
In 10 hours = 128,000
In 12 hours = 256,000
The answer would be D. 12 hours.
Triangles Q R S and X Y Z are shown. Angles Q S R and X Z Y are right angles. Angles Q R S and X Y Z are congruent. The length of Y Z is 9, the length of X Z is 12, and the length of hypotenuse X Y is 15. Given △QRS ~ △XYZ, what is the value of tan(Q)? Three-fifths Three-fourths Four-fifths Four-thirds
Answer:
Three-fourths
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
∠QSR≅∠XZY ---> given problem
∠QRS≅∠XYZ ---> given problem
so
△QRS ~ △XYZ ----> by AA Similarity theorem
Remember that, if two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent
That means
[tex]\frac{QS}{XZ}=\frac{QR}{XY}=\frac{RS}{YZ}[/tex]
∠Q≅∠X
∠R≅∠Y
∠S≅∠Z
In the right triangle XYZ
Find the tangent of angle X
[tex]tan(X)=\frac{YZ}{XZ}[/tex] ---> opposite side angle X divided by adjacent side angle X
substitute the given values
[tex]tan(X)=\frac{9}{12}[/tex]
Simplify
[tex]tan(X)=\frac{3}{4}[/tex]
Remember that
∠Q≅∠X
so
[tex]tan(Q)=tan(X)[/tex]
therefore
[tex]tan(Q)=\frac{3}{4}[/tex] ---->Three-fourths
The manufacturerâs suggested retail price (MSRP) of a certain item is $60. Store A sells the item for 20 percent more than the MSRP. The regular price of the item at Store B is 30 percent more than the MSRP, but the item is currently on sale for 10 percent less than the regular price. If sales tax is 5 percent of the purchase price at both stores, what is the result when the total cost of the item at Store B is subtracted from the total cost of the item at Store A?A. $0B. $0.63C. $1.80D. $1.89E. $2.10
Answer:
C. $1.80
Step-by-step explanation:
Given:
MSRP of a certain item = $60
Percentage of Selling by store A = 20%
Percentage amount will be = [tex]\frac{20}{100}\times 60 = \$12[/tex]
Sales tax = 5% on purchase price
Amount of sales tax = [tex]\frac{5}{100}\times 60 = \$3[/tex]
Total Cost of item at Store A is MSRP of a certain item plus Percentage amount of selling the item plus Amount of Sales tax.
framing in equation form we get;
Total Cost of item at Store A = [tex]60+12+3 =\$75[/tex]
Also Given:
Percentage of Selling by store B = 30%
Percentage amount will be = [tex]\frac{30}{100}\times 60 = \$18[/tex]
Regular price of item at store B = [tex]60+18 = \$78[/tex]
Percentage of discount of item at store B = 10% on regular price
Amount of discounted item at store B = [tex]\frac{10}{100}\times78 = \$7.8[/tex]
Now regular price of item with discount at store B will be equal to Regular price of item at store B minus Amount of discounted item.
regular price of item with discount at store B = [tex]78-7.8 = \$70.2[/tex]
Sales tax = 5% on purchase price
Amount of sales tax = [tex]\frac{5}{100}\times 60 = \$3[/tex]
Total Cost of item at Store B is regular price of item with discount at store B plus Amount of Sales tax.
framing in equation form we get;
Total Cost of item at Store B = [tex]70.2+3 =\$73.2[/tex]
The result when total cost of the item at Store B is subtracted from the total cost of the item at Store A = $73.2 - $75 = $1.80.
Hence the result is $1.80.
can someone help me right now!!!
Which of the following is a solution to this inequality?
y less than two thirds times x plus 2
(0, 3)
(−3, 1)
(3, 5)
(1, 2)
Final answer:
To determine which of the given points satisfy the inequality y < (2/3)x + 2, substitute the x and y values of each point into the inequality and check if it is true. The only solution to the inequality is (0, 3).
Explanation:
To determine which of the given points satisfy the inequality y < (2/3)x + 2, we can substitute the x and y values of each point into the inequality and check if the inequality is true.
Taking the first point (0, 3), we substitute x=0 and y=3 into the inequality: 3 < (2/3)(0) + 2. Simplifying, we have 3 < 2, which is true. Therefore, (0, 3) is a solution to the inequality.
Next, let's check the other points: (-3, 1), (3, 5), and (1, 2).
By substituting the x and y values of each point into the inequality, we find that (-3, 1), (3, 5), and (1, 2) are not solutions to the inequality. Therefore, the only solution to the inequality is (0, 3).
A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
Answer:
(4 √ 3 + 4 √2 ) m
Step-by-step explanation:
The insect can travel from one corner directly to opposite corner in four different ways
each can be calculated using Pythagoras theorem
firstly for one face we need to calculate the diagonal
H² = 1² + 1² = 2
H = √2
then we calculate the diagonal opposite a corner
for example
A to H where A is at the bottom and H opposite A in another plane at the top in the cubical box
(Interior diagonal for A to H)² = √2² + 1² = 3
Interior diagonal from A to H = √ 3
there are four such corners, the fly will travel 4 √ 3 and it could also go 4 √2 diagonally to the the other corners
maximum possible length in meters = (4 √ 3 + 4 √2 ) m
In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt
Answer:
Total number of ways will be 20
Step-by-step explanation:
We have given three identical green shirts and three identical red shirts
So total number of shirts = 3+3 = 5
We have to distribute these shirts to 6 children so that each children got one shirt
Number of ways will be equal to [tex]=\frac{6!}{3!3!}=20[/tex] ( Here we divide by 3!3! because three green shirts and 3 red shirts are identical )
Statistical hypothesis testing allows you to: a.Estimate the standard deviation of your sample distribution b.Use sample statistics to make decisions about population parameters c.Eliminate all potential errors in your research d.Use non-probability samples to make inferences about a population
Answer:
Option B) Use sample statistics to make decisions about population parameters
Step-by-step explanation:
Statistical hypothesis testing
Hypothesis testing is a statistical method that is used in making statistical decisions using a sample that is taken from the population. Statistical hypothesis testing is basically an assumption that we make about the population parameter. with the help of sample statistic.Calculations are performed on samples to gather more information about the population.This helps us to know that the particular sample belongs or does not belong to the population and estimate the parameters of the population with the help of this claim.Probability sampling involves random selection from the population and non-probability sampling relies on the selection based on judgement of the researcher.Thus, Option B) Use sample statistics to make decisions about population parameters is the correct answer.
A standard deck of cards has 52 cards, 4 of each type (Ace, King, Queen, Jack, 10,...,2). From a well-shuffled deck, you are dealt a hand of 5 cards (without replacement).
(a) What is the probability that you are dealt at least one face card (that is a king, queen or jack)?
(b) What is the probability that you are dealt with both; at least one ace and at least one face card?
Answer:
a)%85,5
b)%92,2
Step-by-step explanation:
a) To determine the probability of getting a hand with at least one face, we need to calculate the probability of the hand without any face firstly.
[tex](36/52)*(35/51)*(34/50)*(33/49)*(32/48)=0,145[/tex]
Then, we need to deduct this value from the probability 1
[tex]1-0,145=0,855[/tex]
The probability of the hand with at least one face is %85,5.
b)To determine the probability of a hand with at least one ace and face we will track the same road again.
[tex](32/52)*(31/51)*(30/50)*(29/49)*(28/48)=0,0775[/tex]
Then, we need to deduct this value from the probability 1
[tex]1-0,0775=0,922[/tex]
The probability of the hand with at least one ace and one face is %92,2.
Rebecca moves to a large city that had two event stadiums. The larger stadium can hold 4.5 times 10^4 people, which is about 15 times as many people as the smaller stadium can hold. Which expression could represent the approximate amount of people the smaller stadium can hold?
Answer:
3,000 people
Step-by-step explanation:
Total number of people larger stadium can hold equals 4.5 times [tex]10^{4}[/tex] people,
which equals 45,000 people.
Now,
it is given that larger stadium can hold 15 times more people than small stadium .
So, smaller stadium will hold 15 times less people than the larger stadium ,
Which equals , [tex]\frac{45000}{15}[/tex] = 3,000 people.
Thus ,
Smaller stadium can hold a total of 3,000 people.
Determine whether the value is a discrete random variable, continuous random variable, or not a random variable. a. The number of fish caught during a fishing tournament b. The number of textbook authors now sitting at a computer c. The political party affiliation of adults in the United States d. The square footage of a house e. The number of free dash throw attempts before the first shot is made f. The weight of a Upper T dash bone steak
Answer:
a. Discrete
b. Discrete
c. Not a random variable
d. Continuous
e. Discrete
f. Continuous
Step-by-step explanation:
The discrete random variable is countable while continuous random variable is measurable.
a. The number of fish caught is a discrete random variable because these are countable.
b. The number of text book authors are also countable so it is a discrete random variable.
c. The political party affiliation of adults is not a random variable because the political party affiliation depends on person's interest and it cannot be randomly assigned to person. Also random variable is the numerical outcome of random experiment whereas political affiliation is the categorical variable that results in non numerical responses such as Democrat, Republicans etc.
d. The square footage of a house is measurable and so it is a continuous random variable.
e. The number of free dash throw attempts are countable so it is a discrete random variable
f. The weight of Upper T dash bone steak is measurable and so it is a continuous random variable.
The number of fish caught during a fishing tournament is a discrete random variable, while the square footage of a house is a continuous random variable. The other examples are not random variables.
Explanation:a. The number of fish caught during a fishing tournament is a discrete random variable. The values of the variable are obtained by counting the number of fish caught.
b. The number of textbook authors now sitting at a computer is not a random variable. It is a fixed number and does not vary.
c. The political party affiliation of adults in the United States is not a random variable. It can be determined by surveying adults or analyzing existing data.
d. The square footage of a house is a continuous random variable. The values of the variable are obtained by measuring the size of the house.
e. The number of free-throw attempts before the first shot is made is a discrete random variable. The values of the variable are obtained by counting the number of attempts.
f. The weight of an Upper T-bone steak is a continuous random variable. The values of the variable are obtained by measuring the weight of the steak.
A student with two summer jobs earns $10 per hour at a café and $8 per hour at a market. The student would like to earn at least $800 per month.
A. Write and graph an inequality to represent the situation. Include clear labels on the graph.
B. The student works at the market for 60 hours per month and can work at most 90 hours per month. Can the student earn at least $800 each month? Explain how you can use your graph to determine this.
Help
Answer:
A.) View Image
B.) Not possible. If you look at the graph,the student must work at the cafe at least 80 hours and at the the market for at least 100 hours to earn the minimum $800 they wanted.
60 hours is below the minimum required time of both place. 90 hours can only satisfy the minimum work hour of one of the place, they need to satisfy the minimum of both places.
In other word, they must work at least 180 hours to earn the $800+ they wanted
Step-by-step explanation:
Set up your equation.
let c be hour worked at cafe and m be hours worked at market.
solve for any of the variable. I solved for c because it looked easier. solving for m will give you the same graph as well.
Graph your equation like usual. Since it's a ≥ sign then you must shade above the line. The shaded part represents the hours that the student can work at both place to earn at least $800
For one month, Gilda's cell phone bill costs $40 plus an additional 8 cents per text message. Is she can only afford a total bill of $50 for a month, find the maximum number of text messages she can send in one month.
Answer:
Gilda can send 125 messages in one month.
Step-by-step explanation:
Given,
Total bill of 1 month = $50
Fixed charge of 1 month = $40
Charge of 1 message = 8 cents
∵100 cents = $1
∴8 cents = $0.08
Therefore Charge of 1 message = $0.08
We have to find out the number of messages that Gilda can send in 1 month.
Solution,
Let the total number of messages that Gilda can send in 1 month be 'x'.
Total bill of the month is the sum of fixed charge and charge of 1 message multiplied with total number of messages.
So, framing the above sentence in equation form, we get;
Total bill of 1 month =Fixed charge of 1 month+Charge of 1 message× total number of messages
On substituting the given values, we get;
[tex]40+0.08x=50\\\\0.08x=50-40\\\\0.08x=10\\\\x=\frac{10}{0.08}=125[/tex]
Hence Gilda can send 125 messages in one month.
The values of m and n are whole numbers greater than 1. Which is true about the quotient m/n÷1/n? The expression will always equal n. The expression will always equal m. The expression will equal n only when m > n. The expression will equal m only when m > n.
Answer:
The value of the quotient is always equal to m
Step-by-step explanation:
The values of m and n are whole numbers greater than 1.
We are given a quotient that is [tex]\frac{\frac{m}{n} }{\frac{1}{n}}[/tex].
In this we have fractions in both numerator and denominator.
The given quotient will be the same if we multiply the numerator with the inverse of the denominator.
So we multiply the numerator [tex]\frac{m}{n}[/tex] with the inverse n.
So the quotient will become = [tex]\frac{m}{n} \times n[/tex] = m
Hence the value of the quotient is always equal to m.
Yuan and his friends went to the movies. They ate 8 buckets of popcorn, 4 ice cream bars, and 6 boxes of chocolate. How many snacks did they eat in all?
Answer:
18 snacks
Step-by-step explanation:
Here we are considering that popcorn,icecreams,chocolate are snacks and each quantity represents each unit of snack.
Given that they ate 8 buckets of popcorn, 4 ice cream bars, and 6 boxes of chocolate
So the total number of snacks they ate are 8 + 4 + 6 = 18
QUESTION 4 of 5: You have insurance premiums of $250 due quarterly. How much will you pay annually?
Answer:
$1000
Step-by-step explanation:
quarterly is 1/4
12/4 is 3
so every 3 months you pay $250
annually is the whole year so 250*4
Answer:
1000
Step-by-step explanation:
Yesterday Nadia consumed 250 grams of carbohydrate, 75 grams of protein, and 60 grams of fat. What percentage of Calories of her day's intake came from fat?
Final answer:
To calculate the percentage of calories from fat, multiply the grams of fat by the calories per gram of fat and divide by the total calories consumed.
Explanation:
To calculate the percentage of calories from fat in Nadia's daily intake, we need to know the total number of calories she consumed. Let's assume it was 2000 calories. First, we calculate the number of calories from fat by multiplying the grams of fat consumed (60g) by the number of calories per gram of fat (9 calories/g). This gives us 540 calories from fat. Then, we calculate the percentage by dividing the calories from fat (540 calories) by the total calories consumed (2000 calories) and multiplying by 100. So, the percentage of calories from fat in Nadia's day's intake is 27%.
What is the Median for the following set of numbers? ​21 23 76 47 55 135 45 30 17
Answer:
Median = 45
Step-by-step explanation:
We are given the following data set:
21, 23, 76, 47, 55, 135, 45, 30, 17
Median is the number that divides the data into two equal parts.
Formula:
[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]
Sorted data:
17, 21, 23, 30, 45, 47, 55, 76, 135
Sample size = 9, which is odd
Median =
[tex]\dfrac{9+1}{2}^{th}\text{ term} = \dfrac{10}{2}^{th}\text{ term} = 5^{th}\text{ term}\\\\= 45[/tex]
The median of given set of numbers is 45.
A 85cm snowman melts and loses 3cm in height for every hour the sun shine and the sun shine only 6.5 hours a day. How many days will it take for the snowman to melt
Answer:
Step-by-step explanation:
The initial height of the snowman is 85 cm. snowman melts and loses 3cm in height for every hour the sun shine and the sun shine only 6.5 hours a day. This means that the height that the snowman loses in a day would be
3 × 6.5 = 19.5 cm
Therefore, in a day, the snowman loses 19.5 cm in height. Therefore,
the number of days that it will take the snowman to melt would be
85/19.5 = 4.36 days
On a final exam, 75 percent of a class had scores that were greater than 70, and 60 percent of the class had scores that were less than 85. What percent of the class had scores that were greater than 70 but less than 85 ?
Answer:
The answer is 35.
35% of the class had scores that were greater than 70 but less than 85.
Step-by-step explanation:
Let's assume the number of students in the class =100
If 75% of the class scored greater than 70, then, it means 25% (100-75) of the scores were less than or equal to 70
In other words, 25 scores were less than or equal to 70.
If 60% of the class had scores that were less than 85, then, it means 40%(100-60) of the scores were greater than or equal to 85
Therefore, of the 100 scores, 25 scores were LESS THAN OR EQUAL to 70, and 40 scores were GREATER THAN OR EQUAL to 85.
We've now accounted for 65 (25+40) scores, which means the remaining 35 (100-65) scores must be between 70 and 85.
Answer:
The answer is 35.
35% of the class had scores that were greater than 70 but less than 85.
Step-by-step explanation:
Let's assume the number of students in the class =100
If 75% of the class scored greater than 70, then, it means 25% (100-75) of the scores were less than or equal to 70
In other words, 25 scores were less than or equal to 70.
If 60% of the class had scores that were less than 85, then, it means 40%(100-60) of the scores were greater than or equal to 85
Therefore, of the 100 scores, 25 scores were LESS THAN OR EQUAL to 70, and 40 scores were GREATER THAN OR EQUAL to 85.
We've now accounted for 65 (25+40) scores, which means the remaining 35 (100-65) scores must be between 70
A number consists of two digits whose sum is 12. If 16 is added to the number, then one of the digits is four times the value of the other. What is the number?
Answer: The number is 12.
Step-by-step explanation:
Since we have given that
Sum of two digits = 12
Let the one's digit be 'x'
Let the ten's digit be '12-x'
So, Original number would be
[tex]x+10(12-x)\\\\=x+120-10x\\\\=-9x+120[/tex]
If 16 is added to the number, then one of the digit is four times the value of the other.
According to question, we get that
[tex]-9x+120+16\\\\=-9x+136[/tex]
so, the one's digit be 'x'.
Let the ten's digit be '4x'.
so, New number would be
[tex]10x+4x=14x[/tex]
According to question, it becomes,
[tex]14x=-9x+136\\\\14x+9x=136\\\\23x=136\\\\x=\dfrac{136}{23}=12[/tex]
So, the number would be
[tex]-9x+120=-9\times 12+120=-108+120=12[/tex]
Hence, the number is 12.
Answer:
66
Step-by-step explanation:
Brute force:
2 digit numbers that has sum of 12:
39, 48, 57, 66
Add 16 to the numbers:
55, 64, 73, 82
Find the number that shows a digit is 4 times the value of other:
82 (2x4 =8)
When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by 9?
Final answer:
When a positive integer x, which leaves a remainder of 5 when divided by 9, is multiplied by 3, the remainder when 3x is divided by 9 is 6.
Explanation:
When the positive integer x is divided by 9, the remainder is 5. To determine the remainder when 3x is divided by 9, we should remember that when two positive numbers multiply, the result has a positive sign. So multiplying x by 3, we get 3x. The initial information tells us that x = 9n + 5 for some integer n. Thus, 3x equals 3(9n + 5), which simplifies to 27n + 15. This expression can be further broken down as 27n + 9 + 6, which is the same as 9(3n + 1) + 6. Therefore, when 3x is divided by 9, it is the same as dividing this expression by 9, which leaves us with a remainder of 6. Hence, the remainder when 3x is divided by 9 is 6.
In how many ways can 8 people be seated in a row if there are no restrictions on the seating arrangement?
persons A and B must sit next to each other?
there are 4 men and 4 women and no 2 men or 2 women can sit next to each other?
there are 5 men and they must sit next to one another?
there are 4 married couples and each couple must sit together?
Answer:
a ) P₈ = 40320
b) P = 576
c) P = 1440
d) P = 24
Step-by-step explanation:
a ) 8 people sitting in a row without restrictions
simple Total ways = P₈ = 8!
P₈ = 8*7*6*5*4*3*2*1
P₈ = 40320
b) There are 4 men and 4 women and no two men or 2 women can sit next to each other
Let letters e women and numbers be men we have something like this
1 a 2 b 3 c 4 d
So we have the permutations of the four digits (men)
P₄ = 4!
P₄ = 4*3*2*1 = 24
And the permutations of the 4 women too equal to 24
Then total ways of sitting in a row in this case is
P = 24*24 = 576
c) There are 5 men and they have to sit next to each other
In this case we have 2 groups:
Group 1 5 men Group 2 3 women
We have two groups and the ways are
group men first group of women second 1 way
group of women first group of men second 2 way
Permutations within men group
P₅ = 5! = 5*4*3*2*1
P₅ = 120
Permutations within the women group
P₃ = 3! = 3*2*1
P₃ = 6
Total ways case c) T = 6*120*2 =
P(c) = 1440
d) There are four couples and each couple must be together
P₄ = 4! = 4*3*2*1
P₄ = 24
Write down a 3-digit number where digits are in decreasing order. Reverse the digits and subtract it from the first number.
Step-by-step explanation:
987 → 789
987 - 789 = 198
876 → 678
876 - 678 = 198
765 → 567
765 - 567 = 198
654 → 456
654 - 456 = 198
543 → 345
543 - 345 = 198
432 → 234
432 - 234 = 198
321 → 123
321 - 123 = 198
Generally it is always like that:
3 ≤ x ≤ 9
100x + 10(x-1) + (x - 2) → 100(x - 2) + 10(x - 1) + x
(100x + 10(x - 1) + (x - 2)) - (100(x-2) + 10(x-1) + x)
= 100x + 10(x - 1) + (x - 2) - 100(x - 2) - 10(x - 1) - x cancel 10(x - 1)
= 100x + x - 2 - 100x + 200 - x cancel 100x and x
= 100 - 2
= 198
Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, \small \frac{2}{3} dislike lima beans; and of those who dislike lima beans, \small \frac{3}{5} also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans?
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How I got that answer:
Create a two-way table showing the breakdown of students who like or dislike either vegetable. This is shown in the attached image below. Anywhere there is a letter indicates we aren't able to determine that value. Luckily we dont need to know those values to answer the question.
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Let's say we had 3000 students. I picked some large number that is a multiple of 3. That way when we multiply by 2/3, we get a whole number.
Take 2/3 of 3000 and you should get 2000. So there are 2000 students who dislike lima beans. Write "2000" at the end of the "dislike lima beans" row which is where the total is. Above that we'll have 1000 people who like lima beans, but we dont need to use this value.
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Of those 2000 students, 3/5 of them also dont like sprouts either. So (3/5)*2000 = 0.6*2000 = 1200 students do not like both veggies. Furthermore, 2000 - 1200 = 800 like sprouts but dont like lima beans.
Alternatively, 2/5 of the 2000 students like sprouts but dont like lima beans, so (2/5)*2000 = 0.4*2000 = 800.