Answer:
16
Step-by-step explanation:
Let us assume the square is (m-a)²
We have to find a
(m-a)² = m² -2am + a²
We are given
m² -8m
Comparing terms
2a = 8
a = 4
so the square is (m-4)²
which is m² - 8m + 16
josh was preparing a speech about home real estate and wanted to give an idea how home prices had fallen in all seven recognized neighborhoods in his city. he found that the rounded percentage decreases were 3, 3, 5, 6, 7, 7, and 19. why would using the mean, or average, decrease be misleading?
The mean of the decrease in rounded percentage of home is 7.14 which would used as misleading the about data as raw data very far away from 7.14.
As given in the question,
Given rounded percentage decrease in home price is:
3, 3, 5, 6, 7, 7, 19
Mean of the rounded percentage
= ( Sum of all the values)/ (Number of values)
= ( 3 + 3 + 5 + 6 + 7 + 7 + 19) / 7
= 50 /7
=7.1428
= 7.14
7.14 is very far away from 3 and 19.
Mean is not representing the correct information of the home price.
Using mean is misleading the correct information about percentage decrease.
Therefore, value of the mean is equal to 7.14 which mislead the decrease percentage.
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A given triangle has vertices at (-3,1) (2,4) and (5,-3) . If the triangle were rotated 180 degrees counterclockwise, the new coordinates would be ?
The new coordinates when we rotate the triangle counterclockwise are (3, -1), (-2, -4), (-5, 3).
Define transformations.A function, f, that maps to itself are known as a transformation, or f: X → X. The transformation results in the pre-image X becoming the picture X. This transformation can utilize any operation, or a combination of operations, including translation, rotation, reflection, and dilation.
Given, the triangle vertices (-3, 1) (2, 4), and (5, -3).
When the triangle turned 180 degrees counterclockwise, we must determine the new coordinates;
When we rotate 180 degree counterclockwise, the plane changes not the point.
That is,
(-3, 1) becomes (3, -1)
(2, 4) becomes (-2, -4)
(5, -3) becomes (-5, 3)
Therefore, the new coordinates when we rotate the triangle counterclockwise are (3, -1), (-2, -4), (-5, 3).
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if you are using 5.25% (mass/mass) bleach solution and you determine that the density of the bleach solution is 1.08 g/ml, how many moles of sodium hypochlorite are present in a 4.00 ml sample?
0.00305 moles of sodium hypochlorite are present in a 4.00 ml sample.
A mole is the mass of a substance made up of the same number of fundamental components. Atoms in a 12-gram example are identical to 12C. Depending on the substance, the fundamental units may be molecules, atoms, or formula units.
A mole fraction shows how many chemical elements are present. The value of 6.023 x 1023 is equal to one mole of any substance (Avagadro number). It can be used to quantify the chemical reaction's byproducts. The symbol for the unit is mol.
The number of moles formula is denoted by the following expression.
The mass of material / Mass of one mole equals the number of moles.
moles of NaClO = mass of sample×density of bleach solution×mass of bleach solution×1 mol/ mass of NaClO
= 4.00×1.08×0.0525/74.4 g of NaClO
= 0.00305.
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in 2006, 68.1 million people subscribed to cable television. in 2013, that number had decreased to 57.1 million. find the percent decrease in the number of cable tv subscribers from 2006 to 2013.
The percent decrease in the number of cable TV subscribers from 2006 to 2013 is 17.92%.
In the given question, in 2006, 68.1 million people subscribed to cable television. in 2013, that number had decreased to 57.1 million.
We have to find the percent decrease in the number of cable TV subscribers from 2006 to 2013.
The number of subscriber in 2006 is 68.1 million.
The number of subscriber in 2013 is 57.1 million.
Then the percent decrease in the number of cable tv subscribers from 2006 to 2013 is
Percent decrease in the number of cable TV subscribers = (subscriber in 2006 - subscriber in 2006)/subscriber in 2006 *100
Percent decrease in the number of cable TV subscribers = (61.4-50.4)/61.4 *100
Percent decrease in the number of cable TV subscribers = 11/61.4 *100
Percent decrease in the number of cable TV subscribers = 0.1792*100
Percent decrease in the number of cable TV subscribers = 17.92%
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Question 1 of 10 Which statements describe the function f(x) = 2(x-4)4? A. It has 3 zeros and at most 4 relative maximums or minimums. B. It is a translation of the parent function 4 units to the left. C. It has 4 zeros and at most 3 relative maximums or minimums. D. Both ends of the graph of the function go up. E. The left end of the graph of the function goes up, and the right end goes down. F. It is a translation of the parent function 4 units to the right. SUBMiT
Answer:
C. It has 4 zeros and at most 3 relative maximums or minimums. D. Both ends of the graph of the function go up.F. It is a translation of the parent function 4 units to the right.Step-by-step explanation:
You want the statements that describe the function f(x) = 2(x -4)⁴.
DegreeThe degree of the function is the exponent: 4. This means the function has 4 zeros.
The maximum number of relative extrema is 1 less than the degree. The function has at most 3 relative maximums or minimums.
The even degree means both ends of the graph go in the same direction.
Leading coefficientThe leading coefficient of 2 is positive, so the ends of the function will go up.
Horizontal translationIn the parent function f(x) = x⁴, the value of x has been replaced by (x-4). This causes the graph to be translated 4 units to the right.
__
Additional comment
This function has a zero at (4, 0) with multiplicity 4. There is a flat spot where the graph touches, but does not cross, the x-axis. As a consequence, there is only one minimum and no relative maximums.
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the franchisee estimates that customers will not wait in line more than five minutes for a car wash. a longer time will cause robot to lose the gasoline sales as well as th car wash sale. if the estimate of customer arrivals is 10 per hour, which wash unit should be selected?
Using unit I, calculate the average waiting time of customers in the wash line (μ for unit I = 12 per unit).
For unit II at 15 per hour
if waiting time is the only criterion, unit II should be purchased. But before we make the final decision, we must look at the profit differential between both units.
With unit I, some customers would balk and renege because of the 5 minute wait. And, although this greatly complicates the mathematical analysis, we can gain some estimate of lost sales with unit I by increasing Wq = 5 minutes and solving for λ. This would be the effective arrival rate of customers.
Therefore, because the original estimate of λ was 10 per hour, an estimated 2 customers per hour will be lost.
Lost profit of 2 costumers per hour × 14 hours × 1/2 ($0.70 fill-up profit + $0.40 wash profit) = $15.40 per day.
Because the additional cost pf unit II over unit I is only $4 per day, the loss of $15.40 profit obviously warrants installing unit II.
The original five-minute maximum wait constraint is satisfied by unit II.
Therefore, unit III is not considered unless the arrival rate is expected to increase.
Given question is incomplete.
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If (x-2)(x-4)= (x - h)² + k, then what is the value of k?
Use the grid to help you fill in the blank.
0.9 × ___ = 0.27
Answer:
0.3
Step-by-step explanation:
0.9 times 0.3 equals 0.27
Can someone help me with this
Answer:
Step-by-step explanation:
What are you supposed to do?
x + 6 + 3x = 5x - 4
I need help because I don’t really know how to do this. This is for geometry
Answer: x = 10
Step-by-step explanation:
x + 6 + 3x = 5x - 4
First you want to group your like values together
6 +4 = 5x - 3x - x
Note that when any value crosses the = sign, The sign in front of it (+/-) changes. (e.g. -4 changes to +4 after it crosses the = sign)
Then work out your values
10 = x
Also note that x is the same as 1x. So you minus 3x from 5x to get 2x and 2x minus 1x is equal to 1x which is the same as x.
How many times does the digit 9 appear in the list of all integers from 1 - 500?
9 cannot be in the first postion.
2nd position but not third
5*1*9=45
3rd position but not second
5*9*1=45
2nd and third position
5*1*1=5 2 each so that is 10
45+45+10=100
9 appears in all integers from 1 to 500, 95 times.
9 total from
1-9: 1 time
10-19: 1 time
20-29: 1 time
30-39: 1 time
40-49: 1 time
50-59: 1 time
60-69: 1 time
70-79: 1 time
80-89 : 1 time
among 90 and 99
Between 1991 and 1998, one will occur.
then in 1999, 2 9.
9 from 90 to 99 is 10, thus
9+10 = 19 is the total number of 9s from 1 to 99.
In 1 to 99, there are so 19 9s.
similarly
the following:
19 9's in 100 to 199
19 9's in 200 to 299
19 9's in 300 to 399
19 9's in 400 to 499
Total 9s will therefore be
19+19+19+19+19+19 = 95 times
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a shipment of 15 televisions contains 6 regular and 9 deluxe models. the manufacturer failed to mark the model designation on the cartons. if 3 cartons are selected at random, what is the probability that exactly 2 of them are the deluxe model? (round your answer to three decimal places.)
The probability that exactly 2 of them are the deluxe model =0.474
given that
there is a shipment of 15 televisions contains 6 regular and 9 deluxe models
now 3 cartons are selected at random
C(15,3) =[tex]\frac{15!}{(15-3)!3!}[/tex]
C(15,3) = [tex]\frac{13*14*15}{3*2*1}[/tex]
C(15,3) = 455
2 out of 9 deluxe models be picked
C(9,2) = [tex]\frac{9!}{(9-2)!2!}[/tex]
C(9,2) = [tex]\frac{8*9}{2*1}[/tex]
C(9,2) = 36
Each of these can be matched with 1 of 6 regular models
36 × 6 = 216
the probability that exactly 2 of them are the deluxe model
= [tex]\frac{216}{455}[/tex]
=0.474
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a car is purchased for . each year it loses of its value. after how many years will the car be worth or less? (use the calculator provided if necessary.)
Answer: I calculated the value after 5 years to be 5,734.40 or 5,734
Step-by-step explanation:
What number for C would make the equation below a perfect square trinomial.
x^2 +22x+c
(A)121
(B)72
(C)81
(D)96
The expression x^2 + 22x + c becomes a perfect square trinomial if the value of c is 121, so the correct option is A.
What value of C makes the expression a perfect square trinomial?A perfect square trinomial is written as:
(a + b)^2 = a^2 + 2ab + b^2
Here we have the expression:
x^2 + 22x + c
We can rewrite the second term in such a way that we get:
x^2 + 2*11*x + c
comparing this with the form of the general perfect square trinomial we can see that:
a = x
b = 11
Then the value of c must be 11^2 = 121
Then we have:
x^2 + 2*11*x + 121 = (x + 11)^2
The correct option is A.
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3.9y^2+9.5y-8.2=0 solve using the quadratic formula
Solving the equation using quadratic formula the value of y in the quadratic equation is y= 0.675 or y= -3.11
Quadratic EquationMathematically, quadratic formula can be written as
[tex]y= \frac {-b + or - \sqrt{b^{2}-4ac } }{2a}[/tex]
from the expression, a= 3.9 , b=9.5, c= -8.2
substitute the value into the quadratic equation
[tex]y= \frac{-b^{} + or -\sqrt{b^{2}- 4ac } }{2a}[/tex]
y= [tex]\frac{-9.5^{} + or - \sqrt{9.5^{2}-4(3.9)(-8.2) } }{2(3.9)}[/tex]
y= [tex]\frac{-9.5^{} + or -\sqrt{90.25+127.92} }{7.8}[/tex]
y= [tex]\frac{-9.5^{} + \sqrt{218.17} }{7.8}[/tex] or y= [tex]\frac{-9.5^{} - \sqrt{218.17} }{7.8}[/tex]
y = [tex]\frac{-9.5 + {14.77} }{7.8}[/tex] or y= [tex]\frac{-9.5 - {14.77} }{7.8}[/tex]
y = [tex]\frac{5.27}{7.8\\}[/tex] or y= [tex]\frac{-24.27}{7.8}[/tex]
y= 0.67 or y= -3.11
Therefore the value of y in the quadratic equation is y= 0.67 or y= -3.11
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can someone help me really quick
(please Explain how you know this is correct)
Answer:
$12.50
Step-by-step explanation:
Look at the graph down below \/
U can also mark where the pounds is so right at 25 pounds it would be about $12~
Write an equation of a line in slope intercept form with a slope of 3/2 that passes through point (4,7).
Answer:
y
=
−
3
2
x
+
13
Explanation:
First write it using point-slope form:
y
−
7
=
−
3
2
(
x
−
4
)
Now simplify to slope-intercept:
Add 7 to each side:
y
=
−
3
2
(
x
−
4
)
+
7
Distribute
−
3
2
to each term in the parentheses:
y
=
−
3
2
x
+
6
+
7
y
=
−
3
2
x
+
13
hope I helped you:)
A baker makes a cake that requires 44 ounces of ingredients. The ratio of flour to sugar is 6:5. She didn’t like how the cake tasted so for the new cake, she changed the flour to sugar ratio to 7:6. If the amount of ounces she used increased to 91, how many more ounces of sugar did she add to the new cake? ( BE FAST PLEASE)
The amount of ounces of sugar she added to the new cake is 22 ounces
What is Proportion?
The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.
Given data ,
Let the total ounces of ingredients in the cake be = 44 ounces
The ratio of flour to sugar is = 6 : 5
So , the amount of flour in 44 ounces = 6x
The amount of sugar in 44 ounces = 5x
And , the equation is
6x + 5x = 44
On simplifying the equation , we get
11x = 44
Divide by 11 on both sides , we get
x = 4
So , the total amount of sugar in 44 ounces of ingredients = 5x
Substitute the value for x = 4 , we get
Total amount of sugar in 44 ounces of ingredients = 20 ounces
And ,
Let the total ounces of ingredients in the new cake be = 91 ounces
The ratio of flour to sugar is = 7 : 6
So , the amount of flour in 91 ounces = 7y
The amount of sugar in 91 ounces = 6y
And , the equation is
7y + 6y = 91
On simplifying the equation , we get
13y = 91
Divide by 13 on both sides , we get
y = 7
So , the total amount of sugar in 91 ounces of ingredients = 6y
Substitute the value for y = 7 , we get
Total amount of sugar in 91 ounces of ingredients = 42 ounces
Therefore , the extra ounces of sugar the baker added to the new cake is =
Total amount of sugar in 91 ounces of ingredients - Total amount of sugar in 44 ounces of ingredients
The extra ounces of sugar the baker added to the new cake is = 42 - 20
Extra ounces of sugar the baker added to the new cake is = 22 ounces
Hence ,
The amount of ounces of sugar she added to the new cake is 22 ounces
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f(x) = 2x²+ 3x - 14.
Answer:
Step-by-step explanation:
domain = all real numbers
range = {-121/8,∞)x-intercepts (2,0)
y-intercepts (0,-14)
vertex = minimum (-3/1 - 121/8)
Answer: Okay so I got,
1) x intercept = (2, 0), (-7/2, 0) and y intercept (0, -14)
2) 'x' equals to..
-Exact Form: 2, -7/2
-Decimal Form: 2, -3.5
Hope this helps!
In cell I5, insert the AVERAGEIFS function to calculate the average SAT score that meets two conditions: Scores in the range I11:I67 are greater than or equal to 3500 and Final Decisions in the range J11:J67 are Early Admissions (cell E3). Use mixed references appropriately.
The term in cell I5 will be for the AVERAGEIFS function:
AVERAGEIF( I11:I67 ,"<3500",J11:J67 ,"E3")
For the given question the termed AVERAGEIFS is the function formula in excel which returns the average of all the cells available in the sheet in a range that meets a given condition.the syntax for the term is AVERAGEIF(range, criteria, [average_range]), where the range is Required like One or more cells to get the average. Criteria os the basically the form of an expression, number, or cell reference, and at last the average range is optional it is just the set of cells to average.
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Kadita counted 30 pigs and chickens on the farm. Jack counted a total of 68 legs for the animals. How many pigs were there on the farm?.
Kadita has 4 pigs and 26 chickens in on his farm.
Given, Kadita counted 30 pigs and chickens on the farm. Jack counted a total of 68 legs for the animals.
Let the number of pigs be, x
and the number of chickens be, y
x + y = 30
4x + 2y = 68
Now, we have to two equations,
x + y = 30
2x + y = 34
On subtracting both the equations, we get
x = 4
and y = 30 - 4
y = 26
So, Kadita has 4 pigs and 26 chickens in on his farm.
Hence, Kadita has 4 pigs and 26 chickens in on his farm.
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Last year, Pinwheel Industries introduced a new toy. It cost $2500 to develop the toy and S25 to manufacture each toy. Fill in the blanks below as appropriate.
A.) Give a linear equation in the form Cmn + b that gives the total cost, C, to produce n of these toys Preview
B.) The total cost to produce n 3000 toys is S
C.) With $32500, a total of toys can be produced.
Equation is C=25n + 2500 And The total cost to produce 3000 toys is $77,500 And With $32500, a total of 1200 toys can be produced
A linear equation is what, exactly?
An algebraic equation with only a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
A) Lets have linear equation y=mx+b,
C (total cost) = y and n (number of toys) =x.
The original development cost plus the production cost of each toy would equal the overall cost of creating n toys.
Because m is multiplied by n, the number of toys, the cost to produce a toy will equal m.
Due to the fact that b is a constant and its value is unaffected by the quantity of toys, it will always equal the development cost.
Therefore, C=25n + 2500
B) by above, equation put n=1750
C= 25(3000) + 2500
C= $77,500
C) Now, again by same equation, but C instead of n.
=> 32500= 25n +2500
=> 30000=25n
=> n = 1200
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Response time is an important statistic for measuring the effectiveness of a fire department, and is measured as the difference between the time a fire house receives a call and the time the first piece of fire equipment leaves the station. The response times for fire departments in a large city are found to have an approximately Normal distribution, with a mean of 4.5 minutes and a standard deviation of 1.2 minutes. What percentage of fire station response times are between 4 and 6 minutes?
37.7%
44.3%
55.7%
62.3%
Step-by-step explanation:
55.7% of fire station response times are between 4 and 6 minutes. This can be calculated using the z-score formula, which is (x-μ)/σ, where x is the value of interest (in this case, 4 and 6 minutes), μ is the mean (4.5 minutes), and σ is the standard deviation (1.2 minutes). The z-score for 4 minutes is -0.42 and the z-score for 6 minutes is 0.42. Using a standard
5. A retail store selling soaps, lotions, and candles must ship products to each of their locations. Each shipping box must contain 40 items and weigh exactly 25 pounds (400 ounces) Each body spray weighs 8 ounces and each shower gel weighs 12 ounces. The graph shows a system of equations for this scenario where z is the number of body sprays and is the number of shower gels in one shipping package In parts a-d determine which point(s) best match each description
The solution of the system of equations is given as follows:
x = 20 -> 20 body sprays.y = 20 -> 20 shower gels.How to obtain the amounts with the system of equations?
The variables of the system of equations are given as follows:
Each shipping box must contain 40 items, hence:
x + y = 40.
The box weighs 400 ounces, hence, considering the weight of each item, we have that:
8x + 12y = 400.
From the first equation, we have that:
x = 40 - y.
Then the number of shower gels is obtained as follows:
8(40 - y) + 12y = 400
4y = 80
y = 80/4
y = 20.
The number of body sprays is of:
x = 40 - 20 = 20.
Missing InformationThe problem is incomplete, hence we build and solve the equations, which will help you obtain the other information.
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14.2: Prove Pythagoras Right 15 min
Elena is playing with the equivalent ratios she wrote in the warm-up. She rewrites
=as a² = xc. Diego notices and comments, "I got b2 = yc. The a² and remind me
of the Pythagorean Theorem." Elena says, "The Pythagorean Theorem says that
a² + b² = ².1 bet we could figure out how to show that."
a
h
1. How did Elena get from = to a² = xc?
2. What equivalent ratios of side lengths did Diego use to get b² = yc?
In the following passage we have proven the Pythagoras' theorem.
What is Pythagoras' theorem?
The hypotenuse's square is equal to the sum of the squares of the other two sides if a triangle has a straight angle (90 degrees), according to the Pythagoras theorem. Keep in mind that BC2 = AB2 + AC2 in the triangle ABC signifies this. Base AB, height AC, and hypotenuse BC are all used in this equation. The longest side of a right-angled triangle is its hypotenuse, it should be emphasized.
Here given in the question that one side of the wall 5 feet
So as it is a rectangular bedroom it can be split into 2 right-angled triangles.
So the hypotenuse of the triangle is 13 feet
Now using the Pythagoras theorem we can find out the length of the other wall
b²=h²-p²
AB² + BC² = AD × AC + CD × AC
AB² + BC² = AC (AD + CD)
Since, AD + CD = AC
Therefore, AC² = AB² + BC²
So we have p² + b² = h².
Hence, we prove the Pythagoras theorem using two triangle.
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A machine is used to fill several bags each of 20-ounces of spicey
chips. After each bag is filled, another machine weighs them. If the
bag weighs .25 less or more than the desired weight, the bag is
rejected. Write an equation to represent the maximum weight for a
bag of chips that the machine will approve.
Answer: 19.75oz < x > 20.25oz
Step-by-step explanation: let x represent the weight of the bag of chips.
This equation states that the. weight must not exceed 20.25 but it must not be below 19.75oz.
Answer: 19.75oz < x > 20.25oz
Step-by-step explanation:
each of two pesticide brands is applied to 100 containers of fire ants. brand1 killed all ants in 65 of 100 containers within two hours, and brand 2 killed all ants in 58 of 100 containers within two hours.
The pValue is 0.15625 > 0.05 when brand1 killed all ants in 65 of 100 containers within two hours, and brand 2 killed all ants in 58 of 100 containers within two hours in hypothesis.
Let A represent the new pesticide and B be the Ant Killer. Then, you are given that
Brand1 kills ants in 65 of 100 containers:
Xa = 65/100 = 0.65
Brand2 kills ants in 58 of 100 containers:
Xb = 58/100 = 0.58
You know need to find sample variance for both,
s²A =∑(Xi - Xa)
= 65(1-0.65)² + 35(0 - 0.65)²
= 22.7
s²B =∑(Xi - Xb)²
= 59(1-0.59)² + 41(0 - 0.59)²
= 24.19
You can know do a test for significance
t = (Xa - Xb) / √(sA /nA) +(sB/nB)
and derive the conclusion from there
Here, Null and alternative hypothesis are generally,
Alternative hypothesis
Here, there are three samples are assumed. They are:
i) Samples are taken independently
ii) Samples must be selected randomly
iii) n₁p₁ ≥ 5 , n₁q₁ ≥ 5, n₂p₂ ≥ 5, n₂q₂ ≥ 5
i.e., p₁ = 0.65 ; p₂ = 0.58 , n₁= 100
q₁ = 0.35 ; q₂ = 0.42 , n₂=100
Corresponding p-value at Z = 1.01722
p-value = 0.15625 > 0.05
Hence, the pValue is 0.15625 > 0.05 when brand1 killed all ants in 65 of 100 containers within two hours, and brand 2 killed all ants in 58 of 100 containers within two hours in hypothesis.
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if the original recipe calls for 50 portions, 6oz each and the new recipe calls for 100 portions at 5oz each, what is the recipe conversion factor?
By using multiplication and division of integers, it can be calculated that
Conversion factor of the recipe = [tex]\frac{500}{300}[/tex] = 1.6
What is multiplication and division of integers?
At first it is important to know about integers
Integers are those numbers which has no fractional part. Integer can be positive, negative and zero is also an integer.
Repeated addition is called multiplication. Multiplication is used to find the product of two or more integers.
What is division?
Division is the process by which value of single unit can be calculated from the value of multiple unit.
The number to be divided is known as dividend, the number by which the dividend is divided is the divisor, the result obtained is the quotient and the remaining part is the remainder.
There is a well known formula for division
Divisor x Quotient + Remainder = Dividend.
This is a word problem where multiplication and division of integers is required
The original recipe calls for 50 portions, 6oz each
Required yield of the original recipe = 50 [tex]\times[/tex] 6 = 300 oz
The new recipe calls for 100 portions at 5oz each
Required yield of the new recipe = 100 [tex]\times[/tex] 5 = 500 oz
Conversion factor of the recipe = [tex]\frac{500}{300}[/tex] = 1.6
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An equation in slop intercept form the y intercept -5 the slope -1/2
Answer:
y=-1/2x-5
Step-by-step explanation:
ab³ (5a²-4ab+b²) multiply
Answer: The correct answer is 5a^3b^3 − 4a^2b^4 + ab^5
Step-by-step explanation:
ab^3 (5a^2 − 4ab + b^2)
Multiply together:
(ab^3)*(5a^2 + −4ab + b^2)
(ab^3)*(5a^2) + (ab^3)*(−4ab) + (ab^3)*(b^2)
= 5a^3b^3 − 4a^2b^4 + ab^5
Answer:
[tex]5a^3b^3-4a^2b^4+ab^5[/tex]
Step-by-step explanation:
Given expression:
[tex]ab^3(5a^2-4ab+b^2)[/tex]
Apply the distributive property a(b - c + d) = ab - ac + ad :
[tex]\implies ab^3 \cdot5 a^2-ab^3 \cdot 4ab+ab^3 \cdot b^2[/tex]
Collect like terms:
[tex]\implies 5aa^2b^3-4aabb^3+ab^2b^3[/tex]
Apply the exponent rule a = a¹ :
[tex]\implies 5a^1a^2b^3-4a^1a^1b^1b^3+a^1b^2b^3[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]
[tex]\implies 5a^{(1+2)}b^3-4a^{(1+1)}b^{(1+3)}+a^1b^{(2+3)}[/tex]
Simplify:
[tex]\implies 5a^3b^3-4a^2b^4+ab^5[/tex]