The smaller eigenvalue is -4 with multiplicity 2 and larger eigenvalue is 6 has multiplicity 1
If A is a square matrix of order n x n and v is a non-zero column vector of order n x 1 such that Av = λv , then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector v.
According to the question,
We have a 3 x 3 matrix that has only two real distinct eigenvalues
Let two distinct eigen values be a and b
Its given that tr(A) = - 2
Det(A) = 96
If we add all the eigen values of a matrix it gives us trace of this and product of eigen values result to the determinant
Therefore , 2a + b = -2 --------(1)
a²b = 96 --------(2)
Using substitution method for solving these equations
=> b = -2 - 2a
Substituting the value of b in equation (2)
=> a²( -2 -2a) = 96
=> -2a² - 2a³ = 96
=> a²(1 + a) + 48 = 0
=> a³ + a² + 48 =0
Solving the cubic equation
a = -4 , 1.5-3i , 1.5+3i (but eigen values are only real)
therefore , a = -4
b = -2 -2(-4)
=> -2+8
=> 6
hence, the eigen values = -4 and 6
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Pls answer its for my final
The equivalent expression is (x + 2√3i) (x - 2√3i).
What is a Polynomial Equation?
The equations formed with variables, exponents and coefficients are called as polynomial equations. It can have different exponents, where the higher one is called the degree of the equation.
Given polynomial is,
x² + 12
We need to find it's equivalent expression for that see the options
Solve the option (a)
(x + 2√3i) (x - 2√3i)
Now, solve this if we get equal to x² + 12 that means that option is answer
x(x - 2√3i) + 2√3i(x - 2√3i)
x² - 2xi√3 + 2xi√3 - 2√3i * 2√3i
x² - 12 i²
we know, i² = -1
so, we get
x² + 12
We get the answer by 1st option. If you solve other options you'll get different answer.
Hence, the equivalent expression is (x + 2√3i) (x - 2√3i).
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express 2/(1-(x^2)) into partial fractions
2/ ( 1 + x) ( 1 - x) is express into partial fractions.
With an example, what is a partial fraction?
The division of an algebraic fraction into smaller, "partial fractions," components is possible. Think about the algebraic fraction (3x+5) (2x2-5x-3). This formula can be broken down into its simplest form, [2/(x - 3)] - [1/(2x + 1)]. Partial fractions are the more straightforward pieces [2/(x - 3)] and [1/(2x + 1)].
The four categories of partial fractions are linear factors, degree 2 irreducible factors, repeated linear factors, and repeated irreducible degree 2 factors.
= 2/(1-(x^2))
= 2 / (1)² - (X)²
= 2/ ( 1 + x) ( 1 - x)
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6 Find the value of the unknowns in each of the following.
(a) 2(w 2 5 z)+3(2 x -2 -6)=4(4 7 y 3)
The required value of the unknowns in the given sum of the matrix are w = -5, z = 15, and y = 1.
Given,
To determine the value of the unknowns in the matrix,
A matrix is a collection of numbers that are arranged in rows and columns to form a rectangular array. The numbers are referred to as matrix elements or entries. Matrices are widely used in engineering, physics, economics, and statistics, as well as in many disciplines of mathematics.
Here,
[tex]2\left[\begin{array}{ccc}w&2\\5&z\end{array}\right] +3\left[\begin{array}{ccc}2&x\\-2&-6\end{array}\right] = 4\left[\begin{array}{ccc}4&7\\y&3\end{array}\right][/tex]
From the above matrix,
Equation formed,
2w + 6 = 16,
w = 5,
2z -18 = 12
z = 15
10 - 6 = 4y
y = 1
Thus, the required value of the unknowns in the given sum of the matrix are w = -5, z = 15, and y = 1.
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I Need Help For This TGA
Comparing the given expression as per the solution:
(8 + 3² - 2·6)/(7 - 1·2) > (3/4 + 1/8) + 1/8 - 2²
The expressions are given in the question as:
(8 + 3² - 2·6)/(7 - 1·2) and (3/4 + 1/8) + 1/8 - 2²
As per the PEMDAS rule, the required solution would be as:
⇒ (8 + 3² - 2·6)/(7 - 1·2)
⇒ (8 + 9 - 12)/(7 - 2)
⇒ (17- 12)/5
⇒ 5/5
⇒ 1
⇒ (3/4 + 1/8) + 1/8 - 2²
⇒ (6 + 1)/8 + 1/8 - 4
⇒ 7/8 + 1/8 - 4
⇒ (7 + 1)/8 - 4
⇒ 8/8 - 4
⇒ 1 - 4
⇒ -3
Here 1 > -3
Thus, comparing the given expression as per the solution:
(8 + 3² - 2·6)/(7 - 1·2) > (3/4 + 1/8) + 1/8 - 2².
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Answer:Pemdas
Step-by-step explanation:Comparing the given expression as per the solution:
(8 + 3² - 2·6)/(7 - 1·2) > (3/4 + 1/8) + 1/8 - 2²
The expressions are given in the question as:
(8 + 3² - 2·6)/(7 - 1·2) and (3/4 + 1/8) + 1/8 - 2²
As per the PEMDAS rule, the required solution would be as:
⇒ (8 + 3² - 2·6)/(7 - 1·2)
⇒ (8 + 9 - 12)/(7 - 2)
⇒ (17- 12)/5
⇒ 5/5
⇒ 1
⇒ (3/4 + 1/8) + 1/8 - 2²
⇒ (6 + 1)/8 + 1/8 - 4
⇒ 7/8 + 1/8 - 4
⇒ (7 + 1)/8 - 4
⇒ 8/8 - 4
⇒ 1 - 4
⇒ -3
Here 1 > -3
Thus, comparing the given expression as per the solution:
(8 + 3² - 2·6)/(7 - 1·2) > (3/4 + 1/8) + 1/8 - 2².
5 customers entered a store over the course of 15 minutes. Fill out a table of equivalent ratios and plot the points on the coordinate axes provided.
THE CORRECT ONE GETS BRAINLIEST
Given that 58% of all gold dealers believe next year will be a good one to speculate
in South African gold coins, in a simple random sample of 150 dealers, what is the
probability that between 55% and 60% believe that it will be a good year to
speculate?
Tisha’s mom wants to make a smoothie using this recipe. How many cups of blueberries will she need if he uses 15 bananas?
Answer:
5 cups
Step-by-step explanation:
6bananas =2cups
6(2cups)+6(2cups)+3(1cup)= 15 bananas
5cups of blueberries
Van is 12 yrs older than john. Two yrs ago. Van was twice as old as john. Find thier present age.
Answer:
John = 14 ; Van = 26
Step-by-step explanation:
V = 12 + J
V - 2 = 2(J-2)
12 + J - 2 = 2J - 4
J + 10 = 2J - 4
2J - J = 10 + 4
J = 14
V = 12 + 14
V = 26
Which expression is equivalent to (ab7)4?
ab28
ab11
a5b11
a4b28
The expression that is equivalent to the given expression, (ab⁷)⁴, is a⁴b²⁸. The correct option is the last option a4b28
Writing an equivalent expressionFrom the given information, we are to determine the expression that is equivalent to the given expression
The given expression is
(ab⁷)⁴
To determine the expression that is equivalent to this expression, we will expand the given expression
Expanding the expression
(ab⁷)⁴
Applying the distributive exponent law of indices,
(a)⁴ × (b⁷)⁴
Simplifying
a⁴ × b²⁸
= a⁴b²⁸
Hence, the equivalent expression is a⁴b²⁸
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Answer:
I think it's a4b28
Step-by-step explanation:
The table shows the depths (relative to sea level) of several scuba divers.
What is the difference between Oscar's depth and Annabelle's depth? Write your answer as a mixed number in simplest form
Answer:(-6 5/6)
Step-by-step explanation:
-12.5=-12-1/2
-5 2/3= -5-2/3
=(-12 3/6)-(-5 4/6)
=(-6 5/6)
gimme brainlliest
The city council voted on a new tax. 38 council members voted in favor of the tax. The council has 76 members. What percentage of the council members voted in favor of the tax
The percentage of the council members who voted in favor of the tax is 50%.
What is the percentage?The percentage is a ratio that can be expressed as a fraction of 100.
Given that, the council has 76 members out of which 38 members voted in favor of the tax.
The percentage amount of members who favor the tax is:
(38/76) × 100%
= 50%
Hence, the percentage of the council members who voted in favor of the tax is 50%.
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An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such that it would produce a mean pressure of 8.0 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 24 engines and the mean pressure was 8.4 pounds/square inch with a standard deviation of 1.0 A level of significance of 0.05 will be used. Assume the population distribution is approximately normal. Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places.
The decision for rejecting the null hypothesis is μ = 8.4
In statistics the term hypothesis refers a tentative statement about the relationship between two or more variables.
Here we have given that engineer designed a valve that will regulate water pressure on an automobile engine and he is designed the valve such that it would produce a mean pressure of 8.0 pounds/square inch. Then it is believed that the valve performs above the specifications. Where the valve was tested on 24 engines and the mean pressure was 8.4 pounds/square inch with a standard deviation of 1.0 Here the level of significance of 0.05 will be used.
Here we have to determine the decision for rejecting the null hypothesis.
While we looking into the the given question we have identified the following values,
Standard deviation = 1.0
Mean pressure = 8.4 pounds/square
Level of significance = 0.05
Sample size = 24
According to these values the null and alternative hypothesis is written as,
H0 = μ = 8.4
Hx = μ ≠ 8.4
Therefore, H0 = μ = 8.4 is the decision for this one.
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Miles is renting a meeting room at a college. There is a reservation fee of $21 and an additional fee of $4 per hour. He needs to spend no more than $85. What is the possible number of hours for which Miles could rent the meeting room? Use t for the number of hours.
t ≤26.5
t<4
t ≤ 16
t<21.25
The equation represents that, Miles could rent the meeting room for t ≤ 16 hours.
What is inequality?
The word inequality means a mathematical expression in which the sides are not equal to each other. Basically, an inequality compares any two values and shows that one value is less than, greater than, or equal to the value on the other side of the equation.
The total cost for Miles to rent the meeting room for t hours is given by the equation C = 21 + 4t. We are trying to find the number of hours t for which C is equal to or less than $85.
We can solve this problem by setting up the inequality 21 + 4t ≤ 85 and solving for t.
Subtracting 21 from both sides gives us 4t ≤ 64.
Dividing both sides by 4 gives us t ≤ 16.
Thus, Miles can rent the meeting room for t hours, where t is less than or equal to 16. This means that he can rent the room for 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, or 16 hours.
Hence, the equation represents that, Miles could rent the meeting room for t ≤ 16 hours.
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The function f(x) is a cubic function and a limited table of values is provided below. Write the equation of the cubic polynomial in standard form.
The equation of the cubic polynomial is f(x) = x³ - x² -9x + 9.
What is a polynomial function?
Polynomial functions are defined for all values of x and have a finite number of roots (x-values at which the function is equal to 0). The roots of a polynomial function can be found by solving the equation f(x) = 0.
By using the given table, we can plot the points on the graph and we can draw a graph
• Fitted coefficients:
a = 9
b = -9
c = -1
d = 1
Then we can prepare the polynomial as
y = 9 - 9x - x² + x³
Now write down in the standard form, we get
f(x) = x³ - x² -9x + 9
Hence, the equation of the cubic polynomial is f(x) = x³ - x² -9x + 9.
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The weight of 10-week-old male children is distributed given a normal with a standard deviation of 87 grams.
How much data is enough to estimate, with a 95% confidence, the average weight of this population with an error not superior to 15 grams?
To determine how much data is enough to estimate the average weight of this population with a 95% confidence and an error not superior to 15 grams, we would need to use a statistical sample size calculator or consult a statistical table. Factors that would need to be considered include the desired confidence level, the desired margin of error, and the standard deviation of the population.
Using a sample size calculator, we can input these values and determine the required sample size. For example, with a 95% confidence level, a margin of error of 15 grams, and a standard deviation of 87 grams, the required sample size would be approximately 38 children. This means that we would need to measure the weight of at least 38 male children who are 10 weeks old in order to estimate the average weight of the population with a 95% confidence and an error not superior to 15 grams.
It is important to note that this is just an estimate and the actual sample size may vary depending on the specific characteristics of the population being studied. Additionally, it is always a good idea to consult with a statistician or other expert to ensure that the appropriate sample size is used in any statistical analysis.
Could someone help??
Answer:
C.
Step-by-step explanation:
For the first letter there are 26 choices, then, since one has been used, for the second letter there are only 25 choices left. And so on. You multiply them to get the total number of combinations.
For the digits, the initial choice is from 10 digits (0..9), which invalidates answer B.
Given the following exponential function,
identify whether the change represents
growth or decay, and determine the
percentage rate of increase or decrease.
y = 970(1.042)*
The change represents growth and rate of increase is 4.2%
What is an exponential function?Exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change.
Given that, an exponential function y = 970(1.042)ˣ
When the growth rate is greater than 1, then the function represents a growth.
The general form equation is:
y(x)= a(1+r)ˣ such that r is the growth percent.
So,
1+r = 1.042
r = 0.042
r = 4.2%
Hence, the growth rate is 4.2%
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A wire of length 15 is cut into two pieces which are then bent into the shape of a circle of radius r and a square of side s . Then the total area enclosed by the circle and square is the following function of s and r If we solve for s in terms of r , we can reexpress this area as the following function of r alone:
Thus we find that to obtain maximal area we should let r= To obtain minimal area we should let r=
Since s = (15 - 2πr ) / 4 and s > 0, the maximum r is 15/(2π). This same r will also maximize the area.
Since the length of 15 is divided between the circumference of the circle and the perimeter of the square,
2πr + 4s = 15 ⇒ s = (15 - 2πr ) / 4
A = πr2 + s2 = πr2 + [ (15 - 2πr ) / 4]2 = πr2 + (225 - 60πr + 4π2r2) / 16
A = π(1 + π/4)r2 - (15/4)πr + 225/16
dA/dr = 2π(1 + π/4)r - (15/4)π = 0 ⇒ r = (15/4)π / [2π(1 + π/4)] = 15 / (8 + 2π)
Since A(r) is concave up, it has a minimum at
r = 15 / (8 + 2π)
Since s = (15 - 2πr ) / 4 and s > 0, the maximum r is 15/(2π). This same r will also maximize the area.
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Help me thank y very much
Answer: 1/2
Step-by-step explanation:
Use rise over run. Up 1, over 2.
And so on. Find nice points on the graph to practice slope with!
A line segment on a coordinate plane representing the
distance a bicyclist is from home has a slope of 35/2
On the line segment, x represents the time, in hours,
that the bicyclist has been riding and y represents the
bicyclist's distance, in miles, from home. The bicyclist
began his ride at home and rides his bicycle at a
constant rate. How long will it take until the bicyclist is
140 miles from home?
A. 2 hours
B. 4 hours
C. 7 hours
D. 8 hours
Time taken by the bicyclist is 8 hours.
What is slope?Slope is calculated mathematically as "rise over run" (change in y divided by change in x).
Given slope = 35 / 2 = 17.5
Distance = 140 miles.
then time is calculated by 140 /17.5
= 8 hours
The slope expresses how quickly the dependent variable changes when the independent variable changes. The Greek capital letter D or D is frequently used by mathematicians and economics to represent change. Slope contrasts the change in the vertical axis, y, with the change in the horizontal axis, x.
As x grows, there are 4 different sorts of slopes: negative, positive, zero, and undefined. The "average rate of change" can also be thought of as the slope of a line.
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Write a polynomial function with
zeros: 4, 5i
PLEASE HELP ASAP ITS DUE TMR
Answer:
[tex]85 \sqrt{5} [/tex]
Write as a logarithmic equation. 2^1/2= square root 2
Answer:
1
log₂√2 = ------
2
Step-by-step explanation:
I'm assuming the equation looks like this:
2¹⁽² = √2
1
log₂√2 = ------
2
I hope this helps!
Write the missing numbers
2. 200 300
-600 +400
_____ _____
3. 3 hours= ____minutes
_____hours= 1 day
_______months= 2 years
The missing numbers are 3 hours = 180 minutes 24 hours = 1 day
24 months = 2 years
What are numbers ?
A number is an arithmetic value that is used to calculate and represent a quantity. Numerical symbols, such as "3," are written to represent numbers. A number system is a logical way of writing numbers that uses digits or symbols to represent them.
An arithmetic value used to indicate amount is called a number. A number is a mathematical concept that is used for counting, measuring, and labeling. Thus, mathematics is built on numbers. For instance, this is a single butterfly, and these are four.
1. 5 × x = 45
45 / 5 = x
x = 9
45 ÷ 5 = 9
2. 200 + 600 = 800
300 + 400 = 700
3. 3 hours = 180 minutes
3 * 60 minutes
24 hours = 1 day
24 months = 2 years
12 * 2 = 24
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What does (5x2−14x+24)÷(x−4) equal?
I dont need explanation or anything just answer pls
The value of (5x2−14x+24)÷(x−4) is 5x + 6
How to calculate value of (5x2−14x+24)÷(x−4)?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement.
Given ,
5x2 - 14x - 24
x - 4
5x2 multiplied by 14x, then 24.
The first term is 5x2, and 5 is its coefficient.
The coefficient of the middle term, which is -14x, is -14.
"The constant," the final term, is -24.
Step 1: Multiply the first term's coefficient by the constant 5 • -24 to get a result of -120.
Find two factors of -120 whose combined value is equivalent to the middle term's coefficient, which is -14, in step two.
-120 + 1 = -119
-60 + 2 = -58
-40 + 3 = -37
-30 + 4 = -26
-24 + 5 = -19
-20 + 6 = -14 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -20 and 6
5x2 - 20x + 6x - 24
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (x-4)
Add up the last 2 terms, pulling out common factors :
6 • (x-4)
Step-5 : Add up the four terms of step 4 :
(5x+6) • (x-4)
Therefore The value of (5x2−14x+24)÷(x−4) is 5x + 6
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Simplified term is 5x + 6. Simplification is the process of reducing an expression/fraction/problem to a simpler form.
What is simplification method?A mathematical expression is simplified when it is replaced with a simpler, typically shorter, equivalent statement.
Mathematical expressions are made up of different numbers and operations. To simplify them, we must first understand the rule known as the order of operations. It specifies the order in which operations must be performed when simplifying a mathematical expression.
Factoring 5x2 - 14x - 24
The first term is, 5x2 its coefficient is 5 .
The middle term is, -14x its coefficient is -14 .
The last term, "the constant", is -24
Multiply the coefficient of the first term by the constant 5 • -24 = -120
Find two factors of -120 whose sum equals the coefficient of the middle term, which is -14 .
-120 + 1 = -119
-60 + 2 = -58
-40 + 3 = -37
-30 + 4 = -26
-24 + 5 = -19
-20 + 6 = -14
Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -20 and 6
5x2 - 20x + 6x - 24
Add up the first 2 terms, pulling out like factors :
5x • (x-4)
Add up the last 2 terms, pulling out common factors :
6 • (x-4)
Add up the four terms of step 4 :
(5x+6) • (x-4)
Which is the desired factorization
Cancel out (x-4) which appears on both sides of the fraction line.
5x + 6
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Determine whether the statement is always, sometimes, or never true.
When the measure of ∠4 is 120° , the measure of ∠1 is 60°
always true
sometimes true
never true
Answer:
always true , because 120+60=180
18x=6(z-1) solve for z
Answer: z=3x+1
Step-by-step explanation:
1. Distribute 6 through the parentheses
18x=6z-6
2. Move the z variable to the left side and x variable to the right
-6z= -6-18x
3. Divide both sides by -6 in order to get z by itself
z=1+3x
4. Reorder the terms using commutative property
z=3x+1
(2x-123):3=33 ;200-8.(2x+7)=112
Answer:
x = 222:2 and x = 2
Step-by-step explanation:
(2x-123):3=33
(2x – 123):3 = 33
2x – 123 = 33.3
2x – 123 = 99
2x = 99 + 123
2x = 222
x = 222:2
x = 111
200-8.(2x+7)=112
200 – 8.(2x + 7) = 112
-8.(2x + 7) = 112 – 200
-8.(2x + 7) = -88
2x + 7 = (-88):(-8)
2x + 7 = 11
2x = 11 – 7
2x = 4
x = 4:2
x = 2
I hope my answer helps you.
Mary wishes to estimate the mean height of women aged 18-24. She picks a sample of 100
women aged between 18 and 24 and constructs a 99% confidence interval for the
population mean. If she were to repeat this procedure 200 times in total, she would obtain
200 different confidence intervals. How many of these intervals would you expect to
contain the population mean, u? Explain your thinking.
We would expect 198 of the intervals to contain the mean.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Given that Mary wishes to estimate the mean height of women aged 18-24.
picks a sample of 100
Women aged between 18 and 24 and constructs a 99% confidence interval for the population mean.
If Mary were to repeat this procedure 200 times in total, she would obtain
200 different confidence intervals.
A confidence interval of a parameter gives an interval in which the parameter is expected to be found. Because the interval either contains or doesn't contain the parameter, the probability is either 0 or 100%, which is not helpful. A 95% CI says that if we construct 100 intervals, 95% will contain the parameter. That is a true percentage, not a confidence. We don't know which 95, however, since the parameter is still unknown or unknowable.
I would expect 99%×200 or 198 of the intervals to contain the mean.
Hence, would expect 198 of the intervals to contain the mean.
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6 people equally share 1/2 a pan of macaroni and cheesse draw a diagram to represent the situation
Share of the each person is 1/12. And the above situation can be represented in the given picture.
Fractions and diagrams:Fractions are referred to as the components of a whole in mathematics. A single object or a collection of objects might be entire.
When we cut a piece of cake from the entire thing in real life, that could be the best example for Fractions.
To represent a fraction in a diagram we choose a closed polygon like a circle, square, etc.
First, divide the polygon into the number of parts as given in the denominator and shade the number of taken parts that are given in the numerator.
Here we have
6 people equally share 1/2 a pan of macaroni and cheese
As we know 1/2 means half
By the given data, Half of the macaroni and cheese is shared by 6 people which means Half of the pan of macaroni and cheese again divided into 6 parts
=> (1/2) ÷ 6 = [tex]\frac{ \frac{1}{2} }{6}[/tex] = [tex]\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}[/tex]
Therefore,
Share of the each person is 1/12. And the above situation can be represented in the given picture.
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How long will it take 15 machines to complete a task given that it takes 9 days for 5 machines to complete the same task?
Total machine days are 5 9 = 45 thus, it will take 3 days to complete the same test with 15 machines.
What are work and time?Work is the completion of any task for example if you have done your homework in 5 hours then you have done 5 hours.
Another example of work is that if you have done your food by 1 hour it means your work duration is 1 hour so work is basically the time duration of any task which you have done.
As per the given,
5 machines take 9 days,
5 x 9 = 45 machines day
45/15 = 3 days
Hence "Total machine days are 5 9 = 45 thus, it will take 3 days to complete the same test with 15 machines".
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