Show that the following recurring decimal are rational. (1) 0,4 (2) 0,21 (5) 0,124 (6) 0,124 (3) 0,14 (7) -1,124 (4) (8) 19,45 -2,35

The solution to the expression in this problem is given as follows:

x = -1/3 + 1.7i and x = -1/3 - 1.7i.

The rational expression for this problem is defined as follows:

[tex]\frac{2x}{3x - 5} - \frac{x + 1}{3x + 5} = -\frac{4}{9x^2 - 25}[/tex]

Applying the least common factor at the left side, we have that:

[tex]\frac{2x(3x + 5) - (x + 1)(3x - 5)}{9x² - 25} = -\frac{4}{9x^2 - 25}[/tex]

[tex]\frac{6x^2 + 10x - 3x^2 + 5x - 3x + 5}{9x^2 - 25} = -\frac{4}{9x^2 - 25}[/tex]

As the denominators are equal, the solution is obtained equaling the denominators, hence:

3x² + 2x + 5 = -4

3x² + 2x + 9 = 0.

Which is a quadratic function with coefficients given as follows:

a = 3, b = 2, c = 9.

Inserting these coefficients into a calculator, the solutions are given as follows:

x = -1/3 + 1.7i and x = -1/3 - 1.7i.

The rational expression is given by the image presented at the end of the answer.

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(3,2) the eqaition would be y=3/2x+3

In the given word problem, Mr. Garcia total paid $259.75 for his journey.

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

Given that, Mr. Garcia rents a car at a rate of $29.95 per day. There is an additional mileage charge of $20.00 per 100 miles or fractions of 100, and Mr. Garcia drives 550 miles in 5 days,

Since, cost of 100 mile = $20

Therefore, cost for 1 mile = $0.2

Since, Mr. Garcia travelled for 550 miles,

Therefore, cost of 550 miles = $0.2 × 550 = $110

Since, cost for one day = $29.95

Therefore, cost for 5 days = $29.95 × 5 = $149.75

Now,

Mr. Garcia used the car for 5 days and travelled for 550,

Therefore, total cost he paid = $110 + $149.75

= $259.75

Hence, Mr. Garcia total paid $259.75 for his journey.

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Answer: B. x + 52

Step-by-step explanation:

The unknown number is represented as x.

The phrase "52 more than" means adding 52 to something

From this we can say that the expression is B. x + 52

Answer:

3

Step-by-step explanation:

Take the x-intercept (1,0) and y-intercept (0,-3) to find the slope/gradient:

[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-(-3)}{1-0}=3[/tex]

Answer:

Step-by-step explanation:

If a new I phone costs $799 and is estimated to lose 25% of its value every six months after its purchase ,then the value of phone after 2 years is $253  .

The cost of new i phone is = $799 ;

the percent loss in the value after every 6 month is = 25% = 0.25  ;

in 2 years , there are 4 period of six months , so , n = 4  ;

The depreciated value of i phone can be calculated by formula ;

⇒ Value = Cost × (1 - 0.25)ⁿ ;

Substituting value of n =4 and cost = $799 ;

⇒ Value = 799 × (1 - 0.25)⁴ ;

⇒ Value = 799 × (0.75)⁴ ;

⇒ Value = 799 × (0.75)⁴ ;

⇒ Value = 799 × 0.31640625 = 252.8085 ≈ 253 .

Therefore , the value of the phone after 2  years is $253  .

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For given frequency table the relative frequency table with percentage is given in image.

What is a frequency table ?

A frequency table consists of the lists of items in a given data set and the number of times each item occurs in the data set and The number of times a particular data value occurs in a given data set is referred to as its frequency.

e.g.

A, A, A, B, B, B, B, C, C, D

Count the number of times each grade occurs in the above series. Yes, ‘A’ occurs thrice, ‘B+’ occurs four times, ‘C’ occurs twice, and ‘D’ occurs once. Now, according to the definition of frequency, we can note the frequency of each grade as follows:

Frequency of A = 3

Frequency of B = 4

Frequency of C = 2

Frequency of D = 1

Let us list the grades obtained by the students in the above example, in a frequency table. Table is in image.

Now,

To find the percentage for relative frequency table we used (given value/Total people)*100.

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Answer:

Quadratic

Step-by-step explanation:

The path of a firework is modeled by a quadratic function. The firework ascends from point of launch with decreasing speed (negative acceleration due to gravity) until it reaches it maximum height and then starts dropping back to ground at increasing speeds due to positive acceleration exerted by the force of gravity.

The flight path of the parabola can thus be modeled by height as a function of elapsed time

h = f(t) where f(t) is a quadratic equation in t with the general form being

a · t² +b · t  + c

We can easily eliminate the linear model since that implies the firework will ascend at a constant speed forever

The exponential model indicates that at the moment of launch the speed of the firework is constant and then it suddenly accelerates. However we know from observation that the firework has the highest speed at the moment of launch

Do not be confused by the shape of the graph - it seems to indicate the firework shoots down and up.

The note says:
Note: The path may be going in a different direction, perhaps reflected or translated vertically or horizontally while maintaining the same shape.

Indeed the actual path of the firework is a reflection of the graph about the x axis. so that it is a downward facing parabola. The vertex of the parabola is the highest y-value which is the maximum height the firework would reach.

The height would be the y axis and time t the x-axis

When using percent slope to get tree height, the tree is the rise, and the horizontal distance from the tree with the ground is the run. We can simply measure our horizontal distance from the tree, and we have instruments for gauging the percent slope to the top of a tree.

Hence, with those two measures (run and %slope) we can solve for rise.

You Total up all the trees.

let the average of number of trees are:

3+3+8+10+5+2=31

then you subtract the amount of trees under 75 feet in height.

31-8-3-3=14

divide the number of trees 14 under 75 ft in height by the total amount of trees31.

14/31=0.4516

0.4516=45.2

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Answer: To find the values that make the inequality -v <= -8 true, we can first isolate v on one side of the inequality. We do this by adding v to both sides:

-v <= -8

v >= 8

This inequality states that the only values that make it true are those greater than or equal to 8.

Another equivalent inequality in terms of v would be v>=8

So, the values that make the inequality -v <= -8 true are v >= 8

Step-by-step explanation:

Step-by-step explanation:

please see the attached fir details

The answer is 4^2. Step-by-step explanation: This is because 4 squared is 4*4 which equals 16.

The number O is the Hcf of the two numbers L and O. This is only possible if the number O is the highest common factor of the two.

The highest number among all of the common factors of something like the given numbers is known as the HCF (Highest Common Factor) for two or more numbers.

Similarly, for the given question.

The number O is the Hcf of the two numbers L and O. This is only possible if the number O is the highest common factor of the two.

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Correct option is A, (1,2) is the correct option for the given system of equations.

simultaneous equations, system of equations Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system).

The number of equations must match the number of unknowns for a system to have a singular solution.

A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system.

A group of equations comprising one or more variables is known as a system of equations. The variable mappings that satisfy each component equation, or the points where all of these equations cross, are the solutions of systems of equations.

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Question 1:

[tex]\frac{2}{3} n-16=-\frac{5}{6}n+2[/tex], solve for n.

[tex]\frac{2}{3} n-16=-\frac{5}{6}n+2[/tex]

=> [tex]\frac{2}{3} n=-\frac{5}{6}n+18[/tex]

=> [tex]\frac{2}{3} n+\frac{5}{6}n=18[/tex] =>[tex]\frac{4}{6} n+\frac{5}{6}n=18[/tex]

=> [tex]\frac{9}{6} n=18[/tex] => [tex]\frac{3}{2} n=18[/tex]

=> [tex]n=18(\frac{2}{3})[/tex]

=> [tex]n=6(2)[/tex]

Sol: n=12

Question 2:

[tex]5-\frac{1}{3}p=\frac{4}{9}p+12[/tex], solve for p.

[tex]5-\frac{1}{3}p=\frac{4}{9}p+12[/tex]

=>[tex]5=\frac{4}{9}p+\frac{1}{3}p+12[/tex]

=> [tex]5=\frac{4}{9}p+\frac{3}{9}p+12[/tex]

=>[tex]5=\frac{7}{9}p+12[/tex]

=> [tex]-7=\frac{7}{9}p[/tex]

=> [tex]-7(\frac{9}{7}) =p[/tex]

=> [tex]-(9)=p[/tex]

Sol: p=-9

Question 3:

The relationship between ∠2 and ∠7 is that they are alternate interior angles.

Question 4:

The relationship between ∠1 and ∠3 is that they are corresponding angles.

Question 5:

The relationship between ∠4 and ∠5 is that they are alternate exterior angles.

Answer:

c)12:16​

Step-by-step explanation:

We know

The ratio is 6:8; we times 2 and get the ratio of 12:16

So, the answer is C

The number of years that he save this money is 7 years

Simple interest is a method of calculating the interest charge. Simple interest can be calculated as the product of principal amount, rate and time period.

Simple Interest = (Principal × Rate × Time) / 100

Where P is the principal, R is the rate of interest, and T is the time.

Given;

Mike saves $2,000 at a yearly simple interest rate of 2%;

Money he earns in interest= $280

Then the number of years;

$280 = ($2,000 x 2 x T) / 100

$28,000 = $4,000 x T

T = $28,000 / $4,000

T = 7 years

Therefore, numbers of years mike has to pay interest will be 7

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Mrs. Byrne mowed 1/4 of her lawn. Her son mowed 2/7 of it.  Mrs. Byrne mowed most of the lawn. The lawn still needs to be mowed is 13/28.

Mrs. Byrne mowed of his lawn = 2/7

Her son mowed of his lawn = 1/4

We firstly equal the denominator of both fraction by taking the LCM of both numbers.

The LCM of 7 and 4 is 28.

So we multiply and divide by 4 in the fraction 2/7 and by 7 in fraction 1/4. Now,

Mrs. Byrne mowed of his lawn = 2/7 × 4/4 = 8/28

Her son mowed of his lawn = 1/4 × 7/7 = 7/28

Now we compare the both fraction 8/28 and 7/28. The 8/28 is greater than 7/28. So we can say that Mrs. Byrne mowed most of his lawn.

The total lawn is 28/28.

So, the remaining lawn for mowed = 28/28 - (8/28 + 7/28)

The remaining lawn for mowed = 28/28 - (8 + 7)/28

The remaining lawn for mowed = 28/28 - 15/28

The remaining lawn for mowed = (28 - 15)/28

The remaining lawn for mowed = 13/28

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The complete question is:

Mrs. Byrne mowed 2/7 of his lawn. Her son mowed 1/4 of it. Who mowed most? How much of the lawn still needs to be mowed?

The percentage of alcohol in a handy is 3%. 3/5 * 5% = 3% , Since beer is 5% alcohol, then 3/5 of the handy is 5% alcohol.

The ratio of beer to lemonade in a handy is 3:2, which means that 3 parts of the handy is beer and 2 parts of the handy is lemonade. To express it in terms of fractions, 3/5 of the handy is beer and 2/5 of the handy is lemonade. Since beer is 5% alcohol, then 3/5 of the handy is 5% alcohol. Therefore, if you take the proportion of the beer in the handy and multiply it by the alcohol percentage of the beer, you will get the percentage of alcohol in the handy. In this case, 3/5 * 5% = 3%. This means that the percentage of alcohol in a handy is 3%. It's important to note that this assumes that the beer and lemonade are mixed together evenly and there's no spillage or evaporation. Also, it's important to consider the fact that the alcohol percentage of the beer can vary depending on the brand and type of beer.

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The point where the two lines intersect each other is (2,5).

What are simultaneous equations?

Two or more algebraic equations that share variables, such as x and y, are said to be simultaneous equations. Since the equations are solved simultaneously, they are known as simultaneous equations. These equations alone could have an endless number of solutions.

When two lines intersect each other, the point of intersection is a point common to both lines.

This can be found by solving simultaneous equations.

The given equations of the two lines are

3x+y = 11

4x-y = 3

Adding the above equations, we can remove the y variable.

The result is 7x = 14

x = 2

Now we substitute this value of x in either of the above equations to get the value of y.

3 * 2 + y = 11

y = 11 -6 = 5

Therefore the point where the two lines intersect each other is (2,5).

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Answer: (e) (i) To represent the inequalities graphically on a coordinate plane, you can use a scale of 2cm to represent 5 units on each axis. To shade the unwanted regions, you can start by drawing the coordinate plane and labeling the x and y axes. Then, you can graph the inequalities by plotting the lines and shading the regions that do not satisfy the inequalities.

(b) 2x + 3y ≤ 30

This inequality represents a line with the equation 2x + 3y = 30. You can graph this line by plotting a few points and then drawing a straight line through them. The points that are on or below the line satisfy the inequality.

(c) x - 2y ≥ -10

This inequality represents a line with the equation x - 2y = -10. You can graph this line by plotting a few points and then drawing a straight line through them. The points that are on or above the line satisfy the inequality.

(d) y ≤ -2x + 10

This inequality represents a line with the equation y = -2x + 10. You can graph this line by plotting a few points and then drawing a straight line through them. The points that are on or below the line satisfy the inequality.

The set of points that satisfy all three inequalities is the area that is not shaded in the graph.

(ii) To find the maximum number of rabbits the farmer can buy, we need to find the coordinates of the vertex of the feasible region.

The coordinates of the vertex of the feasible region is (x,y) the point where the line 2x+3y=30 and x-2y=-10 intersects.

We can substitute the x-2y=-10 into the 2x+3y=30 equation

2x+3y=30

2x+3y+2y=-10

2x+5y=-10

x=-5y+10

substituting this into one of the equation we have

2(-5y+10)+3y=30

-10y+20+3y=30

-7y=10

y= -10/7

substituting this into x = -5y+10 we have

x = -5(-10/7)+10 = 15

the maximum number of rabbits the farmer can buy is 15

Step-by-step explanation:

The mixed fraction in the simplest form is 21/20, and the snowfall in December was almost 5 times more than the snowfall in January.

A mixed fraction is one that is represented by both its quotient and remainder. A mixed fraction is, for instance, 2 1/3, where 2 seems to be the quotient and 1 is the remainder. An amalgam of a whole integer and a legal fraction is a mixed fraction.

Given that the amount of snowfall in January was:

[tex]1 \frac{1}{20}[/tex]

To convert the mixed fraction to simplest form, multiply the denominator with the value and add it to the numerator:

[tex]21/20 = 1.05[/tex]

The amount of snowfall in December is:

[tex]5 \frac{2}{24} = 5 \frac{1}{12}[/tex]

To convert the mixed fraction to simplest form, multiply the denominator with the value and add it to the numerator:

[tex]\frac{61}{12} = 5.08[/tex]

Hence, the snowfall in December was almost 5 times more than the snowfall in January.

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The problem where the professor states that the average age of students in the university is 21 years is an example of statistical inference.

Statistical inference is the process of using data from a sample to make inferences or conclusions about a population. It allows us to draw conclusions about a larger group of individuals or objects based on information from a smaller sample. It is a fundamental part of statistical analysis and is used in many fields, including research, business, and government.

Statistical inference includes two main types: estimation and hypothesis testing.

Estimation: This involves using sample data to make estimates about population parameters. For example, using the sample mean to estimate the population mean or the sample proportion to estimate the population proportion.

Hypothesis testing: This involves using sample data to test a claim or hypothesis about a population parameter. For example, testing the claim that a coin is fair, based on the proportion of heads in a sample of coin flips.

The goal of statistical inference is to use the information from a sample to make informed decisions or predictions about a population. It allows us to draw general conclusions from a specific set of data, and it is an essential tool for making sense of data in many fields.

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The Interquartile range of the given set is 1.5.

What is Interquartile range ?

The variation between the third and first quartiles is defined by the interquartile range. The numbers that partition the entire series into four equal sections are known as quartiles. There are therefore 3 quartiles. The lower quartile is represented by the letter Q₁, the higher quartile by the letters Q₁, and the middle quartile by the letter Q₃.

Given set : 81, 82, 83, 83, 84, 84, 84, 85

Here, It is already in increasing order but the number of values is 8 which is even.

So, Q₁ part = 81, 82, 83, 83

Q₁ will be the median of 82 and 83.

So, Q₁ = (82+83)/2

          = 82.5

Similarly, Q₃ part =84, 84, 84, 85

So, Q₃ will be the median of 84 and 84.

hence, Q₃ = (84+84)/2

                 = 84

We know that, Interquartile range = Q₃ - Q₁

                                                          =  84 - 82.5

                                                          = 1.5

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The graphic representation of data points arranged into user-specified ranges is called a histogram. The diagram below that is attached shows the histogram diagram.

A histogram is a figure made up of rectangles with widths equal to the class interval and areas proportionate to the frequency of a variable.

From the given information, we have the following:

Length of time (t)     Frequency

0 ≤ t < 10                        5

10 ≤ t < 25                     24

25 ≤ t < 30                    12

30 ≤ t < 50                    8

So, the image that is attached shows how the table is represented in a histogram.

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The complete question is given below:

Liquid with density 0.135 grams over milliliter and volume 182.8 milliliter has weight equal to 24.68 grams.

Let 'm' represents the weight in grams of a liquid

Volume 'V' of the container to fill a liquid is = 182.8milliliter

Density 'ρ'  of the given liquid is 0.135 grams over milliliter

Density =  mass / Volume

Substitute the values we get,

⇒ 0.135 = m / 182.8 ×

⇒ m = 0.135  grams over milliliter ×  182.8milliliter

⇒ m = 24.678 grams

⇒ m = 24.68 grams

Therefore, the weight of a liquid for the given density and the volume is equal to 24.68 grams.

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In dollars and cents, the total projected cost of gas and oil for two years would be $ 756.00

First we will calculate per year how much is spent.

First year, we have :-

= 13000 * 0.028 = 364.00

Second year, we have:-

12000 * 0.028 = 336.00

The total cost in the two years is the sum of both:

= first year + second year

= 364.00 + 336.00 = 700.00

In dollars and cents, the total projected cost of gas and oil for two years would be $ 700.00.

Total cost

Total cost refers to the overall cost of production, which includes both fixed and variable components of the cost.

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complete question is :-

Compute the requested operating costs as indicated, based on the preceding table and the following information.

Choose the correct answer.

Car = 6 cylinder compact

Years driven = second and third

Miles driven = 13,000 miles in year two and 12,000 in year three

In dollars and cents, the total projected depreciation cost for these two years would be $?

A sample of size 40 is a valid sample size and could result in a sample statistic, as explained above, the correct answer is B.

What is the sample statistic?

A sample statistic is a measure calculated from a sample of the population, while a population parameter is a measure calculated from the entire population. Therefore, a sample statistic may differ from a population parameter due to sampling variability.

Out of the options provided, only option B (40) could result in a sample statistic but not a parameter.

This is because a sample of size 40 is a subset of the population, and a statistic calculated from this sample (such as the sample mean or sample proportion) would be a sample statistic.

Option C (4000) would result in both a sample statistic and a population parameter because a sample consisting of the entire population is a census, and any measure calculated from this sample would also be a parameter.

Option A (both 40 and 4000) and option D (neither 40 nor 4000) are not correct because 4000 is not a valid sample size, as it includes the entire population and would therefore be a census.

Hence, a sample of size 40 is a valid sample size and could result in a sample statistic, as explained above, the correct answer is B.

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Answer:

the answer is A

I hope I help you in this ans and tq.