Verify that the following equation is an identity.

(5sinx+5cosx)^2=25sin2x+25

The sample is chosen by dividing the population into voting precincts, and sampling everyone in the precincts selected is cluster sampling.

As stated in the question statement, the sample was obtained by randomly selecting precincts from which to sample the entire population, which is known as cluster sampling.

A population is divided into clusters, such as districts or schools, and some of these clusters are randomly chosen as your sample via the probability sampling technique known as cluster sampling. In a perfect world, each cluster would be a tiny reflection of the whole population.

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The dimensions of the rectangle that will produce the largest area of the window are x = (12 - √3) / 2 and y = 3 / √3.

Let x be the length of the rectangle, and y be the width of the rectangle. The perimeter of the window is 12m, and since the triangle is equilateral, we know that all 3 sides of the triangle are equal to x.

Therefore, we can set up the following equation to represent the perimeter of the window: x + y + x = 12

The area expression is:

A = x*y + (x^2 * √3)/4

After substituting x = (12-y)/2, the area expression becomes:

A = [((12-y)/2)*y] + [(12-y)^2 * √3]/16

Which can be simplified to:

A = 6y - 3y^2 + (9-3y)√3 / 4

To find the maximum value of this expression, we can differentiate it with respect to y and set it to zero:

dA/dy = -3y + (9-3y)√3/4 = 0

and solve for y:

-3y + (9-3y)√3/4 = 0

-3y = (3y-9)√3/4

-3y = 3y-9 / 2√3

3y = 9 / 2√3

y = 3 / √3

To find x, we can substitute this value of y back into x = (12-y)/2:

x = (12 - 3 / √3) / 2 = (12 - √3) / 2

So the dimensions of the rectangle that will produce the largest area of the window are x = (12 - √3) / 2 and y = 3 / √3.

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Each measure in a 4/4 time signature has four quarter note beats; each measure in a ¾ time signature has three quarter note beats; and each measure in a 2/4 time signature has two quarter note beats.

A 4/4 time signature does not imply that each measure contains simply four quarter notes. It signifies that each measure only contains four beats. These beats may comprise half notes, quarter notes, eighth notes, pauses, or whatever the composer desires, but all note and rest values must add up to no more or less than the time signature’s top number (or numerator).

The most popular musical meter is 4/4. Because it is so widespread, it is also known as common time, and the two numerals in the time signature are sometimes substituted with the letter C. The stacking digits in 4/4 indicate that each measure comprises four quarter note beats. To count 4/4 meter, tap the beat every time you tap the beat.

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If the depth of the snow is 7 inches now, the depth one hour from now is equal to 8.5 Inches.

Based on the information provided above, a function which models the depth (D) in inches, of snow in a yard time (t) hours after it started snowing this morning is given by:

D = 1.5t + 4

where:

When the depth (D) in inches is equal to 7 inches now, the time measured in hours would be given by:

D = 1.5t + 4

7 = 1.5t + 4

1.5t = 7 - 4

1.5t = 3

Time, t = 3/1.5

Time, t = 2 hours.

One hour from now, the time measured in hours would be given by:

Time, t = 1 hour + 2 hours

Time, t = 3 hours.

At time, t = 3 hours, the depth can be calculated as follows;

Depth, D = 1.5t + 4

Depth, D = 1.5(3) + 4

Depth, D = 8.5 Inches.

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Using the concept of change of subject of formula, we can express the radius r₁ and r₂ as r₁ = r₂² / 18π

The formula of volume of a cylinder is given as;

V = πr²h

The equation for the volume of the cylindrical tank is V1 = 36πr₁ and the equation for the volume of the neighborhood pool is V2 = 1/2(4/3πr₂²).

Since V₁ = V₂, we can set the two equations equal to each other and solve for r₁:

36πr₁ = 1/2(4/3πr₂²)

To solve for r₁, we can first multiply both sides of the equation by 2/4/3, which gives:

18πr₁ = r₂²

Then we can divide both sides of the equation by 18π to get:

r₁ = r₂² / 18π

So we can say that r₁ is a function of r₂ because it can be expressed in terms of r₂.

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The probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level is approximately 0.04.

What is the binomial distribution?

In a binomial distribution, the probability of getting success must remain the same for the trials we are investigating. For example, when tossing a coin, the probability of flipping a coin is ½ or 0.5 for every trial we conduct, since there are only two possible outcomes.

To approximate the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level, we can use a normal approximation to the binomial distribution.

The sample size is n = 300 and the proportion in the population is p = 0.22. The standard deviation of the sample proportion is:

[tex]\sigma_{p}[/tex] = √(p(1-p)/n) = √(0.22(1-0.22)/300) = 0.0115

We can use the standard normal distribution to approximate the probability that the sample proportion is at least 0.27:

[tex]P(p_{hat} > = 0.27) \approx P( (p_{hat} - p) / \sigma_{p} > = (0.27 - 0.22) / 0.0115 )[/tex]

P(z >= 1.74)

where z is the standard normal variable.

We can use a standard normal table or calculator to find the probability for z >= 1.74 is about 0.04.

Hence, the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level is approximately 0.04.

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The height of the candle after 9 hours can be written as 23.4 cm.

What is a linear relationship?

The linear relationship is the relationship between two variables such that it follows a straight line.

As it is given that the height of the candle is linear to the amount of time. Now, let's the time be on the x-axis and the y-axis be the height of the candle. therefore, the slope of the linear relationship can be represented as,

Point1 of the slope, (x₁, y₁) = (6, 24.6)

Point2 of the slope, (x₂, y₂) = (21, 18.6)

Now, since we know the two points of the linear relationship, therefore, the slope can be written as,

m = (y2 - y1)/(x2-x1)

m = (18.9 - 24.6) / (21 - 6)

m = -0.4

Thus, the slope of the linear relationship is -0.4.

Now, the equation of the linear relationship can be written as,

y = mx + c

y1 = mx1 + c

24.6 = (-0.4 * 6) + c

24.6 = -2.4 + c

Thus, the Linear relationship of the candle height and the amount of time can be written as y = -0.4x + C.

Now, In order to calculate the height of the candle after 9 hours, we will substitute the value of the x as 9.

y = -0.4x + 27.

y = -0.4(9) + 27.

y = -3.6 + 27.

y = 23.4

Hence, the height of the candle after 9 hours can be written as 23.4 cm.

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The equation of a line passing through the points (-3, 3) and (1, 2) is x + 4y = 9

When two points in a plane are given, a line passing through them can be determined. It is the set of all points in the plane that are equidistant from the two given points.

These points form a straight line, which can be represented by an equation with two variables.

The equation of the line can be found using the slope-intercept form or by solving a system of two equations.

The slope of the line is the ratio of the change in the y-coordinate of the two points over the change in the x-coordinate of the two points.

The y-intercept is the point where the line crosses the y-axis. The line can also be expressed in terms of its intercepts on the x and y-axes.

The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

We know that the equation of a line passing through the points (x1, y1) and (x2 , y2) is given by y - y1 = m (x - x1).

m is the slope given by the formula m = (y2 - y1) / (x2 - x1)

Hence on substituting the given points in the equation of a line, we get

y - 3 = m ( x - (-3) ) -------(1)

m =  (y2 - y1) / (x2 - x1)

m = (2 - 3) / (1+3)

m = -1/4

Substituting value of m in equation (1), we get

y - 3 = -1/4 ( x + 3)

4(y - 3) = -x - 3

4y - 12 = -x - 3

x + 4y = -3 +12

x + 4y = 9

Therefore, the equation of a line passing through the points (-3, 3) and (1, 2) is x + 4y = 9

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A base must be raised to a certain exponent or power, or logarithm, in order to produce a specific number.

How do you explain logarithm?

The exponent or power that must be added to a base to produce a particular number is called the logarithm. If bx = n, then the mathematical expression for x is x = logb n, where x is the logarithm of n to the base b. For instance, since 23 = 8, 3 is the logarithm of 8 in base 2, or 3 = log2 8.

The opposite of exponentiation in mathematics is the logarithm. The exponent to which b must be raised in order to obtain a number x is hence the logarithm of a number x to the base b. For instance, since 1000 = 103, log10 of 1000 is 3, or log10 = 3.

Common logarithms, with a base of 10, binary logarithms, with a base of 2, and natural logarithms, with a base of e 2.71828, are the four most popular varieties of logarithms.

Say you have n=4 and k=2. (Next, we might look at n=8 and k=3). Let f[(x-list_),y] be the expected value of the number of random trials needed to get from a list of thus and so many hits already, to completion with y or more hits in every slot. We want f[(0,0,0,0),2].

Note that f[anylist,0]=0, and that f[(0),1]=1.

Now f[0000,2]=1+f[0001,2], because the first random number has to go somewhere and whichever one it lands on is symmetric to the case where it lands on 4.

Next, f[0001,2]=1+(1/4)f[0002,2]+(3/4) f[0011,2]=1+(1/3)f[000,2]+(3/4)f[0011,2], the second step because once one slot is filled, it just takes 4/3 times as long to finish because 1/4 of your shots will be wasted on the saturated target.

By and by, you’ll encounter f[1111,2] among other situations, and this will be equal to f[0000,1].

It’s a finite calculation, it can be written up as a recursive program, and you’ll be able to get an exact answer for any f[(however many zeros), (whatever value counts as saturation)]. The calculation will, unfortunately, run into geometric expansion of time and space as there are going to be a lot of subcases to consider even for f[00000000,3]. Still, it’s nice to have the actual exact answer for a number of representative small cases.

As to the long term situation, once you’ve fired, say, n log n shots, you theoretically might be done, having `wasted no shots’. But consider the number of hits that “1” has got. This is a binomial distribution, with the probability of k hits out of n log n shots each at odds 1/n of hitting being Binomial[n log n, k](1/n)^k((n-1)/n)^(n log n-k). This should work out to roughly a Poisson distribution, with density something like (x/log n)e^{-x/log n} being the probability that you got x hits. The saturated targets would be the ones with x greater than log n, and you’d expect a typical target would need  ( log n - x )  e x / log n dx further hits.

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The function f(x) is 4x³ + 20x² + 5x - 31.

What is a function?

A function in mathematics from a set X to a set Y allocates precisely one element of Y to each part of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealized relationship between two changing quantities.

Here, we have

Given:  function f(x) is divided by 2x + 3, the quotient is 2x² + 7x - 8 and the remainder is -7.

When a polynomial f(x) is divided by any another polynomial d(x) and there is q(x) and r then it can be written as:

f(x) = d(x)q(x) + r

Now putting values of d(x) = 2x+3, q(x) = 2x² + 7x - 8 and r = -7, we get

f(x) = (2x+3)(2x² + 7x - 8) - 7

f(x) = 4x³ + 14x² - 16x + 6x² + 21x - 24 - 7

f(x) =  4x³ + 20x² + 5x - 31

Hence, the function f(x) is 4x³ + 20x² + 5x - 31.

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The approximate height of the tree is feet.

The exact height of the tree cannot be determined without additional information. However, a rough estimate can be made using the following formula:

Height of Tree = Length of Shadow / Tan(Angle of Elevation of Sun)

In this case, the estimated height of the tree would be: Length of Shadow / Tan( ) = feet/tan( ).

The formula to calculate the height of a tree is:

Height of Tree = Length of Shadow * Tan(Angle of Elevation of the Sun)

In this case, Length of Shadow = feet and Angle of Elevation of the Sun

Therefore, the height of the tree can be calculated as follows:

Height of Tree = feet * Tan( )

Height of Tree = feet * 0.839

Height of Tree = feet

Therefore, By  the angle of elevation the approximate height of the tree is feet.

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Answer:

10.8

Step-by-step explanation:

using Sine Law

a/sin A = b/sin B = c/sinC

a = 8

A = 40°

b = ?

B = 120

substitute the values and you will get the answer 10.778370842670, rounding to the nearest 10ths.

10.8 is the measure of the value of x to the nearest tenth.

The given diagram is a triangle with side 8 and angles 40 and 120 degrees.

Applying the sine rule, we will have:

x/sin120 = 8/sin40

Cross multiply to have:

8sin120 = xsin40

x = (8sin120)/sin40

x = 6.9282/0.6428

x = 10.8

Hence the measure of the value of x to the nearest tenth is 10.8.

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So, the trade discount for 15 boxes of computer paper if the unit price is $14.26, and a single trade discount rate of 45% allowed is $96.255.

Trade discount refers to the reduction in list price known as a discount, allowed by a supplier to the consumer while selling the product generally in bulk quantities to the concerned consumer to increase the sales of the business as more customers are attracted when the discount is given on the list price of the product.

We have to calculate the trade discount for 15 boxes of computer paper.

the unit price of computer paper is $ 14.26, and a single trade discount rate is 45% allowed.

The trade discount on one unit computer paper is.

= 45%×14.26

= $6.417

now, trade discount on 15 unit of computer paper box is.

=6.417×15

=$96.255.

So, the trade discount for 15 boxes of computer paper if the unit price is $14.26, and a single trade discount rate of 45% allowed is $96.255.

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Using a system of equations, the number of each type of block Xavier used is as follows:

A system of equations is two or more equations solved simultaneously or concurrently.

A system of equations is also described as simultaneous equations.

Let a represent the number of cement blocks and b the number of clay blocks.

9a + 5b = 113 ... Equation 1

a + b = 17 ... Equation 2

Multiply Equation 2 by 9:

9a + 9b = 153 ... Equation 3

Subtract Equation 1 from Equation 3:

9a + 9b = 153

-

9a + 5b = 113

4b = 40

b = 10

In Equation 2, substitute b = 10:

a + b = 17

a + 10 = 17

a = 7

Thus, Xavier used 7 cement blocks and 10 clay blocks to make the stack of 17 blocks that was 113 centimeters thick.

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Answer: Let's call the smallest of the 3 consecutive natural numbers "n". The other two numbers are "n+1" and "n+2". The sum of these 3 numbers is "3n + 3"

Also, the sum of the two numbers following these 3 consecutive natural numbers is (n+3) + (n+4) = 2n + 7

We know that these two expressions are equal:

3n + 3 = 2n + 7

By solving this equation we can find the value of n.

Subtracting 3 from both sides: 3n = 4

Dividing by 3: n = 4/3

Since n is a natural number, n has to be the integer 1.

So the three consecutive natural numbers are 1, 2, 3 and the sum of the students in the class is 6, which is also the sum of the next two natural numbers 4 and 5.

Therefore, there are 6 students in her class.

Step-by-step explanation:

Step-by-step explanation:

The given system of equations models the scenario where Jason spent all of his money on buying CDs and magazines, and that if he had bought one more CD, he would have run out of money. The system of equations is:

4x + 2y = 32 (Equation 1)

5x + 2y = 36 (Equation 2)

The first equation represents the total amount of money Jason spent on 4 CDs (x dollars each) and 2 magazines (y dollars each). The second equation represents the total amount of money Jason would have spent if he had bought one more CD. By subtracting equation 1 from equation 2, we get:

x = 4 (Equation 3)

This equation tells us that each CD costs $4.

By substituting this value into equation 1, we get:

4(4) + 2y = 32

16 + 2y = 32

2y = 16

y = 8 (Equation 4)

This equation tells us that each magazine costs $8.

So, Jason spent $4 for each CD and $8 for each magazine.

The starting pay was $ 168.6 and has climbed by $ 3,000 annually since 2008 at a consistent rate.

Since it is a percent increase, they will receive a 2% salary raise in year two, which will amount to $30,000. As a result, it would be 1.2, and we'll now try to identify a pattern. We know that the pay will be 30,000 because of the 2% increase because the power in this situation is one and the salary is in the eleventh year.

At this moment, I would have accepted ten times the 2% rise. During the first year, it would not have occurred. If you want to proceed, divide it by one less than 1.2 to get the temperature. After we calculate it, we'll receive a value for their entire compensation for the next 10 years.

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We can conclude that -

Given are the inequality signs as : >, < and =.

An inequality is used to make unequal comparisons between two numbers or expressions.

Therefore, we can conclude that -

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The side length of the square will be 1 unit.

What is a square?

A square is a two-dimensional plane figure with four equal sides and all the four angles are equal to 90 degrees. The properties of rectangle are somewhat similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle is called a square only if all its four sides are of equal length.

The most important properties of a square are listed below:

Now

As given

1=√A (A=area)

Hence,

A=1

Now because A=side^2

Therefore,

Side=√1

side=1

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On applying higher-order thinking skills, it can be seen that Joel does not count the numbers 82 82 62 80 while counting backwards from 79 to 65.

What is Higher-Order Thinking?

Higher-order thinking is a concept of educational reform based on learning taxonomies (like Bloom's taxonomy). Higher-order thinking skills are also known as higher order thinking skills (HOTS). The theory holds that some forms of learning have more universal advantages while requiring more cognitive effort than others.

Joel counts back from 79 to 65.

The numbers in descending order from 79 to 65 is -

79 78 77 76 75 74 73 72 71 70 69 68 67 66 65

The numbers go from right to left on a number line.

The options for the numbers are -

77 72 70 82 70 82 67 62 80

After applying HOTS the numbers which do not lie in between 79 and 65 when counting backwards is -

82 82 62 80

Therefore, the numbers which can be eliminated are 82 82 62 80.

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the cost of direct materials used for the current year i.e the direct material used is $122,000

The computation of the direct material used is shown below:

= Opening raw material inventory + material purchased - ending raw material inventory

January 1 Raw materials inventory $6,000

Add Materials purchased $123,500

Raw material available for production $129,500

($6,000+$123,500)

Less December 31 Raw material Inventory $7,500

Cost of direct materials used $122,000

= $6,000 + $123,500 - $7,500

= $122,000

Hence, the direct material used is $122,000

the cost of direct materials used for the current year i.e the direct material used is $122,000

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The lithosphere is: a. the rigid outer part of the earth, consisting of the crust and upper mantle.

The Lithosphere is the solid outer shell of the Earth that forms the Earth's surface, including the continents and the ocean floor. It is the uppermost layer of the Earth's solid surface, composed mostly of the minerals, such as silicon and oxygen, that form rocks like granite and basalt.

The lithosphere is composed of the Earth's crust and the uppermost part of the Earth's mantle. It is broken up into tectonic plates that are in constant motion and are responsible for the earthquakes, volcanoes, and the creation and destruction of land masses. The lithosphere is also the layer that supports all life on Earth.

Therefore, we can state that the correct description of the lithosphere is: a. the rigid outer part of the earth, consisting of the crust and upper mantle.

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Answer:

H = 3

K = 0

Step-by-step explanation:

To solve the system of linear equations using elimination by multiplication, we can first multiply the first equation by 3. This gives us:

3(H+2k)=3*3=9

3H+6k=9

Now we have the same coefficient (6k) on the second variable in both equations. We can now subtract the first equation from the second equation to eliminate the second variable:

3H+6k=9

-3H-6k=-9

0H=0

This tells us that the system of equations is consistent and has infinitely many solutions. To find the specific solution, we can use either equation and substitute in a value for one of the variables. For example, we can use the first equation and substitute in H=3. This gives:

3+2k=3

2k=0

k=0

Now we can substitute this value for k back into either equation to find the value of H. For example, using the first equation:

H + 2(0) = 3

H = 3

So the solution to the system of equations is:

H = 3, k = 0

Repeating decimals are never rational numbers. Hence the statement of enrique that "0.44444 is a rational number" is incorrect.

Rational numbers are those which can be expressed as the ratio of two integers, but the denominator should not be zero. Rational numbers can be expressed in p/q form. Rational numbers are never repeating decimals. For example 10/2, 3/2, 2/1 are rational numbers.

Irrational number are those which cannot be written in the form of p/q. They are repeating decimal and recurring decimals. The best example of irrational number is the value of π(pie). The value of π is 3.1415...with no repeating pattern of decimals.

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The probability using the same formula as before: P(X>=3) = 1 - P(X<=2), which gives us 1 - 0.8530p = 0.08, or p = 0.0934.

Let p be the probability of a rivet being defective.  In order for 19% of all seams to need reworking, at least 4 of the 23 rivets must be defective.  This means that the probability of at least 4 of the rivets being defective must be 0.19.  We can calculate this probability using the formula P(X>=4) = 1 - P(X<=3), where X is a binomial random variable with parameters n = 23 and p = p.  This gives us 1 - 0.9920p = 0.19, or p = 0.1984.

To ensure that only 8% of all seams need reworking, at least 3 of the 23 rivets must be defective.  This means that the probability of at least 3 of the rivets being defective must be 0.08.  We can calculate this probability using the same formula as before: P(X>=3) = 1 - P(X<=2), which gives us 1 - 0.8530p = 0.08, or p = 0.0934.

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Step-by-step explanation:

a) The probability of getting at least one hit in a game can be found by using the complement rule, which states that the probability of an event happening is equal to 1 minus the probability of the event not happening.

In this case, the event of getting at least one hit in a game is the complement of getting no hits in a game.

To get no hits in a game, the player must get 0 hits out of 5 at bats.

The probability of getting no hits in a game is (1-.3)^5 = 0.168.

Therefore, the probability of getting at least one hit in a game is 1 - 0.168 = 0.832

b) The probability of getting at least one hit in ten consecutive games is found by multiplying the probability of getting at least one hit in one game by itself ten times.

So it would be (0.832)^10 = 0.1588

Alternatively, we could use the binomial distribution formula, where probability of getting a hit k times out of n trials is given by

P(k) = (n choose k) * p^k * (1-p)^(n-k)

If we want to find the probability of getting at least one hit in ten games, we need to find the probability of getting 0 hits in all ten games and subtract it from 1.

1 - (1 - 0.832)^10 = 0.1588

So the probability of getting at least one hit in ten consecutive games is 0.1588.

Answer:

The common denominator will be 70

Step-by-step explanation:

To set the fractions to a common denominator, we need to know what number they both share. In this case, both 7 and 10 share the number 70, since 7 * 10 is 70.

Therefore, we can multiply 2/7 by 10/10 and 3/10 by 7/7:

[tex]\frac{2}{7}[/tex] * [tex]\frac{10}{10}[/tex] = [tex]\frac{20}{70}[/tex]

[tex]\frac{3}{10}[/tex] * [tex]\frac{7}{7}[/tex] = [tex]\frac{21}{70}[/tex]

Answer: Common denominator = 70

Step-by-step explanation: In order to find a common denominator, we need to find what number they have in common. That'd be 70. So, now rewrite the fractions like this:

[tex]\frac{2}{7} = \frac{?}{70}[/tex]

[tex]\frac{3}{10} = \frac{?}{70}[/tex]

Now, we need to find the equivalent fractions on the right. To do that, use this formula for part 1:

Right denominator / left denominator

Then, what you get after dividing those 2, you multiply by the left numerator.

So, let's plug the numbers in:

70 / 7 = 10

10 * 2 = 20

Therefore, [tex]\frac{2}{7} = \frac{20}{70}[/tex]

70 / 10 = 7

3 * 7 = 21

Therefore, [tex]\frac{3}{10} = \frac{21}{70}[/tex]

I hope this helped!