how to multiply fractions

Final answer:

The original basket contained 72 mangoes, determined by working backward from the known remainder of 3 after the sequential distribution between the king, queen, and princes.

Explanation:

From a basket of mangoes, the distribution among the royal family leads to a classic fractional subtraction problem. The solution involves working backward from the known remainder. The youngest child took the remaining 3 mangoes, which indicates that this was what was left after the king, the queen, and the three chief princes took their respective shares.

Let's denote the original number of mangoes as x. The king took 1/6 of x, leaving 5/6x. The queen then took 1/5 of the remainder (5/6x), leaving 4/5 * 5/6x = 4/6x or 2/3x. Successively, the three chief princes took 1/4, 1/3, and 1/2 of their respective remainders, each action further reducing the quantity of mangoes. The problem states that after these distributions, 3 mangoes were left, which were taken by the youngest child.

To find the original number of mangoes, we begin by denoting the final 3 mangoes as the remaining half after the last prince's share, work through the inverse operations for each prior transaction, and solve for x. Through this back calculation, we find that the original number of mangoes was 72.

The correct statement that describes a method that will help the group accurately estimate this percentage is;

⇒ Take a random sample of residents in the town, and ask each resident in the sample whether or not they support a teen curfew. Then calculate the percentage of the total who say "yes."

Random sampling is defined as a sampling method that utilizes randomization of sample selection.

In random sampling, each sample has the same probability of being selected as that of other samples to be selected to serve as a representation of an entire population.

Now, in this question, since the group would like to estimate the percentage of town residents who would support a teen curfew, then the sampling method they should use is random sampling which is in option C.

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The Euclidean Algorithm is used to find the greatest common divisor (GCD) of two numbers, this method has been applied to find the GCDs of multiple pairs of numbers.

The Euclidean Algorithm is basically a method to find the greatest common divisor (GCD) of two numbers. It is based on the principle that the GCD of two numbers is the same as the GCD of the remainder when the larger number is divided by the smaller one.

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Answer:

D) (-1, 2)

Step-by-step explanation:

See the graph below. The indicated point is not in the shaded region, hence not part of the solution.

Answer:

7(5 sqrt x^2y) or C

Step-by-step explanation:

Answer:

Step-by-step explanation:

C) f(4)=5

Answer:

Amount required is [tex]x=\frac{3}{8}[/tex]

Step-by-step explanation:

Given : Cameron used [tex]1\frac{3}{4}[/tex] ounces of hydrochloric acid in one experiment. She used [tex]2\frac{1}{8}[/tex] ounces of the acid in a second experiment.

To find : How much more acid did Cameron use in the second experiment?

Solution :

In one experiment,

Cameron used [tex]1\frac{3}{4}[/tex] ounces of hydrochloric acid

i.e [tex]1\frac{3}{4}=\frac{7}{4}[/tex]

In second experiment,

She used [tex]2\frac{1}{8}[/tex] ounces of the acid.

i.e. [tex]2\frac{1}{8}=\frac{17}{8}[/tex]

Let x amount more she used then experiment 1 in experiment 2.

So, the equation became

[tex]\frac{7}{4}+x=\frac{17}{8}[/tex]

[tex]x=\frac{17}{8}-\frac{7}{4}[/tex]

[tex]x=\frac{17-14}{8}[/tex]

[tex]x=\frac{3}{8}[/tex]

Therefore, Amount required is [tex]x=\frac{3}{8}[/tex]

Answer:

Step-by-step explanation:

[tex]\\\to\:\:\to\:\:\rm\red{40z - 20}[/tex]

[tex]\\\to\:\:\to\:\:\rm\green{20(2z-1)} \\ [/tex]

(Refer the attachment for graph representation)

[tex]\bigstar\:\:\bf\purple{y = 40z-20} \\ [/tex]

The correct equation used to create the function table provided is Option D: y = 2x, which indicates a linear relationship where y is twice the value of x.

The function table given in the question indicates the relationship between values of x and their corresponding y values. To determine which equation was used to create this table, you can compare the differences between these values. In the given pairs (1, 2), (3, 6), (4, 8), (5, 10), and (10, 20), it is observed that the y values are exactly twice the x values. Therefore, the equation representing this relationship must be y = 2x, which is a linear function with a slope of 2.

Looking at the options provided, Option D: y = 2x is the correct one, as it fits the pattern shown in the function table: for every unit increase in x, y increases by 2 units, implying a line with a slope of 2.

To find the weight of an object on Mars, we can use a direct variation equation. By finding the constant ratio from the given values, we can then solve for the weight on Mars when the weight on Earth is known. In this case, the object would weigh 40 pounds on Mars if it weighs 120 pounds on Earth.

To solve this problem, we will use the concept of direct variation. The weight of an object on Mars varies directly with its weight on Earth. This means that the weight on Mars can be found by multiplying the weight on Earth by a constant ratio. Let's represent the weight on Mars as WM and the weight on Earth as WE. According to the problem, an object that weighs 50 pounds on Mars weighs 150 pounds on Earth. This gives us the following direct variation equation:

WM = k * WE

where k is the constant ratio.

We can now substitute the given values: 50 pounds for WM and 150 pounds for WE.

50 = k * 150

To solve for k, divide both sides of the equation by 150:

k = 50 / 150

k = 1 / 3

Now that we have the value of the constant ratio, we can find the weight of an object on Mars when it weighs 120 pounds on Earth. Let's represent the weight on Mars as x and the weight on Earth as 120:

x = (1 / 3) * 120

x = 40

Therefore, the object would weigh 40 pounds on Mars.

The given points are

[tex] P(4,1) , Q(3,-5) [/tex]

First we find the slope using slope formula which is

[tex] m= \frac{y_{2} - y_{1}}{x_{2}- x_{1}} [/tex]

Substituting the values of x1,y1 and x2, y2, we will get

[tex] m = \frac{-5-1}{3-4} = 6 [/tex]

Now we use slope point form, which is

[tex] y-y_{1} = m (x-x_{1}) [/tex]

Using the values of m,x1 and y1, we will get

[tex] y-1=6(x-4) \\
y-1=6x-24 \\
y = 6x-23 [/tex]

The directional derivative of f(x, y, z) = xy + yz + zx at p(1, -1, 7) towards q(2, 4, 5) is found by first calculating the unit vector in the direction from p to q, then the gradient of f at p, and calculating the dot product of the two vectors. The directional derivative comes out to be 16/ √30.

The directional derivative of the function f(x, y, z) = xy + yz + zx at a point p(1, -1, 7) towards the point q(2, 4, 5) can be found by first getting the unit vector in the direction from p to q, then calculating the gradient of the function at p, and finally calculating the dot product of these two vectors (the unit vector and the gradient of the function).

First, calculate the unit vector u = (q - p) / |q - p|. That gives u = (1, 5, -2), and normalize it to get u = 1/ √30 (1, 5, -2).

Next, get the gradient of the function at p(1, -1, 7), which is the vector of first order derivatives. The partial derivatives of f(x, y, z) with respect to x, y, z are y+z, x+z, and x+y respectively. This gives the gradient vector at p as (-8, 8, 0).

Finally, the directional derivative is the dot product of the unit vector and the gradient, which comes out to be 1/ √30 * ((1*-8) + (5*8) + ((-2)*0)) = 16/ √30.

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Final answer:

The lengths of the diagonals in the parallelogram can be calculated using the properties of vectors and the law of cosines with the given side lengths and angle.

Explanation:

To find the length of each diagonal in a parallelogram with sides measuring 60 centimeters and 40 centimeters and one angle of 132 degrees, we can leverage the properties of vectors and the law of cosines. It is known that one diagonal is the vector sum of the sides, while the other diagonal is the vector difference of the sides.

First, let's compute the length of the shorter diagonal, which is the vector sum of the two sides:

Shorter diagonal (Ñ) = A + B

Using the law of cosines for finding the length of a diagonal, we get:

Ѳ = 60² + 40² - 2 * 60 * 40 * cos(132°)

After computing this, we find the length of the shorter diagonal.

Next, we calculate the length of the longer diagonal, which is found by subtracting the vectors of the two sides:

Longer diagonal (D) = A - B

Again using the law of cosines:

D² = 60² + 40² - 2 * 60 * 40 * cos(48°)

After computation, we find the length of the longer diagonal.

Through these calculations using the provided formulae and the law of cosines, both diagonals' lengths can be established.

The base area of the pyramid is 52900 m².

The volume of the pyramid is 2645000 m³.

where

B = base area

h = height

Therefore,

Base area = 230 × 230

Base area = 52900 m²

h = 150m

B = 52900 m²

Therefore,

V = 1 / 3 × 52900 × 150

V = 7935000 / 3

V = 2645000 m³

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There are 3 different sizes of flower boxes that have different dimensions. Then the dimensions will be 2 by 1, 2by 3, and 2 by 4.

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

Mrs. Wilton is planning a rectangular flower box for her front window.

She wants the flower box to hold exactly 16 cubic feet of soil.

Let the height of the flower boxes will be 1 foot.

Then the number of the flower boxes with the whole dimensions and different-size base will be

Volume = Area × 1

Area = 16 square ft.

There are 3 different sizes of flower boxes that have different dimensions. Then the dimensions will be

Area = 16

Area = 2 + 6 + 8

Area = 2 × 1 + 3 × 2 + 2 × 4

Then the dimensions are 2 by 1, 2by 3, and 2 by 4.

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Final answer:

The factors of the number 211 are only 1 and 211 itself, indicating that it is a prime number. Thus, the correct answer is option a (1 and 211).

Explanation:

To find all the factors of 211, you need to determine all whole numbers that can divide 211 without leaving a remainder. Starting with the smallest factor, which is always 1, and ending with the number itself, because any number is divisible by itself, we iterate through the numbers between 1 and 211 to check if they are factors.

Through inspection or a process of trial and error, we find that there are no numbers other than 1 and 211 that can divide 211 evenly, indicating that 211 is a prime number. Therefore, the correct answer to the question 'Find all the factors of 211.' is option a. 1 and 211 are the only factors of 211

Answer:

B. there is only one solution, and it is viable

Step-by-step explanation:

Answer:

it will take 2 minutes for 200 nurses to measure the blood pressure of 200 patients .

Step-by-step explanation:

We know that the work-time and number of labor and efficiency formula is given by:

[tex]Efficiency=\dfrac{Number\ of\ labor\times Time}{Work\ done}[/tex]formula is given by:

Here Number of labor=Number of nurses.

Work=Number of patients whose blood pressure were noted.

As the efficiency will be same this means that:

[tex]\dfrac{2\times 2}{2}=\dfrac{200\times x}{200}[/tex]

where x denotes the number of time taken by 200 nurses.

Hence,

[tex]x=2[/tex]

          Hence, the answer is:

                    2 minutes.

At 0.05 level of significance, there is sufficient evidence to support the alternative hypothesis that [tex]\sigma[/tex] < 6.

To test the hypothesis that the standard deviation ([tex]\sigma = 6[/tex] ) against the alternative that ( [tex]\sigma = 6[/tex] ), we can use a chi-square test.

The test statistic for a chi-square test is given by ( [tex]\chi^2 = \frac{(n-1)s2}{\sigma2}[/tex]), where:

The sample size = [tex]n[/tex]

The sample standard deviation = [tex]s[/tex]

The hypothesized standard deviation = [tex]\sigma[/tex]

The sample size, n = 20

The sample standard deviation, s = 4.51

The hypothesized standard deviation, [tex]\sigma[/tex] = 6

Substituting these values into the formula:

[tex][ \chi^2 = \frac{(20-1)(4.51)2}{62} \approx 14.13 ][/tex]

The degrees of freedom for the test is ( n - 1 = 20 - 1 = 19 ).

The test statistic for our chi-square test is ([tex][ \chi^2 = \frac{(20-1)(4.51)2}{62} \approx 14.13 ][/tex] ).

The critical value for a chi-square distribution with 19 degrees of freedom at the 0.05 level of significance is approximately 30.14.

Since our test statistic ([tex]\chi^2 = 14.13[/tex] ) is less than the critical value (30.14), we reject the null hypothesis that [tex]\sigma[/tex] = 6  at the 0.05 level of significance.

Thus, we have sufficient evidence to support the alternative hypothesis that [tex]\sigma[/tex] < 6, suggesting that the standard deviation of the time required for high school seniors to complete the standardized test is less than 6 minutes.