The shortest interval that contains 95% of the debt values is $9,492.02 to $20,507.98
How do we calculate the interval values?Given that a credit risk study found that an individual with good credit score has an average debt of $15,000 and the debt of an individual with good credit score is normally distributed with standard deviation $3,000.
Then the 95% confidence interval can be calculated as follows:
Upper limit: µ + Zσ
Lower limit: µ - Zσ
Where
µ is the mean ($15,000)Z is the z-scoreσ is the standard deviation ($3,000).The z-score corresponding to a 95% confidence interval can be found using the standard normal distribution table.
The area to the left of the z-score is 0.4750 and the area to the right is also 0.4750.
The z-score corresponding to 0.4750 can be found using the standard normal distribution table as follows:z = 1.96Therefore
Upper limit: µ + Zσ= $15,000 + 1.96($3,000) = $20,880
Lower limit: µ - Zσ= $15,000 - 1.96($3,000) = $9,120.02
The shortest interval that contains 95% of the debt values is $9,492.02 to $20,507.98.
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Use the following circle to solve for x
We know that the product of lengths of the same chord is equal to the product of the other chord intersecting it.. So;
[tex] \purple{ \mathfrak{x \times 6 = 12 \times 5}}[/tex]
[tex] \large \purple{ \mathfrak{x = \frac{12 \times 5}{6}}}[/tex]
[tex] \large \purple{ \mathfrak{x = \frac{ \cancel{12} \times 5}{ \cancel6}}}[/tex]
[tex] \large \purple{ \mathfrak{x = 2 \times 5}}[/tex]
[tex] \large \boxed{ \red{ \mathfrak{x =10}}}[/tex]
A straw that is 15cm long leans against the inside of a glass. The diameter of a glass is
5cm, and has a height of 8cm. How far past the edge of the glass would the straw extend?
Round your answer to the nearest tenth.
The straw will extend past the edge of the glass in a straight line. To find the answer, subtract the diameter of the glass (5cm) from the length of the straw (15 cm): 15 cm - 5 cm = 10 cm. This is the distance the straw will extend past the edge of the glass. To round to the nearest tenth, round 10.0 up to 10.1. Therefore, the straw will extend past the edge of the glass 10.1 cm.
Kate plans to save 5% of her income. She just earned 80.Howmuchmoneyshouldsheputintosavings?
$4
Multiply 80 by 0.05 to get the answer of $4
Answer:
4 (insert the currency needed)
Step-by-step explanation:
To find our answer we have to find 5% of Kate's wages and to do this we have to do 5 divided by 100. Then that answer is multiplied by 80!
5 ÷ 100 = 0.0580 × 0.05 = 4This means she has to put £4 into her savings!
Hope this helps, have a lovely day! :)
an inner city revitalization zone is a rectangle that is twice as long as it is wide. the width of the region is growing at a rate of 32 m per year at a time when the region is 220 m wide. how fast is the area changing at that point in time?
The area is changing at a rate of 28,160 m²/year at that point in time.
The area of the rectangular region is given by:
A = lw
Where l is the length of the rectangular region and w is the width of the rectangular region.
The width of the rectangular region is given to be 220 m. Therefore, we have the width w = 220 m. The length l of the rectangular region can be found knowing that it is twice as long as it is wide. Therefore, the length of the rectangular region is given by:
l = 2w
l = 2 x 220
l = 440
Therefore, the length l of the rectangular region is 440 m.
At the given point in time, the width of the rectangular region is growing at a rate of 32 m per year. Therefore, we have the rate of change of the width dw/dt to be 32 m per year. We need to find how fast the area of the rectangular region is changing at that point in time. Therefore, we need to find the rate of change of the area of the rectangular region dA/dt.
A = lw
dA/dt = w dl/dt + l dw/dt
dA/dt = 220 d/dt(2w) + 440 dw/dt
dA/dt = 220 x 2 dw/dt + 440 dw/dt
dA/dt = 880 dw/dt
Substitute the value of dw/dt to get:
dA/dt = 880 x 32
dA/dt = 28,160 m²/year
Therefore, the area of the rectangular region has a rate of change of 28,160 m² per year at that point in time.
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What’s -9.1 times 3.75
How do you do this I need help please
Answer:
30,000 grams
Step-by-step explanation:
multiply the 30KG by 1,000 (that is the conversion) and you get 30,000g
Answer:
hi I'm really sorry I can't help
Smoothie Activity
6. Using the relative frequency table, create a segmented bar graph by employee type using technology or by hand. If using Excel technology the columns may need to be switched after inserting the chart. Click on the chart and the "Chart Design" ribbon will pop up. Then select "Switch Row/Column." (10 points)
By answering the presented question, we may conclude that I used the following procedures to produce this graph.
What is graphs?Mathematicians use graphs to visually display or chart facts or values in order to express them coherently. A graph point usually represents a connection between two or more items. A graph, a non-linear data structure, is made up of nodes (or vertices) and edges. Glue the nodes, also known as vertices, together. This graph contains vertices V=1, 2, 3, 5, and edges E=1, 2, 1, 3, 2, 4, and (2.5), (3.5). (4.5). Statistical graphs (bar graphs, pie graphs, line graphs, and so on) are graphical representations of exponential development. a logarithmic graph shaped like a triangle.
I used the following procedures to produce this graph:
I classified the personnel as full-time, part-time, and temporary.
I estimated the proportion of employees who assessed the company's work-life balance as "very good" or "excellent" for each employee category, as well as the percentage who rated it as "good" or "fair/poor."
I used the following procedures to produce this graph:
I classified the personnel as full-time, part-time, and temporary.
I estimated the proportion of employees who assessed the company's work-life balance as "very good" or "excellent" for each employee category, as well as the percentage who rated it as "good" or "fair/poor."
I made the segmented bar graph using these percentages.
The graph was made using Excel technology. You may make a similar graph with Excel or any other software that supports segmented bar graphs.
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the dog eats 8 ounces of dog food each day his owner bought 28 pound bag at the 8 ounces cost $3.50 so how much did the owner spend for 28 bag
Answer:
$196
Step-by-step explanation:
1 lb = 16oz
28 lbs x 16 = 448 ozs (in 28 lb bag)
448/8 = 56 (8 oz portions)
56 x $3.50= $196
A man sells an article at rs 600and makes a profit of 20%. Calculate his profit percentage
Answer:
120
Step-by-step explanation:
20 percent of 600 is 120 so he will get 120
Jayden evaluated the expression a + (2 + 1. 5) for a = 14. He said that the value of the expression was 8. 5. Select all the statements that are true. Jayden's solution is incorrect. Jayden added inside the parentheses before dividing. Jayden substituted the wrong value for a. Jayden divided 14 by 2 and then added 1. 5. Jayden added inside the parentheses before multiplying.
It is true that Jayden's solution is incorrect. It is false that Jayden added inside the parentheses before dividing.
It is false that Jayden substituted the wrong value for a. It is true that Jayden divided 14 by 2 and then added 1. 5. Jayden added inside the parentheses before multiplying.
1) The correct solution is
Given,
a ÷ (2 + 1. 5)
Substituting the value of a which is 14
= 14 ÷ (2 + 1. 5)
= 14 ÷ 3.5
= 4
2) As there is no term which needs to be divided so, the second statement is false.
3) Jayden didn't substitute the wrong value of a he just solved the given expression without considering the bracket and divided the 14 which is the value of a by 2.
4) Jyaden divided 14 by 2 and then added 1. 5. Jayden added inside the parentheses before multiplying.
i.e. a ÷ (2 + 1. 5)
14 ÷ 2 + 1. 5
7+1.5
8.5
This is the way Jayden solved the equation due to which he arrived at the wrong solution.
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The Correct question is as below
Jayden evaluated the expression a ÷ (2 + 1.5) for a = 14. He said that the answer was 8.5. Choose True or False for each statement.
1. Jayden's solution is incorrect.
2. Jayden added in the parentheses before dividing.
3. Jayden substituted the wrong value for a.
4. Jayden divided 14 by 2 and added 1.5
One number is 13 less than another number. Let x represent the greater number. What is the sum of these two numbers?
Answer:
2x - 13
Step-by-step explanation:
If x represents the greater number, then the other number is x - 13. The sum of these two numbers is:
x + (x - 13) = 2x - 13
a data set consists of the data given below plus one more data point. when the additional point is included in the data set the sample mean of the resulting data set is 32.083. what is the value of the additional data point?
The value of the additional data point is [tex]$19.17$[/tex].
What is the value of the additional data point?Let us first find the mean of the given data:
[tex]Mean = \frac{\sum_{i=1}^{n} x_i}{n}=\frac{39 + 45 + 43 + 42 + 44}{5}= 42.6[/tex]
Now let's find the value of the additional data point. Let the value of the additional data point be x. Therefore, the new sum of data is
[tex]$(39+45+43+42+44+x)$[/tex].
Total numbers of data are 6 (five given in the set and one additional data point).So, the mean of the resulting data set is given by:
[tex]32.083 = \frac{(39+45+43+42+44+x)}{6}[/tex]
Multiplying both sides of the equation by 6 we get:
[tex]6 \times 32.083 = (39+45+43+42+44+x)[/tex]
We have the value of [tex]$39+45+43+42+44$[/tex] which is [tex]$213$[/tex].
Therefore, substituting all the values, we get:
[tex]193.83 + x = 213[/tex]
On subtracting [tex]$193.83$[/tex] from both sides, we get the value of
[tex]x. x = 213 - 193.83 = 19.17[/tex]
Therefore, the value of the additional data point is [tex]$19.17$[/tex]
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what is the messure or the vertex angle of an isosceles triangle if one of its base angle measures 16 degrees
Guidance Missile System A missile guidance system has seven fail-safe components. The probability of each failing is 0.2. Assume the variable is binomial. Find the following probabilities. Do not round intermediate values. Round the final answer to three decimal places, Part: 0 / 4 Part 1 of 4 (a) Exactly two will fail. Plexactly two will fail) = Part: 1/4 Part 2 of 4 (b) More than two will fail. P(more than two will fail) = Part: 214 Part: 2/4 Part 3 of 4 (c) All will fail. P(all will fail) = Part: 3/4 Part 4 of 4 (d) Compare the answers for parts a, b, and c, and explain why these results are reasonable. Since the probability of each event becomes less likely, the probabilities become (Choose one smaller larger Х 5
The probability of all will fail is the lowest.
The given problem states that a missile guidance system has seven fail-safe components, and the probability of each failing is 0.2. The given variable is binomial. We need to find the following probabilities:
(a) Exactly two will fail.
(b) More than two will fail.
(c) All will fail.
(d) Compare the answers for parts a, b, and c, and explain why these results are reasonable.
(a) Exactly two will fail.
The probability of exactly two will fail is given by;
P(exactly two will fail) = (7C2) × (0.2)2 × (0.8)5
= 21 × 0.04 × 0.32768
= 0.2713
Therefore, the probability of exactly two will fail is 0.2713.
(b) More than two will fail.
The probability of more than two will fail is given by;
P(more than two will fail) = P(X > 2)
= 1 - P(X ≤ 2)
= 1 - (P(X = 0) + P(X = 1) + P(X = 2))
= 1 - [(7C0) × (0.2)0 × (0.8)7 + (7C1) × (0.2)1 × (0.8)6 + (7C2) × (0.2)2 × (0.8)5]
= 1 - (0.8)7 × [1 + 7 × 0.2 + 21 × (0.2)2]
= 1 - 0.2097152 × 3.848
= 0.1967
Therefore, the probability of more than two will fail is 0.1967.
(c) All will fail.
The probability of all will fail is given by;
P(all will fail) = P(X = 7) = (7C7) × (0.2)7 × (0.8)0
= 0.00002
Therefore, the probability of all will fail is 0.00002.
(d) Compare the answers for parts a, b, and c, and explain why these results are reasonable.
The probability of exactly two will fail is the highest probability, followed by the probability of more than two will fail. And, the probability of all will fail is the lowest probability. These results are reasonable since the more the number of components that fail, the less likely it is to happen. Therefore, it is reasonable that the probability of exactly two will fail is higher than the probability of more than two will fail, and the probability of all will fail is the lowest.
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A student takes a multiple-choice test that has 10 questions. Each question has four choices. The student guesses randomly at each answer. Round the answers to three decimal places Part 1 of2 (a) Find P(5) P(5)- Part 2 of2 (b) Find P(More than 3) P(More than 3)
A student attempts a 10-question multiple-choice test where each question presents four options, and the student makes random guesses for each answer. So the probability of (a) P(5)= 0.058 and (b) P(More than 3)= 0.093.
Part 1: Calculation of probability of getting 5 questions correct
(a) P(5)The formula used to find the probability of getting a certain number of questions correct is:
P(k) = (nCk)pk(q(n−k))
Where, n = total number of questions
(10)k = number of questions that are answered correctly
p = probability of getting any question right = 1/4
q = probability of getting any question wrong = 3/4
P(5) = P(k = 5) = (10C5)(1/4)5(3/4)5= 252 × 0.0009765625 × 0.2373046875≈ 0.058
Part 2: Calculation of probability of getting more than 3 questions correct
(b) P(More than 3) = P(k > 3) = P(k = 4) + P(k = 5) + P(k = 6) + P(k = 7) + P(k = 8) + P(k = 9) + P(k = 10)
P(k = 4) = [tex]10\choose4[/tex](1/4)4(3/4)6 = 210 × 0.00390625 × 0.31640625 ≈ 0.02
P(k = 5) = [tex]10\choose5[/tex](1/4)5(3/4)5 = 252 × 0.0009765625 × 0.2373046875 ≈ 0.058
P(k = 6) = [tex]10\choose6[/tex](1/4)6(3/4)4 = 210 × 0.0002441406 × 0.31640625 ≈ 0.012
P(k = 7) = [tex]10\choose7[/tex](1/4)7(3/4)3 = 120 × 0.00006103516 × 0.421875 ≈ 0.002
P(k = 8) = [tex]10\choose8[/tex](1/4)8(3/4)2 = 45 × 0.00001525878 × 0.5625 ≈ 0.001
P(k = 9) = [tex]10\choose9[/tex](1/4)9(3/4)1 = 10 × 0.000003814697 × 0.75 ≈ 0.000
P(k = 10) = [tex]10\choose10[/tex](1/4)10(3/4)0 = 1 × 0.0000009536743 × 1 ≈ 0
P(More than 3) = 0.020 + 0.058 + 0.012 + 0.002 + 0.001 + 0.000 + 0≈ 0.093
Therefore, the probabilities of the given situations are: P(5) ≈ 0.058, P(More than 3) ≈ 0.093.
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How much was one movie ticket in 2008
Answer: 7 dollars 18 cents.
Step-by-step explanation:
25 out of 68 students have vanilla ice cream and the rest have chocolate. What is the ratio of the number of students who have vanilla to the total number of students?
Answer: The total number of students is the sum of the number of students who have vanilla and those who have chocolate:
Total = 25 + (68 - 25) = 43
The ratio of the number of students who have vanilla to the total number of students is:
Vanilla : Total = 25 : 43
This ratio cannot be simplified any further because 25 and 43 do not have any common factors other than 1. Therefore, the ratio of the number of students who have vanilla to the total number of students is:
25 : 43
Step-by-step explanation:
Calculate the amount of interest on $4,000. 00 for 4 years, compounding daily at 4. 5 % APR. From the Monthly Interest Table use $1. 197204 in interest for each $1. 00 invested
The amount of interest earned on $4,000.00 for 4 years, compounding daily at 4.5% APR, is $1,064.08.
To calculate the amount of interest on $4,000.00 for 4 years, compounding daily at 4.5% APR, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.
In this case, we have P = $4,000.00, r = 0.045, n = 365 (since interest is compounded daily), and t = 4. Plugging these values into the formula, we get:
A = $4,000.00(1 + 0.045/365)^(365*4)
A = $4,000.00(1.0001234)^1460
A = $4,889.68
The final amount is $4,889.68, which means that the interest earned is:
Interest = $4,889.68 - $4,000.00 = $889.68
We are given that the monthly interest table shows that $1.197204 in interest is earned for each $1.00 invested. Therefore, to find the interest earned on $4,000.00, we can multiply the interest earned by the factor:
$1.197204 / $1.00 = 1.197204
Interest earned = $889.68 x 1.197204 = $1,064.08
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Complete the recursive formula of the arithmetic sequence -16, -33, -50, -67,. −16,−33,−50,−67,. Minus, 16, comma, minus, 33, comma, minus, 50, comma, minus, 67, comma, point, point, point. C(1)=c(1)=c, left parenthesis, 1, right parenthesis, equals
c(n)=c(n-1)+c(n)=c(n−1)+c, left parenthesis, n, right parenthesis, equals, c, left parenthesis, n, minus, 1, right parenthesis, plus
The following is the recursive formula for the arithmetic sequence in this issue:
c(1) = -16.
c(n) = c(n - 1) - 17.
An arithmetic sequence is a series of numbers where each term is obtained by adding a fixed constant, known as the common difference, to the previous term. For example, in the sequence 2, 5, 8, 11, 14, 17, each term is obtained by adding 3 to the previous term.
The formula for finding the nth term of an arithmetic sequence is: a(n) = a(1) + (n-1)d, where a(1) is the first term, d is the common difference, and n is the term number. For example, to find the 10th term of the sequence 2, 5, 8, 11, 14, 17, we would use the formula a(10) = 2 + (10-1)3 = 29. Arithmetic sequences have many practical applications, such as in finance, where they can be used to calculate the interest earned on an investment over time.
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Use the following function to find d(0)
d(x)=-x+-3
d(0)=
Answer:
d(0) = -3
Step-by-step explanation:
d(x) = -x + -3 d(0)
d(0) = 0 - 3
d(0) = -3
So, the answer is d(0) = -3
what is the value of y in the solution to the system of equations below.
y=-x+6
2x-y=-9
Answer:
I gave a couple solutions as I wasn't sure if you were asking for graphing purposes or substituting y=-x+6 into the second equation 2x-y=-9. So I gave both solutions just in case.
for the first equation y=-x+6, y intercept is (0,6)
for equation two 2x-y=-9, y intercept is (0,9)
In both of the equations the x value is 1.
Solving for y without graphing. Y=9+2x
and x=-1
Step-by-step explanation:substitute i
HOWEVER, if you are saying that the top equation is the value of y, then you substitute it into the bottom equation. 2x--x+6=-9 which would be x=-5
It really depends on what is expected of the question. I wasn't sure which one, so I gave a couple different approaches. If you could give more information, such as, are you graphing, that would be great. I'll keep an eye out for any comments.
y is inversely proportional to the square of x. It is given that y = 8 for a particular value of x. k= When x increases by 300%, find the new value of y,
Answer:
1/2 or .5
Step-by-step explanation:
If y is inversely proportional to the square of x, we can express this relationship using the formula:
y = k/x^2
where k is a constant of proportionality. We are told that y = 8 for a particular value of x, so we can substitute these values into the equation:
8 = k/x^2
To find the value of k, we can solve for it:
k = 8x^2
Now we are asked to find the new value of y when x increases by 300%. This means that the new value of x will be 4 times the original value (since an increase of 300% means an increase by a factor of 3, and we need to add the original value to get the new value). So we can substitute 4x for x in our equation:
y = k/(4x)^2 = k/16x^2
We already know the value of k, so we can substitute it in and simplify:
y = (8x^2)/(16x^2) = 1/2
Therefore, the new value of y is 1/2 when x increases by 300%.
LetR=[0, 4]×[−1, 2]R=[0, 4]×[−1, 2]. Create a Riemann sum by subdividing [0, 4][0, 4] into m=2m=2 intervals, and [−1, 2][−1, 2] into n=3n=3 subintervals then use it to estimate the value of ∬R (3−xy2) dA∬R (3−xy2) dA.Take the sample points to be the upper left corner of each rectangle
The Riemann sum is:Σ(3-xᵢₖ*yᵢₖ²)ΔA, where i=1,2 and k=1,2,3.
We can create a Riemann sum to estimate the value of the double integral ∬R (3-xy²) dA over the rectangular region R=[0, 4]×[-1, 2] by subdividing [0, 4] into m=2 intervals and [-1, 2] into n=3 intervals. Then we can evaluate the function at the upper left corner of each subrectangle, multiply by the area of the rectangle, and sum all the results.
The width of each subinterval in the x-direction is Δx=(4-0)/2=2, and the width of each subinterval in the y-direction is Δy=(2-(-1))/3=1. The area of each subrectangle is ΔA=ΔxΔy=2*1=2.
Therefore, the Riemann sum is:
Σ(3-xᵢₖ*yᵢₖ²)ΔA, where i=1,2 and k=1,2,3.
Evaluating the function at the upper left corner of each subrectangle, we get:
(3-0*(-1)²)2 + (3-20²)2 + (3-21²)2 + (3-41²)*2 = 2 + 6 + 2 + (-22) = -12.
Thus, the estimate for the double integral is -12.
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How do you find height when you are doing volume with cubic units?
Answer:calculate the cube root of a cube's volume.
Step-by-step explanation:
suppose you start at the origin, move along the x-axis a distance of 7 units in the positive direction, and then move downward a distance of 6 units. what are the coordinates of your position? (x, y, z)
The coordinates of your position If we start at the origin, we are moving only along the x-axis of a distance of 7 units in positive direction and then only in the negative y-axis direction and z-coordinate is zero are (7,-6,0).
The origin is the point in space that has a position of (0, 0, 0), which represents the point where the x, y, and z axes intersect.
The first step is to move 7 units in the positive x direction. The positive x direction is the direction in which x values increase. Therefore, we move to the right along the x-axis to the point (7, 0). This means that we have moved 7 units along the x-axis, and our position is now (7, 0, 0).
The second step is to move downward a distance of 6 units. Since we are not moving in the x direction, we are only changing our position along the y-axis. Moving downward in the y direction means decreasing our y-coordinate. Therefore, we move 6 units downward from our current position to the point (7, -6, 0).
Therefore, the coordinates of our position are (7, -6, 0)
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What geometric shapes can you draw that have exactly one pair of parallel sides? Use pencil and paper. Sketch examples for as many different types of shapes as you can. Use appropriate marks to show the pairs of parallel sides.
A. regular pentagon
B. square
C. Trapezoid
D. parallelogram
A coffee maker is on sale for 45$. If the sales tax is 7%, how much will the buyer spend altogether?
Answer: 38 I think if it's not right I'm sorry I'm bad at math that's like the only thing I suck at
Step-by-step explanation:
Will make you brainlist!
Answer:
x = -2 , y = 2
Step-by-step explanation:
label your equations (1) and (2) the question mention to use elimination method and make x the same for both. To do that multiply equation (1) by 2. than label it (3)so 3x becomes 6x adding the equation (2)+(3) cancels out -6x and 6x so you can find value of yuse value of y to find xhope this helps :)
Please help it’s for tmr
Leo has a number of toy soldiers between 27 and 54. If you want to group them four by four, there are none left, seven by seven, 6 remain, five by five, 3 remain. How many toy soldiers are there?
The answer is 48 but I need step by step explanation
Hence, 28 toy soldiers are the correct answer.
In mathematics, how is a group defined?A group in mathematics is created by combining a set with a binary operation. For instance, a group is formed by a set of integers with an arithmetic operation and a group is also formed by a set of real numbers with a differential operator.
Let's refer to the quantity of toy soldiers as "x".
We are aware that x is within the range of 27 and 54 thanks to the problem.
x can be divided by 4 without any remainders.
The residual is 6 when x is divided by 7.
The leftover after dividing x by five is three.
These criteria allow us to construct an equation system and find x.
Firstly, we are aware that x can be divided by 4 without any residual. As a result, x needs to have a multiple of 4. We can phrase this as:
x = 4k, where k is some integer.
Secondly, we understand that the remaining is 6 when x is divided by 7. This can be stated as follows:
x ≡ 6 (mod 7)
This indicates that x is a multiple of 7 that is 6 more than. We can solve this problem by substituting x = 4k:
4k ≡ 6 (mod 7)
We can attempt several values of k until we discover one that makes sense for this equation in order to solve for k. We can enter k in to equation starting using k = 1, as follows:
4(1) ≡ 6 (mod 7)
4 ≡ 6 (mod 7)
It is not true; thus we need to attempt a next value for k. This procedure can be carried out repeatedly until the equation is satisfied for all values of k.
k = 2:
4(2) ≡ 6 (mod 7)
1 ≡ 6 (mod 7)
k = 3:
4(3) ≡ 6 (mod 7)
5 ≡ 6 (mod 7)
k = 4:
4(4) ≡ 6 (mod 7)
2 ≡ 6 (mod 7)
k = 5:
4(5) ≡ 6 (mod 7)
6 ≡ 6 (mod 7)
k = 6:
4(6) ≡ 6 (mod 7)
3 ≡ 6 (mod 7)
k = 7:
4(7) ≡ 6 (mod 7)
0 ≡ 6 (mod 7)
We have discovered that the equation 4k 6 (mod 7) is fulfilled when k = 7. Thus, we can change k = 7 to x = 4k to determine that:
x = 4(7) = 28
This indicates that there are 28 toy troops. Yet we also understand that the leftover is 3 when x is divided by 5. We don't need to take into account any other values of x because x = 28 satisfies this requirement.
28 toy soldiers are the correct response.
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In a 7-sided figure, three of the angles are equal
and each of the other four angles is 150 greater
than each of the first three. Find the angles.
The sum of the angles of an N-sided convex figure is (n-2)*180 - a simple proof of which is just to decompose the figure into triangles, each of which has all of its vertices the same as three of the vertices of the original figure. (Cut a quadrilateral into two triangles along a diagonal, for instance).
So, a 7-sided figure has angles totaling 5*180 = 900. Now set up a simple equation:
3x + 4(x+15) = 900
7x + 60 = 900
7x = 840
x = 120
The figure has three angles of 120 degrees, and four angles of 135 degrees.
