For Q1-Q4 use mathematical induction to prove the statements are correct for ne Z+(set of positive integers). 4) Prove that for all integers n ≥ 2 n2>n+1.
Answers
By mathematical induction, we have shown that for all integers n ≥ 2, \(n^2 > n + 1\).
To prove the statement for all integers n ≥ 2, we will use mathematical induction.
Base Case
First, we will check the base case when n = 2.
For n = 2,
we have \(2^2 = 4\) and 2 + 1 = 3.
Clearly, 4 > 3, so the statement holds true for the base case.
Inductive Hypothesis
Assume that the statement holds true for some arbitrary positive integer k ≥ 2, i.e., \(k^2 > k + 1.\)
Inductive Step
We need to prove that the statement also holds true for the next integer, which is k + 1.
We will show that \((k + 1)^2 > (k + 1) + 1\).
Expanding the left side, we have \((k + 1)^2 = k^2 + 2k + 1\).
Substituting the inductive hypothesis, we have \(k^2 > k + 1\).
Adding \(k^2\) to both sides, we get \(k^2 + 2k > 2k + (k + 1)\).
Simplifying, we have \(k^2 + 2k > 3k + 1\).
Since k ≥ 2, we know that 2k > k and 3k > k.
Therefore, \(k^2 + 2k > 3k + 1 > k + 1\).
Thus,\((k + 1)^2 > (k + 1) + 1\).
Conclusion
By mathematical induction, we have shown that for all integers n ≥ 2, \(n^2 > n + 1\).
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Related Questions
if the coefficient of correlation is 0.4, the percentage of variation in the dependent variable explained by the variation in the independent variable question 4 options: is 40% is 16% is 4% cannot be in dollars
Answers
If the coefficient of correlation is 0.4, the percentage of variation in the dependent variable explained by the variation in the independent variable is 16%. The correct option is (B)
The percentage of variation in the dependent variable explained by the variation in the independent variable can be determined using the coefficient of correlation.
The coefficient of correlation, often denoted as "r," measures the strength and direction of the linear relationship between the variables.
For the percentage of variation explained, we square the coefficient of correlation (r) and multiply it by 100.
This represents the proportion of the variation in the dependent variable that can be attributed to the variation in the independent variable.
We have that the coefficient of correlation is 0.4, we calculate the percentage of variation explained:
Percentage of variation explained = (r^2) * 100
= (0.4^2) * 100
= 0.16 * 100
= 16%
Therefore, the correct answer is: B) 16%.
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PLZZZZZZZZ HLPPPPPPPPP MEEEEEEEEEEEE
Answers
Answer:
D.8
Step-by-step explanation:
A cube's volume is basically the side length multiplied by itself 3 times, so if it was increased by a factor of 2, it would mean 2x2x2 which equals 8.
neep help with iready
Answers
Answer:
I can help you.
Step-by-step explanation:
Have a positive day
If the coordinates of A are (6, -2) and the midpoint of AB is (3,0), determine the coordinates of the other endpoint B. (4.5, -1) (0, 2) (2, 1.5) (3.0.5)
Answers
Let's begin by listing out the information given to us:
\(\begin{gathered} AB=(6,-2) \\ Midpoint(AB)=(3,0) \end{gathered}\)We are to calculate for the coordinates of B
The formula for calculating the midpoint is given by:
\(\begin{gathered} M=(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}) \\ (3,0)=(\frac{6+x_B}{2},\frac{-2+y_B}{2}) \\ 3=\frac{6+x_B}{2},0=\frac{-2+y_B}{2} \\ 3=\frac{6+x_B}{2}\Rightarrow3\cdot2=6+x_B\Rightarrow6=6+x_B_{} \\ x_B=6-6=0_{} \\ x_B=0 \\ \\ 0=\frac{-2+y_B}{2}\Rightarrow2\cdot0=-2+y_B\Rightarrow0=-2+y_B \\ y_B=0+2=2 \\ y_B=2 \\ \\ \therefore B(x,y)=(0,2) \end{gathered}\)help i m not smart at all see alot
Answers
Answer:
-3
Step-by-step explanation:
You have to bring -16 to the other side by adding.
It would make it: -7a=21
Then you hve to diving the -7.
-7a/-7=21/-7
That would equal -3
Answer:
A= -3
Step-by-step explanation:
-7a - 16 = 5
( -7a - 16) + 16 = 5 + 16
-7a - 16 + 16 = 21
-7a = 21
7a/7 = - 21/7
a = -3
Hope this helps you
Brainless please
if p is the number of ways you can place 5 queens on a chessboard randomly and q is the number of ways you can place 5 queens column-wise (as used in backtracking) randomly, then what is the ratio of p / q approximately?
Answers
The factorial or ratio of p / q approximately is 1199.5.
Calculate p (number of ways to place 5 queens randomly on a chessboard):
A chessboard has 64 squares. We can place the first queen in any of the 64 squares, the second queen in any of the remaining 63 squares, and so on.
So, the number of ways to place 5 queens randomly is: p = 64 * 63 * 62 * 61 * 60 However, since the order in which we place the queens does not matter, we need to divide by the number of ways to arrange the 5 queens,
which is 5! (5 factorial). p = (64 * 63 * 62 * 61 * 60) / (5 * 4 * 3 * 2 * 1) 2=8,048,640
Calculate q (number of ways to place 5 queens column-wise):
We need to find the value of q. Here, we place 5 queens column-wise. In the first column, there are 8 possible squares to place the first queen. In the second column, there are only 7 possible squares left because one square is already blocked by the first queen.
Similarly, in the third column, there are only 6 possible squares left, and so on. Therefore, the total number of ways to place the 5 queens column-wise is: 8 × 7 × 6 × 5 × 4= 6,720
Calculate the ratio p / q:
Now that we have values for p and q, we can find the ratio p / q. p / q ≈ ((64 * 63 * 62 * 61 * 60) / (5 * 4 * 3 * 2 * 1)) / (8 × 7 × 6 × 5 × 4) ≈ 1199.5
Therefore, the ratio of p / q approximately is 1199.5.
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Can someone help me with this math homework please!
Answers
Here, we may use Geometric Progression
The sequence follows as:
27, 18,12,...
The first term (a)= 27
27/18 = 18/12
common ratio = 1.5
Let fourth term be x
Therefore , 12/x = 1.5
=> x= 12/1.5 = 8
Answer:
8
Step-by-step explanation:
Let,
t1 = 27
t2 = 18
t3 = 12
t2 / t1
= 27 / 18
= 3 / 2
t3 / t2
= 18 / 12
= 3 / 2
Common ratio = 3 / 2 = 1.5
t4 = t3 / 1.5
= 12 / 1.5
= 12 x 2 / 3
t4 = 8
For each peak, the ball decreases it's height from the ground by 1 / 1.5 times, Using sequence, we can find the height of ball from the ground for different levels of peaks.
Which of the following equations is equivalent to 4x -y=3`?
Answers
Answer:
x-intercept = (3/4,0)
y'intercept = (0,-3)
Step-by-step explanation:
Not sure what conversion you need but here are the intercepts
Describe the translation. y=(x−2)2+5 → y=(x+2)2−3 A. T<4,8> B. T<4,−8> C. T<−4,−8> D. T<−4,8>
Answers
The equation y=(x−2)²+5 has been translated to y=(x+2)²−3 by shifting 4 units to the left (horizontal shift) and 8 units downward (vertical shift).
The correct answer is D.
To determine the translation that maps the equation
y=(x−2)²+5 to y=(x+2)²−3,
we can observe the changes in the equation and the corresponding shifts.
Comparing the two equations, we notice the following transformations:
The x term has changed from (x-2)² to (x+2)², indicating a horizontal shift.
The constant term has changed from +5 to -3, indicating a vertical shift.
Let's analyze each transformation separately:
Horizontal shift:
In the original equation, y=(x−2)²+5, the vertex of the parabola is located at (2, 5).
In the transformed equation, y=(x+2)²−3, the vertex should be located at the same x-coordinate but with a different y-coordinate.
Since the transformation involves adding 2 to the x-values, the shift is to the left by 2 units.
Therefore, the vertex of the transformed equation is (-2, ?), where ? represents the y-coordinate.
Vertical shift:
In the original equation, the constant term is +5, indicating a vertical shift upward by 5 units from the vertex.
In the transformed equation, the constant term is -3, indicating a vertical shift downward by 3 units from the new vertex.
Based on the given options, we need to find a translation that satisfies both the horizontal and vertical shifts described above.
Looking at the options:
A. T<4,8>
B. T<4,−8>
C. T<−4,−8>
D. T<−4,8>
From these options, the only one that satisfies both the horizontal and vertical shifts is option D: T<−4,8>.
This means the equation y=(x−2)²+5 has been translated to y=(x+2)²−3 by shifting 4 units to the left (horizontal shift) and 8 units downward (vertical shift).
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solve the quadratic linear below :
6x²-x-2=0
Answers
Answer:
x = 3/2; x = -1/2
Step-by-step explanation:
6x² - x - 2 = 0
⇔ 6x² - 4x + 3x - 2 = 0
⇔ (6x² - 4x) + (3x - 2) = 0
⇔ 2x(3x - 2) + (3x - 2) = 0
⇔ (3x - 2)(2x + 1) = 0
⇔\(\left \{ {{3x - 2=0} \atop {2x + 1=0}} \right.\)
⇔\(\left \{ {{3x=2} \atop {2x=-1}} \right.\)
⇔\(\left \{ {{x=2/3} \atop {x=-1/2}} \right.\)
What is 6z−2y−8z+5y−4+z written in simplest form?
Answers
Answer:
First answer 3y-z-4
Step-by-step explanation:
Answer:
D. 3y-z-4
Step-by-step explanation:
use undetermined coefficients to find the general solution for y'' 4y = 4x^2 10e^-x
Answers
Combining the complementary and particular solutions, the general solution is y(x) = C1e²ˣ+ C2e⁻²ˣ+ Ax² + Bx + C + De⁻ˣ.
To find the general solution for y'' - 4y = 4x² + 10e⁻ˣ using undetermined coefficients, we first identify the complementary and particular solutions.
The complementary solution, yc(x), is obtained from the homogeneous equation y'' - 4y = 0. This leads to the characteristic equation r² - 4 = 0, which has roots r1 = 2 and r2 = -2. Therefore, yc(x) = C1e²ˣ + C2e⁻²ˣ.
For the particular solution, yp(x), we assume a form of Ax² + Bx + C + De⁻ˣ. Differentiate yp(x) twice and substitute it into the given equation. Then, solve for the undetermined coefficients A, B, C, and D.
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solve this problem
may be easy or nt
Answers
Answer:
Mark me as brainliest ❤️
Q1) If Q is directly proportional to P and Q = 28 when P = 4, (i) express Q in terms of P, (ii) find the value of Q when P = 5, (iii) calculate the value of P when Q = 42.
Answers
Answer: i) Q=7*P ii) if P=5 => Q=35 iii) if Q=42 => P=6
Step-by-step explanation:
Q is directly prportional to P that means that
Q/P=k=const ( where k is the prportionality coefficient)
So k= 28/4=7
So i) Q= 7*P
ii) Q=5*7=35 if P=5 =>Q=35
iii) P=42/7=6 if Q=42 => P=6
FGH, m∠G= 30° and m∠H= 100°. What is the measure of ∠F?
Answers
Answer:
∠F = 50°
Step-by-step explanation:
The sum of all three angles in a triangle = 180°
∠F = 180° - 30° - 100° = 50°
PLEASE HELP WILL GIVE BRAINLEST! Which trigonometric function does the graph below represent?
Answers
Answer:
Answer In this Picture.
Mario is 63 years older than his grandson Jonas. How many years ago was Mario's age 10 times Jonas's age if Mario's age is currently four times Jonas's age?
Answers
Answer:
14 years ago
Explanation:
Let Mario's age be m, Jonas age be j.
From, Mario is 63 years older than Jonas
Equation 1: m = j + 63
From, Mario's age is four times Jonas age
Equation 2: m = 4j
Solve them by substitution:
m = j + 63
4j = j + 63
4j - j = 63
3j = 63
j = 21
Find Mario's age:
m = j + 63
m = 21 + 63
m = 84
Follow this steps:
m - y = 10(j - y)
84 - y = 10(21 - y)
84 - y = 210 - 10y
-y + 10y = 210 - 84
9y = 126
y = 14
The answer is 14 years ago.
Let Mario's age be M, and Jonas' age be J.
The formed equations :
M = J + 63M = 4JTo find their current ages, substitute the 2nd equation in the 1st equation.
4J = J + 63
3J = 63
J = 21
M = 4(21)
M = 84
For Mario to be 10 times Jonas' age, he has to be an age that is a multiple of 10.
Case (1) : Mario's age = 80
Subtract 4 from both their ages.
Mario : 84 - 4 = 80
Jonas : 21 - 4 = 17
17 × 10 ≠ 80
Case (2) : Mario's age = 70
Subtract 14 from both their ages.
Mario : 84 - 14 = 70
Jonas : 21 - 14 = 7
7 × 10 = 70
samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. when she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. in all, it took her 44 minutes to reach her friend's house. in kilometers rounded to the nearest tenth, how far did samia walk?
Answers
Samia walked 2.8 km to reach her friend's house if she walked at the speed of 5 kilometers per hour.
Let's consider the distance that Samia had to travel in total is 2x. We know that the length of the bike ride and how far she walked are equal, so they are both (2x/2) or x.
She bikes at an average speed of 17 kph, so she travels the distance she bikes in (x/17) hours. She walks at a rate of 5 kph, so she travels the distance she walks in (x/5) hours. Now we use an algebraic expression as follows;
The total time is;
(x/17) + (x/5) = 22x/85.
This is equal to (44/60) = (11/15) of an hour.
Now solving for x;
22x ÷ 85 = 11 ÷ 15
22x = 85 × (11 ÷ 15)
22x = 62.33
x = 62.33 ÷ 22
x = 2.8
Since x is the distance of how far Samia traveled by both walking and biking, and we need to know how far Samia walked to the nearest tenth, we have the result that Samia walked about 2.8 Km.
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Please help, please, 20 PTS and Brainly to whoever gets it right
Answers
Answer: 5/12 inches taller
anyone know the answer to this question
Answers
Answer:
A) There is a vertical shift
Step-by-step explanation:
There is clearly a vertical shift because since +6 is outside of the parentheses, you go up 6.
on her first quiz in social studies Meg answers 92% of the questions correctly on her second quiz she answers 27 out of 30 questions correctly on which quiz did Meg have the better score
Answers
Answer:
90
Step-by-step explanation:
I can’t show you because there is no pin
Answer:
first one
Step-by-step explanation:
27/30 =90 so 92
What is the slope of the line that contains the points (7, 12) and (18, 9)
Answers
The slope of the given equation is (-11/3)
Given points,
(7, 12) and (18, 9)
Let the points be A and B i.e., A = (7, 12) and B = (18, 9)
A slope of a line is the change in y coordinate with respect to the change in x coordinate. The net change in y-coordinate is represented by Δy and the net change in x-coordinate is represented by Δx.
i.e., Slope of a line containing (p, q) and (r, s) is (r-p)/(s-q)
Slope = (r-p)/(s-q) (equation 1)
So now slope of line containing A, B is
(Where A = (7, 12) = (p, q) and B = (18, 9) = (r, s))
slope = \(\frac{(18 - 7)}{(9 - 12)}\) (From equation 1)
i.e., slope = \(\frac{11}{-3}\)
So, the slope of the given line is (-11/3)
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jamie drove 61 miles jamie car used 4 gallons of gas, how many miles per gallon did jamie car get
Answers
The number of miles per gallon that Jamie's car get will be 15.25 miles per gallon.
What is the rate?The rate is the ratio of the amount of something to the unit. For example - If the speed of the car is 20 km/h it means the car travels 20 km in one hour.
Jamie drove 61 miles Jamie's car used 4 gallons of gas.
The number of miles per gallon that Jamie's car get will be given as,
Rate = 61 / 4
Rate = 15.25 miles per gallon
The number of miles per gallon that Jamie's car get will be 15.25 miles per gallon.
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Evaluate 12÷s−∣−2∣ if s=0.6
Answers
Answer:
s=20 divided by 12
Step-by-step explanation:
A 3rd3^{\text{rd}}3rd3, start superscript, start text, r, d, end text, end superscript degree binomial with a constant term of 8888
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
8x3+2x+38x^3+2x+38x3+2x+38, x, cubed, plus, 2, x, plus, 3
(Choice B)
B
2x8+32x^8+32x8+32, x, start superscript, 8, end superscript, plus, 3
(Choice C, Checked)
C
x3−x2+8x^3-x^2+8x3−x2+8x, cubed, minus, x, squared, plus, 8
(Choice D)
D
−5x3+8-5x^3+8−5x3+8minus, 5, x, cubed, plus, 8
Answers
Answer:
\(-5x^3 + 8\)
Step-by-step explanation:
See comment for complete question
Given
\(Degree \to 3rd\)
\(Constant \to 8\)
\(Type \to Binomial\)
Required
Which of the options is true
\(Type \to Binomial\)
This implies that the polynomial has just 2 terms
The above shows that (a), (b) and (c) are not true because they have more than 2 terms
\(Degree \to 3rd\)
This implies that the highest power of x is 3
\(Constant \to 8\)
The second term of the polynomial must be +8
Only (d) satisfy the above conditions
Writing a Linear Equation Given Two Points
Write a linear equation for the line that passes through the given points.
Answers
Answer:
y=-3x+5
Step-by-step explanation: y=mx+b
to find slope (m) use the equation \(\frac{y2-y1}{x2-x1} \\\)
so
\(\frac{-10-(-1)}{5-2}\) = -3
to find b
use y=mx+b
-1=-3(2)+b
-1=-6+b
5=b
Hope this helps :)
?????????????????????
Answers
Answer:
\(\sqrt{x+6}\)
Step-by-step explanation:
So, there are a few things we need to go over to graph a function,
When a number is outside of a root, it changes the y value. For example:
y=\(\sqrt{x}+6\)
With the +6, y will always be 6 higher than normal.
If it was -6, then y will always be 6 lower than normal.
What if the number is inside the root? Well, it works a little differently.
Instead of changing the y value, it changes the x value. For example:
y=\(\sqrt{x+6}\)
So if you put a number in for x, lets say -6, wht would you get?
You would get -6+6=0
The square root of 0 is 0, so when x=-6, y=0
Normally, x would have to equal 0 for the y value to be 0.
So basically, when we see the number isnide of the root, we can think the our x coordinate being subtracted by that number.
This makes since, because if we subtract the +6 from x:
x-(+6)= x-6, and -6 is our x coordinate.
If it was -6 at the start, this would also work:
x-(-6)= x+6. So our x coordinate would start at 6.
Now, lets look at our graph.
As we can see, the x values start at -6, and the y values starts at 0.
This eliminates A and D, since the +6 would change the y value, not the x.
Remember that x-6 would make x a postive 6.
x+6 however, would make x a negative 6.
So we need x+6 in a square root.
This eliminates B, since it has a x-6, making the x coordinate postive 6, not negative 6.
So c is our answer.
Hope this helps!
A. Frequency Distribution
Here is a hypothetical list of the number of caramel popcorn cans sold in 70 scout troops in Maryland. Use Excel's capacity (= FREQUENCY()) to create a frequency distribution
Number of Caramel popcorn cans sold in each Troop
174 105 103 105 148 158 121 130 118 157
153 147 132 110 115 192 158 196 149 140
183 199 174 107 179 183 129 194 119 150
198 171 120 163 163 108 184 134 186 175
180 114 107 107 152 137 184 200 189 103
190 175 156 143 142 152 182 126 142 160 183 119 165 134 172 145 184 168 170 113
Minimum [] Use the MIN () function to find the lowest number of popcorn cans sold
Maximum [ ] Use the =MAX () function to find the highest number of popcorn can sold
Subtract the lowest number of cans sold from the highest number of cans sold
The range of number of cans sold is []
B. Decide how many bins to use. Study the data and pick a number between (5 and 10)
In one paragraph below explain why you selected the numbers of bins (between 5 and 10)
C. Now divide the range of the cans sold by the number of bins to find the size that each bin should be
Size of each bin is []
Answers
The frequency distribution can be created as below using the capacity = FREQUENCY (): Popcorn Cans | Frequency 103-123 | 15 124-144 | 24 145-165 | 17 166-186 | 10 187-207 | 4
A. Frequency Distribution
Number of Caramel popcorn cans sold in each Troop174 105 103 105 148 158 121 130 118 157153 147 132 110 115 192 158 196 149 140183 199 174 107 179 183 129 194 119 150198 171 120 163 163 108 184 134 186 175180 114 107 107 152 137 184 200 189 103190 175 156 143 142 152 182 126 142 160 183 119 165 134 172 145 184 168 170 113Minimum [103] Use the MIN () function to find the lowest number of popcorn cans sold.
Maximum [200] Use the =MAX () function to find the highest number of popcorn can sold.
Subtract the lowest number of cans sold from the highest number of cans sold
The range of number of cans sold is [97]
B. To decide how many bins to use, you should study the data and select a number between 5 and 10 based on the number of caramel popcorn cans sold in 70 scout troops in Maryland.
You may use five bins, based on the amount of popcorn sold, with the ranges of can sales being between 103-123, 124-144, 145-165, 166-186, and 187-207.
In the data collection, the lowest number of cans sold was 103 and the highest was 200.
There are a total of 97 cans sold within the range. If we use ten bins, there will be about 10 numbers in each bin. We only use five bins because this data set is somewhat small,
so we don't want to divide the data too much or too little.
As a result, we choose 5 bins to make the data more meaningful and easier to understand.
C. Now divide the range of the cans sold by the number of bins to find the size that each bin should beThe range of numbers of cans sold is [97].
Since we want to use 5 bins, divide 97 by 5.
Size of each bin is [19.4]
(20 cans in each bin, except for the last bin that will have 17 cans).
Thus, the frequency distribution can be created as below using the capacity = FREQUENCY (): Popcorn Cans | Frequency 103-123 | 15 124-144 | 24 145-165 | 17 166-186 | 10 187-207 | 4.
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Write the following equations as quadratics in standard form (with = 0). Make sure to remove any brackets and fractions present.
Question 1:
\(\frac{x}{2} = x^2 - \frac{3}{2}\)
Question 2:
\(\frac{5}{x} + 1 = x\)
Answers
Answer:
\(-2x^2+x+3=0\)
\(-x^2+x+5=0\)
Step-by-step explanation:
Algebraic Manipulation
Consists of applying basic rules of algebra to simplify or transform algebraic expressions into others.
We are required to rewrite two expressions as quadratics in standard form.
Question 1:
\(\displaystyle \frac{x}{2} = x^2 - \frac{3}{2}\)
To get rid of the denominators, multiply by 2:
\(x=2x^2-3\)
Move all the terms to the left side:
\(x-2x^2+3=0\)
Order by degree:
\(\boxed{-2x^2+x+3=0}\)
Question 2:
\(\displaystyle \frac{5}{x} + 1 = x\)
Multiply by x (x must be different from 0):
\(5+x=x^2\)
Move all the terms to the left side:
\(5+x-x^2=0\)
Order by degree:
\(\boxed{-x^2+x+5=0}\)
Multiply. Write your answer as a fraction in simplest form.7x5201Х5?
Answers
5 can be written as 5/1, then:
\(\frac{7}{20}\cdot5=\frac{7}{20}\cdot\frac{5}{1}\)When multiplying fractions, numerators multiply each other and denominators multiply each other, that is:
\(\frac{7}{20}\cdot\frac{5}{1}=\frac{7\cdot5}{20\cdot1}=\frac{35}{20}\)This result can be simplified by dividing numerator and denominator by 5, as follows:
\(\frac{35}{20}=\frac{\frac{35}{5}}{\frac{20}{5}}=\frac{7}{4}\)7/4 is the final answer