Fibonacci follies: suppose you are playing a round of Fibonacci nim with a friend. You start with 15 sticks. You start by removing two sticks; your friend then takes one; you take two; your friend takes one. What should your next move be? Can you make it without breaking the rules of the game? Did you make a mistake at some point? If so, where?

Answer with step-by-step explanation:

We are given that the recurrence relation

[tex]f_n=5f_{n-4}+3f_{n-5}[/tex]

for n=5,6,7,..

Initial condition

[tex]f_0=0,f_1=1,f_2=1,f_3=2,f_4=3[/tex]

We have to show that Fibonacci numbers satisfies the recurrence relation.

The recurrence relation of Fibonacci numbers

[tex]f_n=f_{n-1}+f_{n-2}[/tex],[tex]f_0=0,f_1=1[/tex]

Apply this

[tex]f_n=(f_{n-2}+f_{n-3})+f_{n-2}=2f_{n-2}+f_{n-3}[/tex]

[tex]f_n=2(f_{n-3}+f_{n-4})+f_{n-3}=3f_{n-3}+2f_{n-4}[/tex]

[tex]f_n=3(f_{n-4}+f_{n-5})+2f_{n-4}=5f_{n-4}+3f_{n-5}[/tex]

Substitute n=2

[tex]f_2=f_1+f_0=1+0=1[/tex]

[tex]f_3=f_2+f_1=1+1=2[/tex]

[tex]f_4=f_3+f_2=2+1=3[/tex]

Hence, the Fibonacci numbers satisfied the given recurrence relation .

Now, we have to show that [tex]f_{5n}[/tex] is divisible by 5 for n=1,2,3,..

Now replace n by 5n

[tex]f_{5n}=5f_{5n-4}+3f_{5n-5}[/tex]

Apply induction

Substitute n=1

[tex]f_5=5f_1+3f_0=5+0=5[/tex]

It is true for n=1

Suppose it is true for n=k

[tex]f_{5k}=5f_{5k-4}+3f_{5k-5}[/tex] is divisible 5

Let [tex]f_{5k}=5q[/tex]

Now, we shall prove that for n=k+1 is true

[tex]f_{5k+5}=5f_{5k+5-4}+3f_{5k+5-5}=5f_{5k+1}+3f_{5k}=5f_{5k+1}+3(5q)[/tex]

[tex]f_{5k+5}=5(f_{5k+1}+3q)[/tex]

It is multiple of 5 .Therefore, it is divisible by 5.

It is true for n=k+1

Hence, the [tex]f_{5n}[/tex] is divisible by 5 for n=1,2,3,..

The Fibonacci sequence with its initial conditions does satisfy the given recurrence relation. With this, it can also be demonstrated that f5n is divisible by 5 for all positive integers n.

The problem involves a specific series in mathematics known as the Fibonacci series. We are given the Fibonacci numbers initial conditions: f0 = 0, f1 = 1, f2 = 1, f3 = 2, and f4 = 3. From these initial conditions, we can calculate the next values of the Fibonacci numbers.

Using the given Fibonacci numbers recurrence relation: fn = 5fn−4 + 3fn−5, we get:

So, fn = 5fn−4 + 3fn−5 is valid for n = 5, 6, 7, and presumably, for greater integers as well.

We then use this recurrence relation to show that f5n is divisible by 5, for n = 1, 2, 3, etc. This is immediately obvious for n = 1, because f5 = 5. But for n greater than 1, we can see that f5n = f(5*(n-1)+5) = 5f(5*(n-1)) + 3f(5*(n-1)-1), which will be divisible by 5 because 5 is a factor in the first term and the second term is also divisible by 5 according to our earlier results. This confirms that f5n is divisible by 5 for n = 1, 2, 3, and so on.

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Answer:

yes

Step-by-step explanation:

Answer:

Yes

Step-by-step explanation:

18 : 5

36 : 10

36/2 : 10/2

18 : 5

The value of y =14.45689 when x=5 and y= 21.74137 when x= 9.

An estimate of the line that depicts the actual, but unidentified, linear relationship between the two variables is called a regression line. When the value of the explanatory variable is known, the regression line's equation is used to predict (or estimate) the value of the response variable.

Because it is the line that fits the points the best when drawn through them, the regression line is occasionally referred to as the "line of best fit." It is a line that minimises the difference between the projected and actual scores.

Given Data:

(−1,0),(0,9), (4,13), (8,20), (10,23).

Sum of X = 21

Sum of Y = 65

Mean X = 4.2

Mean Y = 13

Sum of squares (S[tex]S_x[/tex]) = 92.8

Sum of products (SP) = 169

Regression Equation = ŷ = bx + a

b = SP/S[tex]S_x[/tex] = 169/92.8 = 1.82112

a = [tex]M_y[/tex]- [tex]bM_x[/tex]= 13 - (1.82x4.2) = 5.35129

ŷ = 1.82112x + 5.35129

For x= 5

ŷ = 1.82112(5) + 5.35129

ŷ = 14.45689

and, when x= 9

ŷ = 1.82112(9) + 5.35129

ŷ = 21.74137

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Final answer:

The least-squares regression line equation is ŷ = -1.72 + 2.18x. For x = 5, ŷ is approximately 9.88. For x = 9, ŷ is approximately 18.62.

Explanation:

To find the least-squares regression line, we need to use the formula ŷ = b0 + b1x, where b0 is the y-intercept and b1 is the slope. Using the coordinates of the given points, we can calculate the values of b0 and b1. Substituting these values into the formula, we get ŷ = -1.72 + 2.18x. For x = 5, we can plug this value into the equation: ŷ = -1.72 + 2.18(5) = 9.88. Similarly, for x = 9, we can substitute x into the equation: ŷ = -1.72 + 2.18(9) = 18.62.

Answer:

The answer to your question is Area = 1024/243 or 4 52/243

Step-by-step explanation:

Data

length = 1 7/9

width = 3/4 of its length

Area = ?

Formula

Area of a rectangle = length x width

Process

1.- Convert the mixed fraction to improper fraction

      1 7/9 = (9 + 7) / 9 = 16/9

2.- Get the width

      16/9 / 3/4 = (16 x 4) / (9 x 3)

                       = 64 / 27

3.- Get the area

    Area = (16/9)(64/27)

             = 1024/243

             = 4 52/243

Answer:

2.37 square inches

Step-by-step explanation:

l = 16/9 = 1.78

b = 16/9 *3/4 = 1.33

Area = l * b

Area = 1.78 * 1.33

Area = 2.37 square inches

The range of the body temperature is [tex]2\leq x\leq 35[/tex]

Explanation:

Let x denote the body temperature of the bees.

The maximum body temperature of the bee is 35° Celsius.

When they sleep, the body temperature can drop by up to 2° Celsius.

Thus, the minimum body temperature of the bee is 2° Celsius.

Thus, the range of the body temperature is from 35° Celsius to 2° Celsius.

The range of body temperature is given by:

[tex]2\leq x\leq 35[/tex]

Thus, the range in body temperature of the bees is [tex]2\leq x\leq 35[/tex]

An equation that represents the relationship between t and p is [tex]\rm Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex].

Given

Mary buys p peaches at the farmer's market for d dollars each.

She spends a total of t dollars on peaches.

The cost of each peach is;

[tex]\rm Cost \ of \ each \ peaches = \Total \ number \ of \ peaches \times Amount \ of \ each \ peaches\\\\Cost \ of \ each \ peaches= p \times d[/tex]

Therefore;

An equation that represents the relationship between t and p is; ​

[tex]\rm Cost \ of \ each \ peaches = \dfrac{Total \ number \ of \ peaches \times Amount \ of \ each \ peaches} {Total \ money\ spend \ on \ peaches}\\\\Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex]

Hence, An equation that represents the relationship between t and p is [tex]\rm Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex].

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Answer:

Tomi's slowest possible walking speed is 80 meters per minute

Step-by-step explanation:

Tope swims at a speed of 50 meters per minute

Distance back and forth the long side of the pool is 50+50 meters = 100 meters

Time required to complete one round (back and forth) is 100 / 50 = 2 minutes

Perimeter of the pool is 50+50+30+30 = 160 meters

To walk around the perimeter and meet Tope at the starting point, Tomi has to walk 160 meters in 2 minutes

Tomi’s speed = 160 meters / 2 minutes or 160 / 2 = 80 meters per minute

Tomi's slowest possible speed should be 80 meters per minute

Answer:

The zeros of the given function are not real. The zeros of the function is at :

[tex]x=\frac{1}{2}+\frac{5i}{2}[/tex]    and  [tex]x=\frac{1}{2}-\frac{5i}{2}[/tex]

Step-by-step explanation:

Given quadratic function:

[tex]f(x)=2x^2-2x+13[/tex]

To  find the zeros of the function using quadratic formula.

Solution:

Applying quadratic formula:

[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

For the given function:

[tex]a=2, b=-2\ and\ c=13[/tex]

Thus, we have:

[tex]x=\frac{-(-2)\pm\sqrt{(-2)^2-4(2)(13)}}{2(2)}[/tex]

[tex]x=\frac{2\pm\sqrt{4-104}}{4}[/tex]

[tex]x=\frac{2\pm\sqrt{-100}}{4}[/tex]

[tex]x=\frac{2\pm\sqrt{100}i}{4}[/tex]

[tex]x=\frac{2\pm10i}{4}[/tex]

[tex]x=\frac{2+10i}{4}[/tex]      and  [tex]x=\frac{2-10i}{4}[/tex]

[tex]x=\frac{2}{4}+\frac{10i}{4}[/tex]   and  [tex]x=\frac{2}{4}-\frac{10i}{4}[/tex]

[tex]x=\frac{1}{2}+\frac{5i}{2}[/tex]    and  [tex]x=\frac{1}{2}-\frac{5i}{2}[/tex]

Thus, the zeros of the given function are not real. The zeros of the function is at :

[tex]x=\frac{1}{2}+\frac{5i}{2}[/tex]    and  [tex]x=\frac{1}{2}-\frac{5i}{2}[/tex]

In two rolls, 219.8 square inches of wall she will paint

Solution:

Given that,

Jenny uses a roller to paint a wall

The roller has a radius of 1.75 inches and a height of 10 inches

We have to find the area of the wall that she will paint in two rolls

Find the lateral surface area of cylinder

[tex]A_L = 2 \pi r h[/tex]

Where, "r" is the radius and "h" is the height

From given,

r = 1.75 inches

h = 10 inches

Substituting the values we get,

[tex]A_L = 2 \times 3.14 \times 1.75 \times 10\\\\A_L = 6.28 \times 1.75 \times 10\\\\A_L = 109.9[/tex]

Thus lateral surafce area is 109.9 square inches

For two rolls,

Area = 2 x 109.9 = 219.8

Thus in two rolls, 219.8 square inches of wall she will paint

Answer:

The answer is C.

Step-by-step explanation:

810 (goal) - 510 (amount saved so far) = 300 (Balance left to achieve goal) divided by 5( months remaining to achieve goal) = $60 a month

The statement is Sheila needs to save $60 per month to achieve her goal, the option is C.

We are given that;

Monthly saving= $45

Bathroom remodel amount= $810

Now,

To find the answer, we need to calculate how much more Sheila needs to save and how many months she has left.

We can subtract the amount she has saved from the amount she needs to save:

810 - 510 = 300

So, Sheila needs to save $300 more. We can also subtract the number of months she has saved from the number of months in her plan:

18 - 13 = 5

So, Sheila has 5 months left. To find the monthly amount she needs to save, we can divide the remaining amount by the remaining months:

300 / 5 = 60

Therefore, by algebra the answer will be $60 per month.

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Answer: there are 11 birds and 5 cats

Step-by-step explanation:

Let x represent the number of birds in the pet store.

Let y represent the number of cats in the pet store.

A bird has one head and a cat also has one head. The store manager says that there are 16 animal heads in the store. It means that

x + y = 16

A bird has 2 feet and a cat has 4 feet. The store manager says that there are 42 animal feet in the store. It means that

2x + 4y = 42 - - - - - - - - - - - - 1

Substituting x = 16 - y into equation 1, it becomes

2(16 - y) + 4y = 42

32 - 2y + 4y = 42

- 2y + 4y = 42 - 32

2y = 10

y = 10/2 = 5

x = 16 - y = 16 - 5

x = 11

Answer:

A) 61.68

Step-by-step explanation:

This is an exponential decay of 0.875. The way to know that is to get that is[tex]\frac{right}{left} \\[/tex] (Pick a random number and divide by the number to the left of it) From there, there is a 12 hour difference from 7 am to 7pm! Divide 306.25/0.875 and divide it by 0.875 by the answer of that each time 12 times. The exact answer is 61.68402914

Answer: option A is the correct answer.

Step-by-step explanation:

Looking at the table, the amount of medicine in her system is decreasing constantly with time. This constant amount by which this amount of medicine is decreasing is the decay constant.

Decay constant = 350/400 = 306.25/ 350 = 0.875

Let y represent the amount of medicine in her system after t hours.

Let a represent the initial amount of medicine in her system.

Let b represent the decay constant.

Let t represent the number of hours. Therefore, the function representing the exponential decay would be

y = 400b^t

When she wakes up at 7.00 am, the time from 5.00 pm would be 14 hours. Therefore

y = 400 × 0.875^14

y = 61.7 mg

Applying the tangent ratio, the value of x in the given image is calculated as B. x = 15.92

The tangent ratio is a trigonometric ratio that represents the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle in a right-angled triangle. It is denoted as tan(θ), where θ is the angle. Mathematically, tan(θ) = opposite/adjacent.

Thus, using the tangent ratio, we have:

tan 53 = x/12

x = tan 53 * 12

x ≈ 15.92

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Step-by-step explanation:

a) √ 8/10

= to get the bench mark we have to look for perfect square that are closer to the numbers 8 and 10

= for 8, it is 9 ; for 10 it is also 9

Therefore, √ 8/10 is about√ 9/9

= 3/3 = 1

b) √17/5

= to get the bench mark we have to look for perfect square that are closer to the numbers 17 and 5

= for 17, it is 16; for 5 it is 4

Therefore, √ 17/5 is about √ 16/4

= 4/2= 2

c) √7/13

= to get the bench mark we have to look for perfect square that are closer to the numbers 7 and 13

= for 7, it is 9; for 13 it is 16

Therefore, √ 7/13 is about √ 9/16

= 3/4

d) √29/6

= to get the bench mark we have to look for perfect square that are closer to the numbers 29 and 6

= for 29, it is 25; for 6 it is 4

Therefore, √ 29/6 is about √ 25/4

= 5/4

Answer:

Option a)  Has an above average price-to-earning ratio                                            

Step-by-step explanation:

We are given the following in the question:

The price-to-earning ratio for firms in a given industry is distributed according to normal distribution.

For a particular firm the ratio x has a standard normal variable has a value,

z = 1

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

[tex]1 = \dfrac{x - \mu}{\sigma}\\\\\sigma = x - \mu\\x = \mu + \sigma[/tex]

Thus, the firm has an above average price-to-earning ratio as the ratio is one standard deviation above the mean.

Option a)  Has an above average price-to-earning ratio

If a firm in a given industry has a standard normal variable value of Z=1, it signifies that the firm has an above average price-to-earning ratio.

A student has asked about the interpretation of a standard normal variable value, Z, in the context of a firm's price-to-earning (P/E) ratio in a given industry. If a firm has a Z-value of 1, it means that the firm's P/E ratio is one standard deviation above the mean for the industry. Since the standard normal distribution has a mean of 0 and a standard deviation of 1, a Z-value of 1 corresponds to performance that is better than average. Therefore, a correct interpretation of this scenario is a. Has an above average price-to-earning ratio.

The price-to-earning ratio for firms in a given industry is distributed according to a normal distribution. A firm with a standard normal variable value of Z=1 would have an above average price-to-earning ratio.

This is because the standard normal distribution has a mean of 0 and a standard deviation of 1. A Z-score of 1 corresponds to a value that is one standard deviation above the mean. Since the mean represents the average price-to-earning ratio in the industry, a firm with a Z-score of 1 would have a ratio that is above average.

Area of the window [tex]=49\frac{5}{6}\ \text{ft}^2[/tex]

Solution:

Width of the window = [tex]8\frac{2}{3}[/tex] feet

Height of the window = [tex]5\frac{3}{4}[/tex] feet

To find the area of the window:

The window is in rectangular shape.

Formula for the area of the rectangle = base × height

Substitute the given values in the formula.

Area of the window = [tex]8\frac{2}{3}\times 5\frac{3}{4}[/tex]

Convert mixed fraction into improper fraction.

                                [tex]$=\frac{26}{3}\times \frac{23}{4}[/tex]

Multiply the numerators and denominators separately.

                               [tex]$=\frac{598}{12}[/tex]

Divide the numerator and denominator by the common factor 2.

                               [tex]$=\frac{598\div2}{12\div2}[/tex]

                               [tex]$=\frac{299}{6}[/tex]

Now convert improper fraction into mixed fraction.

                              [tex]$=49\frac{5}{6}\ \text{ft}^2[/tex]

Area of the window [tex]=49\frac{5}{6}\ \text{ft}^2[/tex].

Answer:

49 5/6 will be your answer

Step-by-step explanation:

Answer:

0.726 is the probability that the project will be completed in 33 days or less.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 30 days

Variance = 25 days

Standard Deviation,

[tex]\sigma = \sqrt{\text{Variance}} = \sqrt{25} = 5[/tex]

We assume that the distribution of path length is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P(completed in 33 days or less)

[tex]P( x \leq 33) = P( z \leq \displaystyle\frac{33 - 30}{5}) = P(z \leq 0.6)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(x \leq 33) = 0.726 = 72.6\%[/tex]

0.726 is the probability that the project will be completed in 33 days or less.

Answer:

20

Step-by-step explanation:

nPr = n!/(n-r)!

5P2 = 5!/(5-2)!

= 5!/3!

= 5×4×3!/3!

= 5×4

= 20

Answer:

C) [tex]\frac{2}{4}, \frac{2}{3}, \frac{7}{10}, \frac{4}{5}[/tex]

Step-by-step explanation:

Given fractions:

[tex]\frac{2}{4}, \frac{4}{5}, \frac{7}{10},\frac{2}{3}[/tex]

To arrange the fractions from least to greatest.

Solution:

In order to arrange the fractions from least to greatest, we need to make the denominators common by taking LCD.

LCD of 4,5,10,3 can be found using their multiples.

4= 4,8,12,16,20,24,28,32,36,40,.........60

5= 5,10,15,20,25,30,.........60

10= 10,20,30,40,50,60

3= 3,6,9..........................60

So, 60 is the LCD.

Making the denominators common by multiplying same numbers to numerator and denominator.

[tex]\frac{2}{4}=\frac{2\times 15}{4\times 15}=\frac{30}{60}[/tex]

[tex]\frac{4}{5}=\frac{4\times12}{5\times 12}=\frac{48}{60}[/tex]

[tex]\frac{7}{10}=\frac{7\times 6}{10\times 6}=\frac{42}{60}[/tex]

[tex]\frac{2}{3}=\frac{2\times 20}{3\times 20}=\frac{40}{60}[/tex]

Comparing the numerators we can arrange the fractions.

[tex]\frac{2}{4}, \frac{2}{3}, \frac{7}{10}, \frac{4}{5}[/tex]

Therefore the cost of each CD is =$2.30

Step-by-step explanation:

Given , Kairi spent $40 .18 no CDs. The sale tax was $2.33.Kairi also used a coupon for $1.00 0ff his purchase.

Total cost price of The CDs is = $(40+2.33-1.00)

                                                =$41.33

Therefore the cost of each CD is =$(41.33÷18)

                                                       =$2.30

Answer:

option a (2,2,5)

Step-by-step explanation:

First we have to notice that it tells us that the scale is 1/5.

This means that the measures of the new triangle will be 5 times smaller than those of the other triangle.

Now we just have to take the measurements of the sides of the triangle and multiply them by 1/5 or divide them by 5

AB = 10      DE = 10/5 = 2

BC = 10      EF = 10/5 = 2

AC = 25     DF = 25/5 = 5

DE = 2

EF = 2

DF = 5

(2, 2, 5)

Answer:

This is 7.408 kilometres far.

Step-by-step explanation:

Given:

A ship's mast is sighted just over the horizon at 4 nautical miles.

Now, to find the distance in kilometers.

As, given ship's mast is sighted just over the horizon at 4 nautical miles.

So, by using conversion factor we get the nautical mile into kilometers:

1 nautical mile = 1.852 kilometer.

Thus, 4 nautical miles

= [tex]4\times 1.852[/tex]

=  [tex]7.408\ kilometers.[/tex]

Therefore, this is 7.408 kilometres far.

Answer:

7.4

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

Firstly, she gets on the elevator at the 6th floor. This means she was originally at the 6th floor.

She then stopped in the next three floors. This means she got off at floor 9 where we had Doris.

From Doris, she has to take 7 floors down to meet Julio. This means she went back to floor 2.

Now since her starting position was 6 and she is presently at the 2nd floor, this means she is four floors away from her starting position on the sixth floor.

Answer:24, it depend upon the picture

Step-by-step explanation:

if the ac and bc are diagonals the area of rhombus equal to 24.because area of the rhombuss = pq/2where p and q are  the diagonals

Answer:

45°

Step-by-step explanation:

From the figure;

Triangle WVX is a right-angled triangle

WV= 2

WX = 1

We are required to determine, m∠X

We need to determine the appropriate trigonometric ratio to use;

Therefore, since WV is the opposite and WX is the adjacent to m∠x, then the appropriate trigonometric ratio is tangent.

That is;

Tan m∠X = WV/WX

               = 2/1

tan m∠X = 1

Thus,

angle m∠X = tan⁻¹ 1

                  = 45°

Thus, m∠X = 45°

Answer: the cost of one box of envelope is $3.75.

the cost of one box of notebook paper is $2.85

Step-by-step explanation:

Let x represent the cost of one box of envelope.

Let y represent the cost of one box of notebook paper.

The cost of 3 boxes of envelopes and 4 boxes of notebook paper is $22.65. This means that

3x + 4y = 22.65 - - - - - - - - - - -1

Two boxes of envelopes and 6 boxes of notebook paper cost $24.60. This means that

2x + 6y = 24.6 - - - - - - - - - - -2

Multiplying equation 1 by 2 and equation 2 by 3, it becomes

6x + 8y = 45.3

6x + 18y = 73.8

Subtracting, it becomes

- 10y = - 28.5

y = - 28.5/-10

y = 2.85

Substituting y = 2.85 into equation 1, it becomes

3x + 4 × 2.85 = 22.65

3x + 11.4 = 22.65

3x = 22.65 - 11.4 = 11.25

x = 11.25/3 = 3.75

Answer:

B. n(t) = 0.072t^2 - 0.15t + 2.73

Step-by-step explanation:

Plot data as pair of coordinates where t =x and n(t)=y

Use a graph tool to plot the coordinate points and join the points with a smooth curve

From the answers, test for the function that fits the points as plotted on the tool.

In this case, the function that fits the plot of the data is ;

n(t) = 0.072t^2 - 0.15t + 2.73

See attached;

The function that best used to model this situation is n(t) = 0.072t² - 0.299t + 2.57

Quadratic function is in the form:

where a, b, c are constants.

Let n(t) represent the number of text at time t months.

It is given by:

n(t) = at² + bt + c

At point (5, 3):

At point (20, 27):

At point (40, 112):

a = 0.072, b = -0.29, c = 2.57

n(t) = 0.072t² - 0.299t + 2.57

The function that best used to model this situation is h(t) = -4.9t² + 295 + 2

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Answer:

There were 180 votes for blue.

Step-by-step explanation:

Given:

Riverside elementary school is holding a schoolwide election to choose a color. 5/8 of the votes were for blue. 5/9 of the remaining votes were for green and the remaining 48 votes were for red.

Now, to find the votes for blue.

Let the total votes be [tex]x.[/tex]

The votes for blue:

[tex]\frac{5}{8} \ of\ x[/tex]

[tex]=\frac{5}{8} \times x[/tex]

[tex]=\frac{5x}{8}[/tex]

Remaining votes are:

[tex]x-\frac{5x}{8} \\\\=\frac{8x-5x}{8} \\\\=\frac{3x}{8}[/tex]

The votes for green:

[tex]\frac{5}{9} \ of\ \frac{3x}{8} \\\\=\frac{5}{9} \times \frac{3x}{8} \\\\=\frac{15x}{72}[/tex]

The remaining votes for red = 48.

Now, the total votes are:

[tex]\frac{5x}{8} +\frac{15x}{72} +48=x\\\\\frac{45x+15x+3456}{72}=x\\\\\frac{60x+3456}{72}=x[/tex]

Multiplying both sides by 72 we get:

[tex]60x+3456=72x[/tex]

Subtracting both sides by 60[tex]x[/tex] we get:

[tex]3456=12x[/tex]

Dividing both sides by 12 we get:

[tex]288=x\\\\x=288.[/tex]

Thus, the total votes = 288.

Now, to get the votes for blue:

[tex]\frac{5}{8} \ of\ 288\\\\=\frac{5}{8}\times 288\\\\=0.625\times 288\\\\=180.[/tex]

Therefore, there were 180 votes for blue.