Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take

Answers

Answer 1

Answer:

Step-by-step explanation:

2005 AMC 8 Problems/Problem 20

Problem

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24$

Solution

Alice moves $5k$ steps and Bob moves $9k$ steps, where $k$ is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, $14k$, is a multiple of $12$. Since this number must be a multiple of $12$, as stated in the previous sentence, $14$ has a factor $2$, $k$ must have a factor of $6$. The smallest number of turns that is a multiple of $6$ is $\boxed{\textbf{(A)}\ 6}$.

Related Questions

what is 4 + 3i^2 and is it real or imaginary?

Answers

Answer:

4+6i

Step-by-step explanation:

Distribute the 2 to the 3i

You get 6i, you can't add 4 to 6i because 4 is real and 6i is imaginary.

It's a complex imaginary number

Real

[tex]\\ \sf\longmapsto 4+3i^2[/tex]

i^2=-1

[tex]\\ \sf\longmapsto 4+3(-1)[/tex]

[tex]\\ \sf\longmapsto 4-3[/tex]

[tex]\\ \sf\longmapsto 1[/tex]

x + 2y = 7
x - y = 1

Solve for x and y?

Answers

Answer:

y=2

x=3

Step-by-step explanation:

make x - y = 1 into x= y + 1

put that wherever x is in the other equation

so y + 1 + 2y =7. then combine like terms into 3y =6. then divide 6 by 3 so you get y = 2

then substitute 2 into the y in the equation x - y = 1 and you get x = 3

METHOD OF ELIMINATION:

Given pair of linear equations are:

x+2y= 7 – – – Eqn(i)

On comparing with a1x+b1y+c1 = 0

a1 = 1, b1 = 2, c1 = -7

x-y = 1 – – – Eqn(ii)

On comparing with a2x+b2y+c2 = 0

a2 = 1, b2 = -1 , c2 = -1

a1/a2 = 1/1 = 1

b1/b2 = 2/-1 = -2

c1/c2 = -7/-1 = 7

We have,

a1/a2 ≠ b1/b2 ≠ c1/c2

So, Given pair of linear equations in two variables have a unique solution.

Now,

On Subtracting eqn(ii) from eqn(i) then

x+2y = 7

x-y = 1

(-)

_______

0 +3y = 6

_______

⇛ 3y = 6

⇛ y = 6/3

⇛ y = 2

On Substituting the value of y in (1) then

⇛ x-2 = 1

⇛ x = 1+2

⇛x = 3

Answer: Therefore , the value of x and y with be (3,2) respectively.

Additional comment:

If a₁x+b₁y+c₁ = 0 and a₂x+b₂y+c₂ = 0 are pair of linear equations in two variables thenIf a₁/a₂ ≠b₁/b₂ ≠ c₁/c₂ then they are Consistent and independent lines or Intersecting lines and they have a unique solution.If a₁/a₂ = b₁/b₂ = c₁/c₂ then they are Consistent and dependent lines or Coincident lines and they have infinitely number of many solutions.If a₁/a₂ =b₁/b₂ ≠ c₁/c₂ then they are Inconsistent lines or Parallel lines lines and they have no solution.

Please help me with this question

Answers

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

You only need to read as far as ...

y ≤

to choose the correct graph. The values in the solution will be less than or equal to those on the boundary line. That means the graph will be shaded below the line, and the line will be solid (because it is part of the solution).

The graph with a solid line and shading below is graph F.

Factor 26r3s 52r5 â€"" 39r2s4. What is the resulting expression? 13(2r3s 4r5 â€"" 3r2s4) 13r2s(2r 4r3 â€"" 3s3) 13r2(2rs 4r3 â€"" 3s4) 13r2(26r3s 52r5 â€"" 39r2s4).

Answers

The resulting expression from the factorization of 26r³s + 52r^(5) - 39r²s⁴ is; 13r²(2rs + 4r³ - 3s⁴)

We want to factor the algebraic expression;

26r³s + 52r^(5) - 39r²s⁴

Now, we need to first find the highest common factor here and factorize out.

First of all for the letters;

The highest common factor of the letters is r²

Factors of 26 = 1, 2, 13, 26

Factors of 39 = 1, 3, 13, 39

Factors of 52 = 1, 2, 4, 13, 26, 52

The highest common factor for the 3 numbers is 13.

Thus,in total, the highest common factor for the algebraic expression is 13r².

Finally, we have;

13r²(2rs + 4r³ - 3s⁴)

Read more about factorization of algebraic expression at; https://brainly.com/question/11434122

Which graph shows a dilation?

Answers

the 1st and 3rd one shows dilation. mark me brainlist

Can someone help me with this please?!

Answers

Answer:

x < 6

Step-by-step explanation:

1) -5x > -30

2) divide both sides by -5

3) flip the sign as we divided inequality by a negative

4) x < 6

The price of a pair of shoes increases from $54 to $68. What is the percent increase to the nearest percent? [The percent increase is ___%]

Answers

Answer:

25.93%

Step-by-step explanation:

Percent change formula is [tex]\frac{(V_2-V_1)}{|V_1|}[/tex]x 100

v2 = value 2

v1 = Value 1

now you substitute in.

[tex]\frac{68-54}{|54|}[/tex]x 100

[tex]\frac{14}{54}[/tex]x100

0.259259x100

= 25.9259% or 25.93%

Cathy is hiking in the mountains. She hikes 5.9 km the first
day and 4.9 km the second day. What is the average
distance she hiked each day?
5.4 km
B
6.2 km
1.0 km
D
10.8 km

Answers

Answer:

Step-by-step explanation:

d1 = 5.9 km

d2 = 4.9 km

d_av = (5.9 + 4.9)/2 = 10.8/2 = 5.4 km I think that's A

If a team won 2/3 of its games and lost the rest. what was the team's win-loss ratio

Answers

Answer: 2 wins to 1 loss

Step-by-step explanation:

2 wins and one loss.

Now that Yardley Auto Repair has a different owner, its mechanics make $35 per hour. That is 25% more than they originally made. How much did the mechanics use to make per hour?

Answers

Answer:

Step-by-step explanation:

So the Yardley people got a raise! Hurray! They make ALL that they made before and a little bit more.

Let x = original pay

100% of x is what they made before.

25% of x is the amount of raise they got.

100% of x should be written 1x, or just x (convert the % to a decimal)

25% of x should be written .25x

So a math sentence that says their original pay plus their raise is now $35 would be written

1x + .25x = 35

Combine the x terms.

1.25x = 35 Now divide both sides by 1.25 to get the x all by itself.

1.25x/1.25 = 35/1.25

x = 28 Remember what x stood for? x is the original pay. The mechanics used to make $28 per hour.

Which statement about the following equation is true? 3x2 â€"" 8x 5 = 5x2.

Answers

The discriminant is greater than 0, so there are two real roots and it can be determined by using nature of roots.

Given that,

Equation; [tex]\rm 3x^2 - 8x + 5 = 5x^2[/tex]

We have to determine,

Which statement about the following equation is true?

According to the question,

To determine the true statement about the equation following all the steps given below.

Equation; [tex]\rm 3x^2-8x+5=5x^2[/tex]

Simplify the equation for the roots of the equation,

[tex]\rm 3x^2-8x+5=5x^2\\\\\rm5x^2- 3x^2+8x-5=0\\\\ 2x^2+8x-5=0\\\\ 2x^2+8x-5=0\\\\2x^2+8x=5\\\\2x(x+4)=5\\\\Then, \ 2x = 5, \ \ x = \dfrac{5}{2} \\\\And\ \ x+4=5, \ \ x=5-4, \ \ x=1[/tex]

The equation has two real roots and x is 1 and 5/2.

Hence, The discriminant is greater than 0, so there are two real roots.

For more details about Discriminate refer to the link given below.

https://brainly.com/question/10706429

write one in factor form with solutions 3,-1,-3i?

Answers

Answer:

Factored form: [tex](x-3)(x+1)(x^2+9)[/tex]

Expanded form: [tex]x^4-2x^3+6x^2-18x-27[/tex]

Step-by-step explanation:

[tex](x-3)(x+1)(x^2+9)[/tex]

[tex](x^2-2x-3)(x^2+9)[/tex]

[tex]x^4+9x^2-2x^3-18x-3x^2-27[/tex]

[tex]x^4-2x^3+6x^2-18x-27[/tex]

[tex]x-3=0[/tex] has a solution of [tex]x=3[/tex]

[tex]x+1=0[/tex] has a solution of [tex]x=-1[/tex]

[tex]x^2+9=0[/tex] has a solution of [tex]x=\pm 3i[/tex]

Miss Watson runs a distance of 200 meters in 25 seconds.
Work out the average speed of Miss Watson in meters per second?

Answers

Step-by-step explanation:

D = 200m T = 25s

The formula for speed is speed = distance ÷ time

S = 200 ÷ 25

S = 8m/s

the numbers 1 to 9 are randomly arranged to form a 9-digit number. what is the probability that the number obtained is divisible by 18?
(please write the solution)

Answers

Answer:

Hence the probability that the number is divisible by 18 is 4/9 or approx. 0.4444. or even 44.4%

Step-by-step explanation:

If we have 9 digits 1–9 placed randomly to form a 9-digit number without repetition and we want the probability of that number is divisible by 18, then we only need to look at the fact if it is divisible by 18, then it is also divisible by 2 and 9. Adding the digits 1–9 would give us the following (including checking for divisibility by 9):

(9 x 10)/2 =90/2 = 45

45÷9=5

By divisibility of 9, if the sum of the digits is divisible by 9, then the original number itself is divisible by 9. Since the number 45 is divisible by 9, we know that all the possible 9-digit numbers will be as well. The only thing we would have to look at is if it is even since the number has to be divisible by 2.

There are 9 digits of which 4 are even (2, 4, 6 or 8).

So the probability that the number is divisible by 2 is 4/9.

Hence the probability that the number is divisible by 18 is 4/9 or approx. 0.4444.

this question

please help please

m=n+4b

Answers

Answer:

m=n+4b:- b=m-n/4.........

15 POINTS HELP PLZ

Which expression is equal to f(x) - g(x)?

Answers

Answer:

c it's probably c I think I'm not really too good at this but I'm probably sure it's c.

[tex] \frac{5}{5x - 7} = 10[/tex]
solve the equation.

Answers

Answer:

x=1.5

Step-by-step explanation:

5=10(5x-7)0.5=5x-77.5=5xx=1.5

PLZ ANYONE HELP ME PLZ BEFORE THE HOLIDAY BREAK BEGINS!!!!!!!!!!!!!!

The Uber base fare is $1. There is a charge of $1.05 per mile. You also plan to give the

driver a $2 tip.

Task 1 Student Directions:

Part A: Write an expression to represent the amount of money need to take a taxi or take Uber.

(x = the number of miles.)
Taxi Expression: ____________________ Uber Expression: __________________

Part B: Complete the table to determine how much it will cost to go 12 miles, 20 miles and 50 miles.

Taxi Uber

Part C: Which mode of transportation is better, use information from the table to support your answer. BUT PLZ JUST DO THE PART B AND PART C ONLY THANKS.

Answers

Answer:

Part A. (Uber)$1 + $2 + 1.05x

(taxi)

Part B. (Uber)$15.60 (12mi) $24(20mi) $55.50(50mi)

(taxi)

Part C.

Step-by-step explanation:

Part B. Uber

12 miles: $1 + $2 + 1.05(12)

3+12.60

15.60

20 miles: $1 + $2 + 1.05(20)

3+21

50 miles: $1 + $2 + 1.05(50)

3+52.50

55.50

Part B. Taxi

Part C.

PLEASE HELP ME!!! I"L GIVE BRAINLEST

3m + 3

m = -7

-12

-18

-16

24

Answers

Answer:

m = [tex]\frac{-421}{6}[/tex]

Step-by-step explanation:

Let's solve your equation step-by-step.

Step 1: Simplify both sides of the equation.

3m+3m=−7−12−18−(16)x(24)

3m+3m=−7+−12+−18+(−384)

(3m+3m)=(−7+−12+−18+(−384))(Combine Like Terms)

6m=−421

Step 2: Divide both sides by 6.

[tex]\frac{6m}{6} =\frac{-421}{6}[/tex]

m = [tex]\frac{-421}{6}[/tex]

I really hope this help's :)

Find the output, y, when the input,x, is 6.
y =?

Answers

0 because when x is 6 y is on the bottom of the x axis or at 0

Answer:

Y= 0

The standard form of an exponential function is

y = a[tex]b^{x}[/tex]

Use points on the graph to find a and b

Using (0, 5 ) , then

5 = a[tex]b^{0}[/tex] = a

Then

y = 5[tex]b^{x}[/tex]

Using (1, 2 )

2 = 5[tex]b^{1}[/tex] = 5b ( divide both sides by 5 )

[tex]\frac{2}{5}[/tex] = b

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