tastyabananas2ndDad

When combining three bags of marbles with 36%, 32%, and 25% yellow marbles respectively, the resulting proportion of yellow marbles in the bowl would be the weighted average of the percentages:

weighted average = (percentage of bag 1 * weight of bag 1 + percentage of bag 2 * weight of bag 2 + percentage of bag 3 * weight of bag 3)/(total weight)

Assuming each bag has the same weight (equal weight), the calculation becomes:

Weighted average = (0.36 + 0.32 + 0.25) / 3 ≈ 0.31 or 31%

So, approximately 31% of the marbles in the bowl would be yellow.

To solve this, we can set up a proportion equation:

x / 4 = 9 / y

Cross-multiplying gives:

xy = 36

Now, we need to find the pairs of positive integers (x, y) that satisfy this equation. The pairs that work are:

(1, 36), (2, 18), (3, 12), (4, 9), (6, 6), (9, 4), (12, 3), (18, 2), (36, 1)

So, there are 9 pairs of positive integers (x, y) that satisfy the given property.

Therefore, the required pair of positive integers (x,y) satisfied the given property is 9.

We have the number $256

The proper divisors of $256 are $1$, $2$, $4, $8$, $16$, $32$, $64$, and $128$

Now 1 + 2 + 4 + ..... + 128; Here first term = 1 

                                                     common ratio = 2

       = \( {1 (2^8 - 1) \over 2 - 1}\)         [ a + ar + ar² + ... + ar^n-1

                                  = \({a(r^n - 1) \over r - 1}\);(r >1)

       = \( {256 - 1 \over 1}\)

       = 255

Thus the sum of proper divisor of $256$ is $255$

We observe that this is a rational function in which the numerator is 

a polynomial of degree 3 and this polynomial is defined for all real values.

So, this rational function will be undefined at the points where we will have, 
denominator = 0

This gives us,

|x - 3| - |x + 1| = 0 ......... (1)

To remove the absolutes in equation (1), we will have following three intervals: 

x ≤ -1, -1 ≤ x ≤ 3, x ≥ 3

If x ≤ -1, then using (1), we get

-(x - 3) - (-(x + 1)) = 0

        - x + 3 + x + 1 = 0

                            4 = 0

This is not a true statement.

It means that we have no solution when x ≤ -1.

If -1 ≤ x ≤ 3, then using (1), we get

-(x - 3) - (x + 1) = 0

 -x + 3 - x - 1 = 0

         -2x + 2 = 0

               2x = 2

                 x = 2/2

                 x = 1

It means that when -1 ≤ x ≤ 3, then the solution is given by

x = 1 

If x ≥ 3, then using (1), we get

(x - 3) - (x + 1) = 0

    x - 3 - x - 1 = 0

                 - 4 = 0

This is not a true statement.

It means that we have no solution when .

Therefore, the solution of the equation (1), is given by x = 1

Hence, the function g(x) is defined for all real values except x = 1.

Therefore, the domain of the function g(x) is (-∞, 1)U(1, ∞).

Domain (-∞, 1)U(1, ∞)

Given that triangle XYZ is equilateral and X is the circumcenter of triangle OYZ, it implies that angle OXY is 120 degrees. This makes angle OXZ also 120 degrees, making triangle OXZ isosceles.

Since X is the circumcenter of triangle OYZ, XY = XZ. Therefore, angle XYZ = angle XZY = (180 - 120) / 2 = 30 degrees.

Now , we can use the area of the circle

           A = πr²

where r is the radius of the circle centered at o.

Given that the area of the circle is 48π

 we can find the radius

     48π = πr²

        r² = 48

        r = √48

        r = 4√3

Since triangle OXY is isosceles, we can find the height of the triangle (which is also the radius of the circle):

Height = r = 4√3

Now, we can use the forumala for the area of an equilateral triangle A = (side² * √3)/4

Area of XYZ = (XY² * √3)/4

                     = (4√3)² * √3)/4 

                     = 48√3/4

Area of XYZ = 12√3

So, the area of triangle XYZ is 12√3 square units.

There are 15 line segments to be colored:
(1) 6 edges
(2) 3 diagonals that connect opposite vertices
(3) 6 other diagonals

For the appearance of the figure to be unchanaged when it is rotated by 60 degrees, all segments of the same type must be the same one of the two colors.

So the number of different colorings is 2*2*2 = 8.

Note that, area of semicircle = πr²/2, where

                          r - radius of semicircle.

The area of semicircle APC = 7

Let r₁ be radius of semicircle APC.

By above note,

Area of semicircle APC = πr₁²/2

           => πr₁²/2 = 7

           => r₁² = 14/π

                r₁ = √14/π.

 length AC = 2r₁ = 2√14/√π

Again, the area of semicircle CQB = 13

Let r₂ be radius of semicircle CQB.

       By above rule,

Area of semicircle CQB = πr₂²/2

           => πr₂²/2 = 13

           => r₂² = 26/π

                r₂ = √26/π.

length CB = 2r₂ = 2√26/√π

--------------------------------------

Now, in △ACB, MLC = 90°

     By Pythagoras theorem,

          AB² = AC² + CB² 

                  =  4(14)/π + 4(26)/π

        => AB² = 160/π

            AB = √160/π

Therefore, radius of third circle = r₃ = 1/2 √160/π.

Area of third circle = πr₃²/2 

                              = π(1/4 * 160/π)/2

                              = 40/2 = 20 

Therefore area of third circle is 20 square unit.

let No. of tickets Ryan have = x

     No. of tickets Andy have = y

No. of days Andy take when he sells 5 tickets each day = y/5.

then No. of tickets Ryan sells in 5 days = (y/5) * 10 = 2y

Ryan will have 40 tickets left when Andy sells all his tickets 

So, We can write

            No. of tickets Ryan has - No. of tickets Ryan sells = 40

                                                                         x - 2y = 40 ---- (1)

Now If Andy sells 10 tickets a day then 

No. of days Andy take to sell all his tickets y/10

On the same time Ryan sells tickets = y/10 * 5 = y/2

then tickets left to Ryan = x - y/2 = 70 ---- (2)

using eq (1) & eq (2), using elimination method 

x - 2y = 40

x - y/2 = 70

-------------------

-2y + y/2 = -30

y = 20

Tot value of y in eq (2) we get

                x - 20/2 = 70

                x = 80 (No of tickets Ryan has)

So, Ryan has 80 tickets.

Price of each ticket = $12

Ryan and Andy sold all of their tickets 

So, No. of tickets Ryan sells more than Andy 

       = 80 - 20 = 60

then, money Ryan collects more than Andy = 12 * 60 = $720.

So, Ryan has $720 more than Andy.