All Answers 358514 Answers

Let the initial number of $2, $5 and $10-notes be 7x, 2x and 3x respectively. After 5/7 of the $2-notes and 9 $5-notes are taken out, the number of remaining $2-notes and $5-notes are 2x and 3x-9 respectively. The total value of the remaining notes is 2x2 + (3x-9)5 + 3x10 = 307 11x = 262 x = 24 Therefore, the number of $10-notes is 3x = 3*24 = 72. So the answer is 72

We have ab = 20(a + b) = 20a + 20b,.......(*)

so

ab - 20a = 20b,

a(b - 20) = 20 b,

a = 20b/(b - 20).

a is to be positive so b must be greater than 20, and since the roles of a and b can be reversed a must also be greater than 20.

Let a = 20 + m, and b = 20 + n, then, substituting into (*), 

(20 + m)(20 + n) = 20(20 + m + 20 + n),

400 + 20m + 20n + mn = 800 + 20m + 20n.

mn = 400.

So we are looking for two positive integers having a product of 400.

(1)  m = 1, n = 400 from which a = 21, b = 420.

(2)  m = 2, n = 200  ...............  a = 22, b = 220.

(3)  m = 4, n = 100  ...............  a = 24, b = 120.

etc.

There are five others, (fifteen in all, counting reversals).

You have to specify a ceiling as to how high you want to check.

Checking pairs up to 100, you get the the following:

        a    b

1 = (70, 28)
2 = (60, 30)
3 = (45, 36)
4 = (40, 40)
5 = (36, 45)
6 = (30, 60)
7 = (28, 70)

Like this

This is only a section of the graph, the steps keep going.

2 to the power of any number must be greater than 0

3* a number that is greater than 0 is greater than 0

-4 and the outcome must be greater than -4  

Hence the range will be  y>-4  with and asymtote at  y=-4

CHECK

(a,b,c,d,e) = (2,-4,6,1,5)

We have that ab=20(a+b). Solving for b, we get b=20a/(20-a)​. We want b to be an integer, so a−20 must divide 20. The divisors of 20 are 1, 2, 4, 5, 10, and 20. For each divisor d, we can take a=20d, and then b=20. Thus, there are 6​ ordered pairs (a,b).

I claim that the system of equations has no solutions. Assume, for the sake of contradiction, that the equation does have solutions. Subtracting \(q\) from both sides in the second equation yields \(-2r = -2 \rightarrow r=1.\) Subtracting $p$ from both sides in the last equation yields \(r=9,\) contradiction, so the system does not have a solution \((p,q,r).\)

Note that since \(BY\) is both an angle bisector and a median, triangle \(ABC\) is isosceles with \(AB=BC\). Therefore, \(BC = \boxed{16}\). A simple application of the angle bisector theorem (using \(CZ\) as the angle bisector) yields \(\frac{BZ}{AZ} = \frac{BC}{AC} = \frac{1}{2},\) so \(BZ = \frac{1}{3} \cdot 16 = \boxed{ \frac{16}{3}}.\)

NEVER MIND THIS ANSWER.  I READ THE PROBLEM WRONG. 

_.

Hi! Thanks for responding so quickly, I disagree with your solution, however. You can plug in 1/8 as b and a as 2025/2022 but you won't get it to work. I think when you added the roots you did something wrong.

Let \(x\) be the total number of people polled. We have the equation \(\frac{7}{12}x = 280\) which yields the solution \(x=480\) so there are \(480\) people in total. Since \(\frac{7}{12}\) of all people are bored by cable news, \(\frac{5}{12}\) are not, so the answer to the problem is \(\frac{5}{12} \cdot 480 = \boxed{200}.\)

The sum of the roots of a quadratic equation is given by -b/a, and the product of the roots is given by c/a, where a, b, and c are the coefficients of the quadratic equation.

In this case, we know that the sum of the roots is b + 8b^2 + 2, and the product of the roots is 2023.

We can use these values to find a and b.

(b+1)+(8b^2+1)=b+8b^2+2 2b+2=b+8b^2+2 b=8b^2

We know that b is a real number, so b cannot be 0. Therefore, b = 1/8.

We can now find a.

-b/a+c/a=b+8b^2+2 -1/(8a)+2023/a=b+8b^2+2 -1/(8a)+2023/a=1/8+2023/8 2022/(8a)=2025/8 a=2025/2022

Therefore, a + b =2025/2022+1/8 =2051/2022.

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

"The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles  

_.

a = 4, b = 3.

The range is all real numbers.

The coefficient of x^3 is 1 - 7 = -6.

The series can be split into two parts:

1 + 4 + 2 + 8 + 3 + 12 + ... + 24 + 96 25 + 100

The first part is an arithmetic series with first term 1, common difference 2, and 25 terms. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first part is

(1 + 96)/2 * 25 = 2475

The second part is a geometric series with first term 25, common ratio 4, and 2 terms. The sum of a geometric series is equal to the first term divided by 1 minus the common ratio, so the sum of the second part is

25 / (1 - 4) = 100

The sum of the entire series is equal to the sum of the first part plus the sum of the second part, or 2475 + 100 = 2575.

10/49

The two plans will cost the same for 80 minutes.

Go to Desmos graphing calculator here and experiment with it:  https://www.desmos.com/calculator

(1)
a = (-3/7)^(1/3)/2^(2/3)
a = -(3/7)^(1/3)/2^(2/3)
a = -((-1)^(2/3) (3/7)^(1/3))/2^(2/3)

(2)Results
a = -3/2 (sqrt(5) - 1)
a = 3/2 (1 + sqrt(5))