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1 - The first child has 6 choices

2 - The sibling of the first child has 5 - 1=4 choices [since he/she cannot sit directly in front of his /her sibling]

3 - The first sibling of the 2nd pair has 4 choices.

4 - His/her sibling has 3- 1 =2 choices [Same reason as in 2 above]

5 - The first sibling of the 3rd pair and his /her sibling have the remaining 2 chairs.

6 - The total number of ways of seating the 3 pairs of siblings=6 x 4 x 4 x 2 x 2 x 1 =384 ways.

If the altitudes of the triangle are: AD=12  and  BE=12, then that means the triangle ABC is an isosceles triangle.

1/12 + 1/12 > 1/CF, solve for CF

CF > 6 

Which means the minimum altitude of CF = 7 and there is no ceiling for the largest possible value.

(p + 2q) + (6p - 8q) + (17p + 2q)=180, solve for p

From the above eqation we obtain:

p = (q + 45) / 6.      Try q=1, 2, 3..........etc.

The smallest q that balances the equation =3

p =(3 + 45) / 6 ==8 - the smallest value of p that balances the equation.

24^ -1 mod 11^2 ==116 - because:

24*116 mod 11^2 ==1

2,784 mod 11^2 == 1

The modular inverse of 24 mod 121 is 96. This can be found using the Extended Euclidean Algorithm.

def extended_euclidean_algorithm(a, b): if b == 0: return (a, 1, 0) else: q, r = divmod(a, b) x, y, z = extended_euclidean_algorithm(b, r) return (z, y - q * z, x) def modular_inverse(a, m): g, x, y = extended_euclidean_algorithm(a, m) if g != 1: raise ValueError("No modular inverse exists") else: return x % m print(modular_inverse(24, 121))

This prints:

96

So the answer is 96.

In 200 seconds, the little monsters can do 200 / 12 = 16 overhead presses. In 200 seconds, the little monsters can do 200 / 30 = 6.67 squats. Since 6.67 is not a whole number, the little monsters will do 6 squats and 1 squat plus 12 seconds of other activity. During the 12 seconds of other activity, the little monsters will do 1 overhead press. Therefore, the little monsters will do an overhead press and a squat at the same instant 7 times during the 200 second drill. So the answer is 7

Let there be a number x such that multiplying each ratio by x gives the actual number of marbles there are of that color. We know \(1x+2x+3x+4x=180\), and we can simplify like terms to get \(10x=180\). Dividing by 10 gives \(x=18\), so to find the number of blue marbles we multiply \(3 \cdot 18=54\).

There are \(\boxed{54}\) blue marbles in the bag.

We can factor this expression as follows:

(ab + ac + bc)^3 - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(ab + ac + bc)^2 - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(ab^2 + ac^2 + bc^2 + 2abc) - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(a^2b^2 + a^2c^2 + b^2c^2 + 2abc + a^2bc + a^2cb + b^2ac + c^2ab) - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(a^2(b^2 + c^2) + b^2(c^2 + a^2) + c^2(a^2 + b^2) + 2(abc + a^2bc + a^2cb + b^2ac)) - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2(abc + a^2bc + a^2cb + b^2ac)) - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2abc(a + b + c)) - (a^3b + a^3c + b^3a + b^3c + c^3a + c^3b) = (ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2abc(a + b + c) - (a^2b + a^2c + b^2a + b^2c + c^2a + c^2b)) = (ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2abc(a + b + c) - (a^2 + ab + ac + b^2 + bc + c^2)) = (ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2abc(a + b + c) - (a + b + c)(a^2 + b^2 + c^2)).

The expression in the parentheses is a perfect square, so we can factor it as follows:

(ab + ac + bc)(a^2(b + c)^2 + b^2(a + c)^2 + c^2(a + b)^2 + 2abc(a + b + c) - (a + b + c)(a^2 + b^2 + c^2)) = (ab + ac + bc)(a + b + c)((a + b)^2 + (b + c)^2 + (c + a)^2 - (a^2 + b^2 + c^2)) = (ab + ac + bc)(a + b + c)(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc).

Therefore, the complete factorization is:

(ab + ac + bc)(a + b + c)(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc).

Triangle ABC has altitudes AD, BE, and CF.  If AD = 12, BE = 12, and CF is a positive integer, then find the largest possible value of CF.

Answered yesterday.

The perimeter of triangle ABC is 15 + 4*sqrt(3).

i too need help

Good work, Double-oh Seven, but with a simple mistake, I think.  

You lost me at    

                        Substituting the value of T back into Equation 4:

                        (80/3) - B = -235 

                        B = 235 - (80/3)      

Shouldn't this have been      B = 235 + (80/3) ???????  

From that we get                               705  +  80  

                                              B  =  ——–————  

                                                                3  

Which is                                 B  =  785 / 3  =  261.666667  

and that's the same answer that I got when I solved (tried to solve?)  

this problem at  https://web2.0calc.com/questions/system_86906  

I thought maybe there'd been something wrong, either with the problem  

or with my feeble efforts, since problems like this usually come out even.  

_.

\(Expected value = \frac{n_3\times3+n_8\times8}{n_3+n_8}\\ a) 6 = \frac{8\times3+(2+n)\times8}{8+2+n}\)

Solve for n.

Let \(S(n)=\frac{n(n+1)(2n+1)}{6}\)

Then you want  \(S(100)-S(20)\)

Let's assume the cost of one piece of toast is T cents, the cost of one bagel is B cents, and the cost of one muffin is M cents.

From the given information, we can create the following equations:

Equation 1: 2T + B = 315 (Mary's order)

Equation 2: B + M = 350 (Gary's order)

Equation 3: T + 2B + 3M = 815 (Larry's order)

To solve these equations, we can use a method called substitution.

From Equation 2, we can express M in terms of B: M = 350 - B

Now, let's substitute this value of M into Equation 3:

T + 2B + 3(350 - B) = 815

T + 2B + 1050 - 3B = 815

T - B = -235 ...(Equation 4)

We now have two equations with two variables: Equation 1 (2T + B = 315) and Equation 4 (T - B = -235).

Solving these equations simultaneously, we get:

2T + B = 315 ...(Equation 1)

T - B = -235 ...(Equation 4)

By adding Equation 1 and Equation 4 together, we eliminate B:

3T = 80

T = 80 / 3

Substituting the value of T back into Equation 4:

(80/3) - B = -235

B = 235 - (80/3)

Calculating the value of B:

B = 705/3

B = 235

Therefore, one bagel costs 235 cents.

I hope this answer help you to solve your question. 

Thanks for the help! I already tried that answer and it's incorrect.

1. The formula for solving the area of a right triangle is 1/2 b * h. 

The base is 6, and the height is 8. 

That means that the 1/2 b * h = 3 * 8.

Since 3*8 = 24, the area of the triangle is 24 square units.

2. The sum of all the angles in a triangle is 180. 

180 - 135 - 30 = 15 degrees.

a)  If there are 10 slips of paper in the bag, and 8 of them have 3 on them, then the expected value is 3. If we add one 8 to the bag, then the expected value becomes (8 * 1/11) + (3 * 10/11) = 3.63636..., which is still greater than 6. If we add two 8's to the bag, then the expected value becomes (8 * 2/12) + (3 * 10/12) = 5, which is equal to 6. Therefore, we need to add 2​ 8's to the bag to make the expected value equal to 6.  The answer is 2.

Let's assign the bubble gum price the variable x. 

Since the candy bar costs $1 more than the bubblegum, the candy bar is x + 1.

Adding the bubble gum and candy bar obtains 2x+1, which is $1.50.

Solving with algebra, we get that the price of the bubblegum is \(25\) cents.

Thanks for the help! Unfortunately, it's incorrect.

The area of triangle XYD is at most 24 if and only if the height of triangle XYD is at most 4. The height of triangle XYD is at most 4 if and only if point D is within the shaded region below.

[asy] unitsize(0.5 cm);

pair X, Y, Z, D;

X = (0,0); Y = (12,0); Z = (6,6*sqrt(3)); D = (6,3);

draw(X--Y--Z--cycle); draw(X--D);

label("X", X, SW); label("Y", Y, SE); label("Z", Z, NW); label("D", D, S);

draw((0,3)--(12,3)); [/asy]

The shaded region is a right triangle with legs of length 6 and 3. The area of this triangle is (6)(3)/2=9. Therefore, the probability that the area of triangle XYD is at most 24 is the ratio of the area of the shaded region to the area of triangle XYZ, which is 9/(12)(6)=1/8.