Pangolin14

Where k = chickens

g = goats

c = cows:

 c  = 2g

6g = 5k

6c = xk

First, we start by finding the number of cows that equal chickens.

We have that 6g/2g = 3, so 3c = 6g = 5k!

If 3c=5k and

  6c = xk

then x = 10

so 10 cows

:))))

What is the number of solutions to h + 3t = 8?

What is the probability that the frog goes over '8'? What happens if it gets three 3s?

Can you use modular arithmetic?

h+3t = 8

h = 2 (mod 3)

Let h be the number of heads.

Let t be the number of tails.

h = 3n + 2

so, t = 2-n

Ordered pairs of solutions:

(2, 2) ~ (5, 1) ~ (8, 0)

Non solutions that are greater than 8 but no extra leap over:

(0, 3) ~ (1, 3) ... count the rest, being mindful not to go an extra leap after you have reached >8.

Then answer is 3/ (non solutions + 3 )

We can form isosceles triangles with our angles.

We know that RQ is 8, and SQ is 2. 

However, we can't solve this problem without finding another isosceles triangle! This calls for a diagram! 

We can see now that ∆QTR is isosceles, with QT = QR = 8. 

But now we know that QT = 8, so ST = 8 - QS = 8 - 2 = 6!

 By AA similarity, ∆QTR is similar to ∆RST! By similar triangle ratios, RT/8 = 6/RT, and RT^2 = 48.

Thus, RT = 4 √3.

Also, by the base angles being equal, by AA similarity we have ∆PQR is similar to ∆QRT, and thus we have PQ/8 = 8/4√3 , so PQ = 64/4√3. 

Rationalizing the denominators, we have PQ = $\frac{16√3 }{3}$

There's a curious pattern... see if you can find it. 

Hint: 11 *  b+8 = ?

11 * 16 = 176

There are many ways to fill the license plate in. To demonstrate it, we create a template:

_ _ _ _ _ _ = 26 * 26 * 26 * 10 * 10 * 10

There are $26^3$ ways to choose the letters and $10^3$ ways to choose the numbers. Thus, the number of license plates are $26^3 \cdot 10^3= 17576000.$

Surface area of a sphere  $ = 4 \pi r$

Volume = $\frac{4}{3} \pi r$

Because they are three digits and whole, we know that r is divisible by 3. In other words, $r=3n$ where n is all integers.

Looking at the three digit part, we have $r = 300x+30y+3z,$ where x, y, and z are integer digits 0-9. 

It is impossible to determine a unique value for r and thus the question needs more information.

This problem is a common example of clever manipulations. 

The question: $$\frac{1}{a} + \frac{1}{b} \cdot (a-b) = 1 \text{ and } a \cdot b = 5. \text { What is } a^2 + b^2 ? $$

The solution: 

This can be simplified to $(a^2-b^2) = ab$, but we know that $ab=5,$ so $a^2-b^2=5.$ We now know that by substitution, $$a^2+b^2=2a^2+5.$$ 

Be careful asinus, I think you made a mistake!

$a^2+b^2=5\sqrt{5}!$

First, we evaluate the result of the equations without k. We get:

$$y = 6 x - 12, z = 22/3 - (7 x)/3$$

$$ x = 1/7 (22 - 3 z)$$

For $kx+z=3$ to agree with these equations, we need to solve for k.

Solving for k in terms of z, $$ k = \frac{ 7 (z - 3)}{3 z - 22}.$$

However, we want $kx+z \ne 3.$ 

See if you can work more on this on your own!