NotThatSmart

Cake and ladders is more for more basic levels. 

In my opinion, it gives awesome visual representations. 

It's inefficient for big numbers though...I would reccomend prime factorization for those numbers...

We want to try to find 3 numbers with an Least Common Multiple of 12.

We also want these integers to be as small as possible, but can't let it be 1.  

Let's take a look at the numbers \(2, 3,4\). 

The reciprocals of these numbers are

\(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\)

Let's add these fractions up. We get

\(\frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}\)

The common denominator is 12. Converting all fractions to a denominator of 12, we get

\(\frac{6}{12}+ \frac{4}{12}+ \frac{3}{12} = \frac{13}{12}\)

So 2,3 and 4 are our answers. 

Thanks! :)

Relatively prime means when two numbers only have 1 as their common factor. 

We want numbers under 20 that share NO common divisors other than 1. 

Any number that is a factor of 20 won't work, since they are divisble by themselves. 1 doesn't count. This excludes

\(2, 4, 5, 10, 20\)

Any even number also won't work since they are both divisble by 2. This excludes

\(6,8,12,14,16,18,20\)

Lastly, we exclude any numbers divible by 5. This excludes

\(5,10,15\)

This leaves

\(1,3,7,9,11,13,17,19\)

There are 8 numbers. 

So 8 is our answer. 

Thanks! :)

We want to find the LCM of 36, 90, and 385. 

We can use the cake and ladder method to complete this. We basically divide by prime numbers as long as two of the numbers are divisble by it. We write down the result and multiply the numbers in yellow. We get

Multyiplying the numbers in yellow, we get

\(2 * 3 * 3 * 5 * 2 * 77=13,860\)

So they ring the bell at 13,860 seconds. We want to convert into minutes. We get

\(13,860/60 = 231\)

So our answer is 231. 

Thanks! :)

We can do this using a tactic. 

First, let's find the number of ways we can arrange the 7 rolled numbers. 

We just have \(7! = 7*6*5*4*3*2*1\)

However, we can't just end here. Since there are two 6s, we must account for the duplicates.

We do this by dividing the entire thing by \(2!\)

Thus, we just have

\(7! / 2! = 2520 \)

So our answer is just 2520. 

Thanks! :)

First, let's calculate the number of ways to sit each group. 

There are 4 children. We can arrange them in \(4! = 4*3*2*1=24\) ways. 

Then, there are 4 parents. We can arrange them in \(4! = 4*3*2*1=24\) ways.

For every one arrangment of adults, there are 24 ways to arrange children, so we have

\(24*24=576\)

So our answer is 576. 

Let me know if I messed up, my counting is not the best.

Thanks! :)

In order to calculate the area of the resulting polyhedron, we can find the volume of the original cube and then subtract the pyrmaids cut off. 

First off, the area of the original cube can be represented with the equation \(V=s^3\) where s is the sidelenght. Plugging in 6, we get

\(V=6^3=216\)

Next up, let's calculate the volume of each pyramid he sliced off. 

Let's note that each pyramid removed has a BASE area of 1/4 each of the cube and the height is 1/2 the sidelength. 

We know this since all the post-cutoff edges are the same length. 

The volume of the cube is represented with the equation \(\frac{1}{3}bh\) where b is the base and h is the height. 

We know the base is \(b=\frac{1}{4}*6*6=9\) and the height is \(1/2*6 = 3\)

So we have

\(1/3*9*3 = 9\)

There are 8 corners on a cube so there are 8 pyramids, so we have \(9*8=72\)

Doing the subtraction, we get \(216-72=144\)

So our answer is 144. 

Thanks! :)

Tottenham10, you did a great job with simplifying the quadratic, but there was one tiny mistake. 

The descriminant is \(b^2-4ac\), not \(\sqrt{b^2-4ac}\). Other than that, you're good!

So, now we...

First, let's combine all like terms. We get;

\(x^{2}+8x+9\)

Now, the decriminant is \(b^2-4ac\) for any quadratics in the form of \(ax^2+bx+c\)

Thus, plugging in the numbers in the quadratic we have, we get

\(8^2-4(1)(9) \\ = 64- 36\\ =28\)

So 28 is our answer. 

Thanks! :)

Combining all like terms and expanding, we get that

\(6c+3=219c-27\)

Isolating c to one side, we get

\(-213c=-30\)

Dividing both sides by -213, we get

\(c=\frac{10}{71}\)

So 10/71 is our only answer. 

Thanks! :)