Algebra

Oh, dude, you're hitting me with some fractions here. So, five sixths of two is 10/6, and two fifths is 4/5. When you add them together, you get 10/6 + 4/5, which simplifies to 50/30 + 24/30, giving you a grand total of 74/30. Math can be fun, right?

Well, honey, if you're looking to add something to 145 to make it 180, you just need to toss in a 35. It's as simple as that. Just like adding a cherry on top of a sundae, but with numbers.

2x2 ÷ x = 2x

You should write the question out as z= (y/x). To solve this, you would multiple x to the other side to get y alone. Then you would multiply z to the other side.

z= (y/x) Given

xz= y

x= y/z

An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. These numbers have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include the square root of 2, pi, and the golden ratio. Irrational numbers are an essential part of mathematics and play a crucial role in various mathematical proofs and calculations.

Ah, let's take a moment to appreciate the beauty of division. When we divide P by 9, we are simply breaking P into 9 equal parts. It's like sharing a delicious cake with 9 friends - everyone gets a fair slice. Remember, there's no mistakes in math, just happy little numbers!

There are an infinite number of equal parts in the whole. If you divide something into four equal parts, there will be four parts in the whole. If you divide something into 100,000 equal parts, there will be 100,00 parts in the whole.

Oh, dude, it's like basic math time! So, if the perimeter of a square is 2.4, you just divide that by 4 to get the length of each side. That means each side is 0.6. Easy peasy lemon squeezy!

Let us first define a combination and a permutation. Say we have 5 letters and we want to pick two at a time. First let's say the order matters and we allow repetition.

There are 5 letters and we have 5 choices for the first letter and then 5 choices for the second. The total choices is 5x5 or 52. In general if we have n objects and we are picking r of them and allowing for repetition, then there are nr ways to do that. Remember that the order matters here.

If we don't allow for repetitions, then we have n! divided by (n-r)! where ! means factorial. For example, 3! means simply 3x2x1 and 5! means 5x4x3x2x1.

So in the case of the 5 letters taken two at a time where order matters we have

5!/((5-2)! which is 5x4x3x2x1 divided by 3x2x1. Cancel out the 3x2x1 and you are left with

5x4 or 20.

There is no need to remember this formula, just remember there are 5 choices for the first letter, but whatever you pick for the first one, you can't use that one again so there are 4 choices for the second. The total is given by the multiplication rule and we have 5x4=20.

In the case of the entire alphabet with 26 letters, if we allow repetitions, we have 262 permutations. There are 26 choices for the first letter and 26 for the second.

But if we don't allow repetitions, there are 26x25 permutations.

Now combination is the same thing, but order does not matter. So AB is the same as BA.

Let's just worry about either lower case or upper case letters first. Forget about things like Ab or cB for now. Only combinations we are looking at are like ab or AB, but NOT aB.

First let's talk about combinations where we don't allow repeats. This is much easier to understand. I will leave combinations with repetitions for last. So AB is OK, but not AA.

When we figures out how many permutations there were without repeats, we found that if we are taking r objects taken from n of them, we had n!/(n-r)!. Now since ab is the same as ba, for example, if we know how many permutations we have of two letters taken from the alphabet, we could just divide that by 2. This is because ever time we have something like dc and CD, they are the same as a combination since order no longer matters.

So in general we have n!/(n-r)! now divide by r! since if you have r items the number of ways to arrange all r of them is r!. For example, say you pick 5 letters out of the alphabet instead of 2, then there would be 26!/(26-5)! permutations, but each one of the 5 letter permutations can be arranged in 5! ways. For example, 5 choices for the first letter, 4 for the second, 3 for the third etc, which is 5!.

So the final formula for finding how many combinations of r items you have if you take the r of them from a total of n items ( such as 5 letters from 26) is n!/(n-r)!r!.

This is usually written nCr.

Now combinations with repetitions is the toughest one to see. This problem did not say if you have repeats or not. So the answer to the problem using ONLY lower or ONLY upper case an allowing no repeats is 26!/24!2!=325. If you allow upper and lower case letters, think of the the upper and lower case as totally separate things. So your alphabet now has

52 objects and the solutions becomes 52x51/2=1326

Now let's say we allow repetitions of the letters. So AA is ok for example. So we are looking at combinations so order does not matter and we allow repetitions.

Here is one common way to see this that is taught in many stats classes.

The method is commonly called bars and stars. Say we have n objects. Lets say 5.

Write down 5-1 or 4 vertical bars. In the case of n, draw n-1 vertical bars.

The 4 vertical bars look like this: | | | | you can see how they create 5 spaces from left to write. I will label the space in case you don't see it. 1| 2 |3 | 4 |5 where 1 is the first space etc. Now put stars in each space. The number or stars can be anything from 0 ( no stars) to r where you are picking r objects from n. BUT, the total number of stars is r. So for example is I pick 5 letters out of 26 and allow repetitions and only worry about combinations, then I have 25 bars and 5 stars. If I put all 5 stars in the first space, I am done. If I put 2 in the first space, I have 3 more to put anywhere I want.

Here is an example you can see: Example: n=3, r=5,

* | * | *** -> 1 object of type 1, 1 of type 2, 3 of type 3

| * | **** -> zero objects of type 1, 1 object of type 2, 4 objects of type 3.

Now, the number of combinations using bars and stars is equal to the number of rows of n-1 bars and r stars. The total number of bars and stars is n+r, then subtract 1 since we have n-1 bars. So we have n+r-1 items (bars and stars combined) and we are selecting the locations of the r stars out of the n+r-1 locations. The order does not matter. Therefore, the number of such rows is C(n+r-1,r), and this is also the number of combinations with repetitions allowed. It might be hard to remember unless you think of n-1 bars plus r stars is n+r-1 and you are finding where to place the r stars, but it does not matter which space they go in.

So in the case of 2 letter combinations taken from 26 letters and allowing repetitions, we have C(26+2-1,2) or C(27,2) This is 27x26/2=351 The general formula for this comes from just plugging the n+r-1 chose r into the formula for combinations.

This is just considering upper or lower case letters. Now how many combinations are there if you use both? I will leave that one for you!

In this question, the issue of repititions is now mentioned and upper or lower case is not noted either. In math we need precision, so the questions does not have a unique answer without that information.

The equation that has the solution x = -3, is, precisely:x = -3

If you want anything more fancy, you can add some number (the same number to both sides), multiply by some number (the same number to both sides), etc.

The angles of a triangle must add to a total of 180 degrees, so if you have 27 and 58 as two of your angle measurements the rest is easy.

27+58 = 85

180-85 = 95

Your third angle will be 95 degrees.

To convert centimeters to meters, you need to divide by 100 because there are 100 centimeters in 1 meter. Therefore, 650 cm equals 6.5 meters.

An obtuse angle is 130 degrees. Any angle more than 90 degrees is obtuse, any angle under 90 degrees is acute, and 90 degrees is a right angle.

To simplify 8 over 50, you need to find the greatest common factor of 8 and 50, which is 2. Divide both the numerator and denominator by 2 to get 4 over 25. Therefore, 8 over 50 simplified is 4 over 25.

It is: 2*2*5*5 = 100

Well, isn't that just a happy little math problem! When x is multiplied by 8, we get 8x. So, if we want to find the value of 5x, we simply multiply 5 by the value of 8x. That gives us 40x. Just remember, there are no mistakes in math, only happy accidents!

38.00

2+3 = 5 and 2*3 = 6

In the context of a research study, money can be considered an independent variable if it is being manipulated or controlled to observe its effects on other variables. For example, in a study on consumer behavior, researchers may manipulate the amount of money given to participants to see how it influences their purchasing decisions. However, money can also be considered a dependent variable if it is being measured or observed to understand how other factors, such as income level or economic policies, impact its distribution or circulation in an economy.

Oh, dude, you want me to do math? Fine, fine. So, like, all the multiplication problems that equal 54 are 1 x 54, 2 x 27, 3 x 18, 6 x 9... you get the idea. There are, like, several ways to get to 54, but who's counting, right?

Well, isn't that just a happy little math problem we have here! To find n, we can simply divide 450 by 15. When we do that, we find that n equals 30. Just like painting a beautiful landscape, sometimes all we need is a little bit of math to reveal the hidden beauty in numbers.

Well, honey, intersecting lines are like two cars crashing at an intersection - they meet at one point and then go their separate ways. Picture a big ol' plus sign or the letter "X" - those are some classic examples of intersecting lines. Just remember, when lines intersect, it's like a brief encounter in the busy city streets - they cross paths for a moment and then it's on to the next adventure.

Oh, dude, the Sears Tower, now known as the Willis Tower, is approximately 1,450 feet tall, with a base length of about 225 feet and a width of around 145 feet. So, like, it's pretty massive, you know? It's like looking up at a giant metal stick in the sky.

24.5

None!!!! Because ;feet; is a measure of length, and 'square yards' is a measure of area.

However, if you mean 'How many SQUARE feet in 1440 square yards/' Then remember 3 feet = 1 yard ( linear)

3ft x 3 ft = 9 sq.ft. = 1 sq. yd (area)

So multiply 1440 sq.yds x 9 = 12960 sq.ft.