Commutations 1

The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and and 2 trainees, how many different such teams are possible?

This question is an example of a problem that you will need your nCr button on your calculator to solve. First of all, the company can choose 1 experienced plumber out of 4. That's 4 take 1 (4 nCr1), or 4. The company can choose 2 trainees out of 4. That's 4 take 2 (4nCr2), or 6.

Since there are two different types of plumber, you have two different spots to place these numbers: _ times (multiplied by)_. Using blanks and mulitplying is key to commutations.

6 times 4 is 24, the answer.

Difficult functions 1

Let fx be defined as fx= x squared - x for all values of x. if fa = fa-2, what is the value of a?

A. 1

B. 1/2

C. 3/2

D. 6/5

E. 3

For a function problem like this you must plug in the values of a which you have been given. First plug a into the function, and you will find that a squared - a equals fa -2. To find the function of a-2, plug it into the function to get a squared-4a+4 - a + 2.

So now you have an equation to work:

a squared-a = a squared- 4a + 4 - a +2

a squared cancels, now add like terms

-a= -5a +6

add +5a to both sides

4a= 6

divide both sides by 4, then reduce

a= 3/2

3/2 is C, the answer.

Visualizing Cubes 1

The cube shown above has edges of length 2, and A and B are midpoints of the two edges. What is the length of AB (not shown)?

A. square root of 2

B. square root of 3

C. square root of 5

D. square root of 6

E. square root of 10

This question is intimidating, but with intimidating questions you must first gather what information is obvious and then proceed from what you have. All that you may notice now is that if A and B are the midpoints of the edges with lengths of 2, than the lengths from the closest corners to A and B must both have lengths of 1.

Now the question mentions that AB is not shown. All that must be done to show AB is draw a simple line with your pencil:

Wait, couldn't you makes triangle out of that? And the lengths of triangles are usually simples to find. In fact, you already know that the length from the corner to B=1.

Finding the length of A to that very same corner is a bit more complicated. You're going to need to use even more triangles. Notice how the floor of the cube is a square. You've drawn a line on that square from A to the upper corner. That square looks like this:

You know the length of A to the closer corner=1 because A is the midpoint of the side which equals 2, and you know that the length of the side of the square=2 because you were given that information.

Now it's time to use the Pythagorean theorem (a squared+b squared=c squared), which yields to us that the line from A to the corner= the square root of 5.

Now we refer back to the entire cube:

You now know the lengths of two sides of the right triangle (you know that is right because all the angles in a cube or square are right).

Now you must employ the Pythagorean theorem once more. The square root of 5 squared again is simply 5. 1 squared by itself is simply 1. The square root of the sum of these, 6, is the answer.

D's the correct answer.

Inclusive Sums 1

If the sum of the consecutive integers from -22 to x, inclusive, is 72, what is the value of x?

A. 23

B. 25

C. 50

D. 75

E. 94

With problems like this it is important to realize, first of all, that since 22 is negative, all positive integers up until 22 will cancel out. All that remains then is to find the numbers after 22 which have a sum of 72. A number line will help illustrate this concept:

___________________

-3, -2, -1, 0, 1, 2, 3

The sum of the negative numbers from -3 to -1 is -6. The sum of the positive numbers 1 to 3 is 6. -6 and 6 together are 0, so they cancel out.

Back at the original problem, you can cancel out all numbers up to positive 22. However, this doesn't help us, since all the options listed have higher values than 22.

However, you have another way of finding the solution. You can plug in the given answers. The number that is the answer, added with the number values that come before it but are after 22, will have a value of 72.

Is 23 the answer? No, since no numbers can be added to it, it alone has a value of 23 and not 72. Try the next one down, 25. The number values after 22, but before 25, are 23 and 24.

23+24+25=72!

B's the answer.