We know that and .

Because 90 falls is approximately halfway between 81 and 100, the square root of 90 is approximately halfway between 9 and 10, or 9.5.

To place in order, first we must find a common denominator and convert all fractions to that denominator.

have a common denominator of .

have a common denominator of .

have a common denominator of .

Therefore we can use common denominators to make all of the fractions look similar.  Then the ordering becomes trivial.

For percent problems there are verbal cues:

"IS" means equals and "OF" means multiplication.

Then the equation to solve becomes:

We are looking for a number that is not real. 

, , and  are irrational numbers, but they are still real. 

Then,  is equivalent to  by the rules of complex numbers. Thus, it is also real. 

That leaves us with:  which in fact is imaginary (since no real number multiplied by itself yields a negative number) and simplifies to . 

Real numbers can be found anywhere on a continuous number line ranging from negative infinity to positive infinity; therefore, all of the numbers are real numbers. 

How many cards in the deck are either a spade or a 3?

There are thirteen spades, including a 3 of spades.

There are four 3's, including a 3 of spades. 

Since we are counting the same card (3 of spades) twice, there are actually

distinct cards that fit the criteria of being either a spade or a 3.

Since any of the 52 cards is equally likely to be drawn, the probability that it is a spade or a 3, is

To find the distance on a number line:

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

Chenge the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator to get

.

Every matrix has a dimension, which is represented as the number of rows and columns. For example, a matrix with three rows and two columns is said to have dimension 3 x 2. 

The matrix 

has two rows and three columns, so its dimension is 2 x 3. (Remember that rows go from left to right, while columns run up and down.)

Matrix multiplication is defined only if the number of columns in the first matrix is equal to the number of rows on the second matrix. The easiest way to determine this is to write the dimension of each matrix. For example, let's say that one matrix has dimension a x b, and the second matrix has dimension c x d. We can only multiply the first matrix by the second matrix if the values of b and c are equal. It doesn't matter what the values of a and d are, as long as b (the number of columns in the first matrix) matches c (the number of rows in the second matrix).

Let's go back to the problem and analyze the choice .

The dimension of the first matrix is 2 x 3, because it has two rows and three columns. The second matrix has dimension 2 x 2, because it has two rows and two columns.

We can't multiply these matrices because the number of columns in the first matrix (3) is not equal to the number of rows in the second matrix (2). Thus, this product is not defined.

The answer is .