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# Arithmetic for Praxis I: Pre-Professional Skills Test Study Guide (page 4)

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Updated on Jul 5, 2011

### Converting Fractions to Decimals

• To convert a fraction to a decimal, simply treat the fraction as a division problem.
Example
Convert to a decimal.
So, is equal to .75.

### Converting Mixed Numbers to and from Improper Fractions

Rule:

• A mixed number is number greater than 1 that is expressed as a whole number joined to a proper fraction. Examples of mixed numbers are . To convert from a mixed number to an improper fraction (a fraction where the numerator is greater than the denominator), multiply the whole number and the denominator and add the numerator. This becomes the new numerator. The new denominator is the same as the original.
• Note: If the mixed number is negative, temporarily ignore the negative sign while performing the conversion, and just make sure you replace the negative sign when you're done.

Example
Convert to an improper fraction.

Using the conversion formula, .

Example
Convert to an improper fraction.

Temporarily ignore the negative sign and perform the conversion: .

The final answer includes the negative sign: .
• To convert from an improper fraction to a mixed number, simply treat the fraction like a division problem, and express the answer as a fraction rather than a decimal.
Example
Convert to a mixed number.
Perform the division: 23 ÷ 7 = .

### Percents

Percents are always "out of 100": 45% means 45 out of 100. Therefore, to write percents as decimals, move the decimal point two places to the left (to the hundredths place).

Here are some conversions you should be familiar with:

### Absolute Value

The absolute value of a number is the distance of that number from zero. Distances are always represented by positive numbers, so the absolute value of any number is positive. Absolute value is represented by placing small vertical lines around the value: |x|.

Examples
The absolute value of seven: |7|.
The distance from seven to zero is seven, so |7| = 7.
The absolute value of negative three: |–3|.
The distance from negative three to zero is three, so |–3| = 3.

### Positive Exponents

A positive exponent indicates the number of times a base is used as a factor to attain a product.

Examples
Evaluate 25.

In this example, 2 is the base and 5 is the exponent. Therefore, 2 should be used as a factor 5 times to attain a product:

25 = 2 × 2 × 2 × 2 × 2 = 32

### Zero Exponent

Any nonzero number raised to the zero power equals 1.

Examples
50 = 1       700 = 1       29,8740 = 1

### Negative Exponents

A base raised to a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent (absolute value of the exponent).

Examples

### Exponent Rules

• When multiplying identical bases, you add the exponents.
Examples
22 × 24 × 26 = 212
a2 × a3 × a5 = a10
• When dividing identical bases, you subtract the exponents.
Examples
• If a base raised to a power (in parentheses) is raised to another power, you multiply the exponents together.
Examples
(32)7 = 314 (g4)3 = g12

### Perfect Squares

The number 52 is read "5 to the second power," or, more commonly, "5 squared." Perfect squares are numbers that are second powers of other numbers. Perfect squares are always zero or positive, because when you multiply a positive or a negative by itself, the result is always positive. The perfect squares are 02, 12, 22, 32

Perfect squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…

### Perfect Cubes

The number 53 is read "5 to the third power," or, more commonly, "5 cubed." (Powers higher than three have no special name.) Perfect cubes are numbers that are third powers of other numbers. Perfect cubes, unlike perfect squares, can be either positive or negative. This is because when a negative is multiplied by itself three times, the result is negative. The perfect cubes are 03, 13, 23, 33

Perfect cubes: 0, 1, 8, 27, 64, 125…
• Note that 64 is both a perfect square and a perfect cube.

### Square Roots

The square of a number is the product of the number and itself. For example, in the statement 32 = 3 × 3 = 9, the number 9 is the square of the number 3. If the process is reversed, the number 3 is the square root of the number 9. The symbol for square root is and is called a radical. The number inside of the radical is called the radicand.

Examples
52 = 25; therefore, = 5

Because 25 is the square of 5, it is also true that 5 is the square root of 25.

The square root of a number might not be a whole number. For example, the square root of 7 is 2.645751311.… It is not possible to find a whole number that can be multiplied by itself to equal 7. Square roots of nonperfect squares are irrational.

### Cube Roots

The cube of a number is the product of the number and itself for a total of three times. For example, in the statement 23 = 2 × 2 × 2 = 8, the number 8 is the cube of the number 2. If the process is reversed, the number 2 is the cube root of the number 8. The symbol for cube root is the same as the square root symbol, except for a small 3: . It is read as "cube root." The number inside of the radical is still called the radicand, and the 3 is called the index. (In a square root, the index is not written, but it has an index of 2.)

Examples
53 = 125; therefore,

Like square roots, the cube root of a number might be not be a whole number. Cube roots of nonperfect cubes are irrational.

### Probability

Probability is the numerical representation of the likelihood of an event occurring. Probability is always represented by a decimal or fraction between 0 and 1, 0 meaning that the event will never occur, and 1 meaning that the event will always occur. The higher the probability, the more likely the event is to occur.

A simple event is one action. Examples of simple events are: drawing one card from a deck, rolling one die, flipping one coin, or spinning a hand on a spinner once.

### Simple Probability

The probability of an event occurring is defined as the number of desired outcomes divided by the total number of outcomes. The list of all outcomes is often called the sample space.

Examples
What is the probability of drawing a king from a standard deck of cards?

There are four kings in a standard deck of cards, so the number of desired outcomes is 4. There are 52 ways to pick a card from a standard deck of cards, so the total number of outcomes is 52. The probability of drawing a king from a standard deck of cards is . So, P(king) = .

Examples
What is the probability of getting an odd number on the roll of one die?
There are three odd numbers on a standard die: 1, 3, and 5. So, the number of desired outcomes is 3. There are six sides on a standard die, so there are 6 possible outcomes. The probability of rolling an odd number on a standard die is . So, P(odd) = .
Note: It is not necessary to reduce fractions when working with probability.

### Probability of an Event Not Occurring

The sum of the probability of an event occurring and the probability of the event not occurring = 1. Therefore, if we know the probability of the event occurring, we can determine the probability of the event not occurring by subtracting from 1.

Examples
If the probability of rain tomorrow is 45%, what is the probability that it will not rain tomorrow?

45% = .45, and 1 – .45 = .55 or 55%. The probability that it will not rain is 55%.

### Probability Involving the Word Or

Rule:

P(event A or event B) = P(event A) + P(event B) – P(overlap of event A and B)

When the word or appears in a simple probability problem, it signifies that you will be adding outcomes. For example, if we are interested in the probability of obtaining a king or a queen on a draw of a card, the number of desired outcomes is 8, because there are four kings and four queens in the deck. The probability of event A (drawing a king) is , and the probability of drawing a queen is . The overlap of event A and B would be any cards that are both a king and a queen at the same time, but there are no cards that are both a king and a queen at the same time. So the probability of obtaining a king or a queen is .

Examples
What is the probability of getting an even number or a multiple of 3 on the roll of a die?

The probability of getting an even number on the roll of a die is , because there are three even numbers (2, 4, 6) on a die and a total of 6 possible outcomes. The probability of getting a multiple of 3 is , because there are 2 multiples of three (3, 6) on a die. But because the outcome of rolling a 6 on the die is an overlap of both events, we must subtract from the result so we don't count it twice.

P(even or multiple of 3) = P(even) + P(multiple of 3) – P(overlap)

### Compound Probability

A compound event is performing two or more simple events in succession. Drawing two cards from a deck, rolling three dice, flipping five coins, having four babies are all examples of compound events.

This can be done "with replacement" (probabilities do not change for each event) or "without replacement" (probabilities change for each event).

The probability of event A followed by event B occurring is P(A) × P(B). This is called the counting principle for probability.

Note: In mathematics, the word and usually signifies addition. In probability, however, and signifies multiplication and or signifies addition.

Examples
You have a jar filled with three red marbles, five green marbles, and two blue marbles. What is the probability of getting a red marble followed by a blue marble, with replacement?

"With replacement" in this case means that you will draw a marble, note its color, and then replace it back into the jar. This means that the probability of drawing a red marble does not change from one simple event to the next.

Note that there are a total of ten marbles in the jar, so the total number of outcomes is 10.

If the problem was changed to say "without replacement," that would mean you are drawing a marble, noting its color, but not returning it to the jar. This means that for the second event, you no longer have a total number of 10 outcomes; you have only 9, because you have taken one red marble out of the jar. In this case,

### Statistics

Statistics is the field of mathematics that deals with describing sets of data. Often, we want to understand trends in data by looking at where the center of the data lies. There are a number of ways to find the center of a set of data.

### Mean

When we talk about average, we usually are referring to the arithmetic mean (usually just called the mean). To find the mean of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set.

average = (sum of set) ÷ (quantity of set)
Examples
Find the average of 9, 4, 7, 6, and 4.

The mean, or average, of the set is 6.
(Divide by 5 because there are five numbers in the set.)

### Median

Another center of data is the median. It is literally the center number if you arrange all the data in ascending or descending order. To find the median of a set of numbers, arrange the numbers in ascending or descending order and find the middle value.

• If the set contains an odd number of elements, then simply choose the middle value.
Examples
Find the median of the number set: 1, 5, 4, 7, 2.
•

First arrange the set in order—1, 2, 4, 5, 7—and then find the middle value. Because there are five values, the middle value is the third one: 4. The median is 4.

• If the set contains an even number of elements, simply average the two middle values.
Examples
Find the median of the number set: 1, 6, 3, 7, 2, 8.
•

First arrange the set in order—1, 2, 3, 6, 7, 8—and then find the middle values, 3 and 6. Find the average of the numbers 3 and 6: . The median is 4.5.

### Mode

Sometimes when we want to know the average, we just want to know what occurs most often. The mode of a set of numbers is the number that appears the greatest number of times.

Examples
For the number set 1, 2, 5, 9, 4, 2, 9, 6, 9, 7, the number 9 is the mode because it appears the most frequently.