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# Arithmetic for Praxis II ParaPro Test Prep Study Guide (page 4)

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Updated on Jul 5, 2011

### Properties of Arithmetic

While ETS says that the ParaPro Assessment will not test your knowledge of the properties of mathematics, they are very important to know.

Commutative Property: This property states that the result of an arithmetic operation is not affected by reversing the order of the numbers. Multiplication and addition are operations that satisfy the commutative property.

Examples
5 × 2 = 2 × 5
(5)a = a(5)
b + 3 = 3 + b

However, neither subtraction nor division is commutative, because reversing the order of the numbers does not yield the same result.

Examples
5 – 2 ≠ 2 – 5
6 ÷ 3 ≠ 3 ÷ 6

Associative Property: If parentheses can be moved to group different numbers in an arithmetic problem without changing the result, then the operation is associative. Addition and multiplication are associative.

Examples
2 + (3 + 4) = (2 + 3) + 4
2(ab) = (2a)b

Distributive Property: When a value is being multiplied by a sum or difference,multiply that value by each quantity within the parentheses. Then, take the sum or difference to yield an equivalent result.

Examples
5(a + b) = 5a + 5b
5(100 – 6) = (5 × 100) – (5 × 6)

This second example can be proved by performing the calculations:

5(94) = 5(100 – 6)
470 = 500 – 30
470 = 470

### Additive and Multiplicative Identities and Inverses

The additive identity is the value that, when added to a number, does not change the number. For all integers, the additive identity is 0.

Examples
5 + 0 = 5
–3 + 0 = –3
Adding 0 does not change the values of 5 and –3, so 0 is the additive identity.

The additive inverse of a number is the number that, when added to the number, gives you the additive identity.

Example
What is the additive inverse of –3?
This means, "What number can I add to –3 to give me the additive identity (0)?"
–3 + ___ = 0
–3 + 3 = 0

The multiplicative identity is the value that, when multiplied by a number, does not change the number. For all integers, the multiplicative identity is 1.

Examples
5 × 1 = 5
–3 × 1 = –3
Multiplying by 1 does not change the values of 5 and –3, so 1 is the multiplicative identity.

The multiplicative inverse of a number is the number that, when multiplied by the number, gives you the multiplicative identity.

Example
What is the multiplicative inverse of 5?
This means, "What number can I multiply 5 by to give me the multiplicative identity (1)?"
5 × ___ = 1
× 5 = 1

There is an easy way to find the multiplicative inverse. It is the reciprocal, which is obtained by reversing the numerator and denominator of a fraction. In the preceding example, the answer is the reciprocal of 5; 5 can be written as , so the reciprocal is .

Note: Reciprocals do not change signs.

Note: The additive inverse of a number is the opposite of the number; the multiplicative inverse is the reciprocal.

### Factors and Multiples

Factors are numbers that can be divided into a larger number without a remainder.

Example
12 ÷ 3 = 4

The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The common factors of two numbers are the factors that both numbers have in common.

Examples
The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 18 = 1, 2, 3, 6, 9, and 18.

From the examples, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From this list it can also be determined that the greatest common factor of 24 and 18 is 6. Determining the greatest common factor (GCF) is useful for simplifying fractions.

Example
Simplify .

The factors of 16 are 1, 2, 4, 8, and 16. The factors of 20 are 1, 2, 4, 5, and 20. The common factors of 16 and 20 are 1, 2, and 4. The greatest of these, the GCF, is 4. Therefore, to simplify the fraction, both numerator and denominator should be divided by 4.

Multiples are numbers that can be obtained by multiplying a number x by a positive integer.

Example
5 × 7 = 35

The number 35 is, therefore, a multiple of the number 5 and of the number 7. Other multiples of 5 are 5, 10, 15, 20, and so on. Other multiples of 7 are 7, 14, 21, 28, and so on.

The common multiples of two numbers are the multiples that both numbers share.
Example
Some multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36 …
Some multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48 …

Some common multiples are 12, 24, and 36. From the above it can also be determined that the least common multiple of the numbers 4 and 6 is 12, since this number is the smallest number that appeared in both lists. The least common multiple, or LCM, is used when performing addition and subtraction of fractions to find the least common denominator.

Example (using denominators 4 and 6 and LCM of 12)

### Decimals

It is very important to remember the place values of a decimal. The first place value to the right of the decimal point is the tenths place. The place values from thousands to ten thousandths are as follows:

In expanded form, this number can also be expressed as:
1,268.3457 = (1 × 1,000) + (2 × 100) + (6 × 10) + (8 × 1) + (3 × 0.1) + (4 × 0.01) + (5 × 0.001) + (7 × 0.0001)

### Comparing and Ordering Decimals

To compare or order decimals, compare the digits in their place values. It's the same process as comparing or ordering whole numbers. You just need to pay careful attention to the decimal point.

Example
Compare 0.2 and 0.05.

Compare the numbers by the digits in their place values. Both decimals have a 0 in the ones place, so you need to look at the place value to the right. 0.2 has a 2 in the tenths place while 0.05 has a 0 in the tenths place. Because 2 is bigger than 0, 0.2 is bigger than 0.05. You can show this as 0.2 > 0.05.

Example
Order 2.32, 2.38, and 2.29 in order from greatest to least.

Again, look at the place values of the numbers. All three numbers have a 2 in the ones place, so you cannot order them yet. Looking at the next place value to the right, tenths, reveals that 2.29 has the number 2 in the tenths place whereas the other numbers have a 3. So 2.29 is the smallest number. To order 2.32 and 2.38 correctly, compare the digits in the hundredths place. 8 > 2, so 2.38 > 2.32. The correct order from greatest to least is 2.38, 2.32, and 2.29.

### Rounding Decimals

It is often inconvenient to work with decimals. It is much easier to have an approximation value for a decimal. In this case, you can round decimals to a certain number of decimal places. The most common ways to round are as follows:

• To the nearest integer: zero digits to the right of the decimal point
• To the nearest tenth: one digit to the right of the decimal point (tenths unit)
• To the nearest hundredth: two digits to the right of the decimal point (hundredths unit)

In order to round, look at the digit to the immediate right of the digit you are rounding to. If the digit is less than 5, leave the digit you are rounding to alone, and omit all the digits to its right. If the digit is 5 or greater, increase the digit you are rounding by one, and omit all the digits to its right.

Example
Round 14.38 to the nearest whole number.
The digit to the right of the ones place is 3. Therefore, you can leave the digit you are rounding to alone, which is the 4 in the ones place. Omit all the digits to the right.
14.38 is 14 when rounded to the nearest whole number.
Example
Round 1.084 to the nearest tenth.
The digit to the right of the tenths place is 8. Therefore, you need to increase the digit you are rounding to by 1. That means the 0 in the tenths place becomes a 1. Then all of the digits to the right can be omitted.
1.084 is 1.1 to the nearest tenth.

Adding and subtracting decimals is very similar to adding and subtracting whole numbers. The most important thing to remember is to line up the numbers to be added or subtracted by their decimal points. Zeros may be filled in as placeholders when all numbers do not have the same number of decimal places.

Examples
What is the sum of 0.45, 0.8, and 1.36?
Take away 0.35 from 1.06.

### Multiplying Decimals

The process for multiplying decimals is exactly the same as multiplying whole numbers.Multiply the numbers, ignoring the decimal points in the factors. Then add the decimal point in the final product later.

Example
What is the product of 0.14 and 4.3?

Now, to figure out where the decimal point goes in the product, count how many decimal places are in each factor. 4.3 has one decimal place and 0.14 has two decimal places. Add these in order to determine the total number of decimal places the answer must have to the right of the decimal point. In this problem, there are a total of three (1 + 2) decimal places. Therefore, the decimal point needs to be placed three decimal places from the right side of the answer. In this example, 602 turns into 0.602. If there are not enough digits in the answer, add zeros in front of the answer until there are enough.

Example
Multiply 0.03 × 0.2.

There are three total decimal places in the two numbers being multiplied. Therefore, the answer must contain three decimal places. Starting to the right of 6 (because 6 is equal to 6.0), move left three places. The answer becomes 0.006.

### Dividing Decimals

To divide decimals, you need to change the divisor so that it does not have any decimals in it. In order to do that, simply move the decimal place to the right as many places as necessary to make the divisor a whole number. The decimal point must also be moved in the dividend the same number of places to keep the answer the same as the original problem. Moving a decimal point in a division problem is equivalent to multiplying a numerator and denominator of a fraction by the same quantity, which is the reason the answer will remain the same.

If there are not enough decimal places in the dividend (the number being divided) to accommodate the required move, simply add zeros at the end of the number. Add zeros after the decimal point to continue the division until the decimal terminates, or until a repeating pattern is recognized. The decimal point in the quotient belongs directly above the decimal point in the dividend.

Example
What is
To make 0.425 a whole number, move the decimal point three places to the right: 0.425 becomes 425. Now move the decimal point three places to the right for 1.53: You need to add a zero, but 1.53 becomes 1,530.

The problem is now a simple long division problem.

### Fractions

A fraction is a part of a whole, represented with one number over another number. The number on the bottom, the denominator, shows how many parts there are in the whole in total. The number on the top, the numerator, shows how many parts there are of the whole. To perform operations with fractions, it is necessary to understand some basic concepts.

### Simplifying Fractions

To simplify fractions, identify the greatest common factor (GCF) of the numerator and denominator and divide both the numerator and denominator by this number.

Example
Simplify
The GCF of 16 and 24 is 8, so divide 16 and 24 each by 8 to simplify the fraction:

To add or subtract fractions with like denominators, just add or subtract the numerators and keep the denominator.

Example

To add or subtract fractions with unlike denominators, first find the least common denominator or LCD. The LCD is the smallest number divisible by each of the denominators.

For example, for the denominators 8 and 12, 24 would be the LCD because 24 is the smallest number that is divisible by both 8 and 12: 8 × 3 = 24, and 12 × 2 = 24.

Using the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the appropriate factor to get the LCD, and then follow the directions for adding/subtracting fractions with like denominators.

Example

### Multiplying Fractions

To multiply fractions, simply multiply the numerators and the denominators.

Example

### Dividing Fractions

Dividing fractions is similar to multiplying fractions. You just need to flip the numerator and denominator of the divisor, the fraction being divided. Then multiply across, like you would when multiplying fractions.

Example
Solve: .
Flip the numerator and denominator of the divisor and change the symbol to multiplication.
Now multiply the numerators and the denominators, and simplify if necessary.
Because both the numerator and the denominator of can be divided by 2, the fraction can be reduced.

### Comparing Fractions

Sometimes it is necessary to compare the sizes of fractions. This is very simple when the fractions have a common denominator. All you have to do is compare the numerators.

Example
Compare and .
Because 3 is smaller than 5, is smaller than . Therefore, < .

If the fractions do not have a common denominator, multiply the numerator of the first fraction by the denominator of the second fraction. Write this answer under the first fraction. Then multiply the numerator of the second fraction by the denominator of the first one. Write this answer under the second fraction. Compare the two numbers. The larger number represents the larger fraction.

Examples
Which is larger: or ?
Cross multiply.
7 × 9 = 63           4 × 11 = 44
63 > 44; therefore,
Compare and .
Cross multiply.
6 × 6 = 36           2 × 18 = 36
36 = 36; therefore,

### 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 common conversions:

The practice quiz for this study guide can be found at:

Math for Praxis II ParaPro Test Prep Practice Problems