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# Arithmetic for Praxis I: Pre-Professional Skills Test Study Guide (page 2)

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Updated on Jul 5, 2011

#### Prime and Composite Numbers

A positive integer that is greater than the number 1 is either prime or composite, but not both.

• A prime number is a number that has exactly two factors: 1 and itself.
Examples
2, 3, 5, 7, 11, 13, 17, 19, 23…
• A composite number is a number that has more than two factors.
Examples
4, 6, 8, 9, 10, 12, 14, 15, 16…
• The number 1 is neither prime nor composite since it has only one factor.

### Operations

Addition is used when it is necessary to combine amounts. It is easiest to add when the addends are stacked in a column with the place values aligned. Work from right to left, starting with the ones column.

Example
Add 40 + 129 + 24.
1. Align the addends in the ones column. Because it is necessary to work from right to left, begin to add starting with the ones column. The ones column totals 13, and 13 equals 1 ten and 3 ones, so write the 3 in the ones column of the answer, and regroup or "carry" the 1 ten to the next column as a 1 over the tens column so it gets added with the other tens:
2. Add the tens column, including the regrouped 1.
3. Then add the hundreds column. Because there is only one value, write the 1 in the answer.

#### Subtraction

Subtraction is used to find the difference between amounts. It is easiest to subtract when the minuend and subtrahend are in a column with the place values aligned. Again, just as in addition, work from right to left. It may be necessary to regroup.

Example
If Becky has 52 clients and Claire has 36, how many more clients does Becky have?
1. Find the difference between their client numbers by subtracting. Start with the ones column. Because 2 is less than the number being subtracted (6), regroup or "borrow" a ten from the tens column. Add the regrouped amount to the ones column. Now subtract 12 – 6 in the ones column.
2. Regrouping 1 ten from the tens column left 4 tens. Subtract 4 – 3 and write the result in the tens column of the answer. Becky has 16 more clients than Claire. Check by addition: 16 + 36 = 52.

#### Multiplication

In multiplication, the same amount is combined multiple times. For example, instead of adding 30 three times, 30 + 30 + 30, it is easier to simply multiply 30 by 3. If a problem asks for the product of two or more numbers, the numbers should be multiplied to arrive at the answer.

Example
A school auditorium contains 54 rows, each containing 34 seats. How many seats are there in total?
1. In order to solve this problem, you could add 34 to itself 54 times, but we can solve this problem more easily with multiplication. Line up the place values vertically, writing the problem in columns. Multiply the number in the ones place of the top factor (4) by the number in the ones place of the bottom factor (4): 4 × 4 = 16. Because 16 = 1 ten and 6 ones, write the 6 in the ones place in the first partial product. Regroup or carry the ten by writing a 1 above the tens place of the top factor.
2. Multiply the number in the tens place in the top factor (3) by the number in the ones place of the bottom factor (4): 4 × 3 = 12. Then add the regrouped amount: 12 + 1 = 13. Write the 3 in the tens column and the 1 in the hundreds column of the partial product.
3. The last calculations to be done require multiplying by the tens place of the bottom factor. Multiply 5 (tens from bottom factor) by 4 (ones from top factor); 5 × 4 = 20, but because the 5 really represents a number of tens, the actual value of the answer is 200 (50 × 4 = 200). Therefore, write the two zeros under the ones and tens columns of the second partial product and regroup or carry the 2 hundreds by writing a 2 above the tens place of the top factor.
4. Multiply 5 (tens from bottom factor) by 3 (tens from top factor); 5 × 3 = 15, but because the 5 and the 3 each represent a number of tens, the actual value of the answer is 1,500 (50 × 30 = 1,500). Add the two additional hundreds carried over from the last multiplication: 15 + 2 = 17 (hundreds). Write the 17 in front of the zeros in the second partial product.
5. Add the partial products to find the total product:

Note: It is easier to perform multiplication if you write the factor with the greater number of digits in the top row. In this example, both factors have an equal number of digits, so it does not matter which is written on top.

#### Division

In division, the same amount is subtracted multiple times. For example, instead of subtracting 5 from 25 as many times as possible, 25 – 5 – 5 – 5 – 5 – 5, it is easier to simply divide, asking how many 5s are in 25: 25 ÷ 5.

Example
At a road show, three artists sold their beads for a total of \$54. If they share the money equally, how much money should each artist receive?
1. Divide the total amount (\$54) by the number of ways the money is to be split (3). Work from left to right. How many times does 3 divide 5? Write the answer, 1, directly above the 5 in the dividend, because both the 5 and the 1 represent a number of tens. Now multiply: since 1(ten) × 3(ones) = 3(tens), write the 3 under the 5, and subtract; 5(tens) – 3(tens) = 2(tens).
2. Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide 24? Write the answer, 8, directly above the 4 in the dividend. Because 3 × 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0.

#### Remainders

If you get a number other than zero after your last subtraction, this number is your remainder.

Example
9 divided by 4.
1 is the remainder.
The answer is 2 r1. This answer can also be written as , because there was one part left over out of the four parts needed to make a whole.

### Working with Integers

Remember, an integer is a whole number or its opposite. Here are some rules for working with integers:

Adding numbers with the same sign results in a sum of the same sign:

(positive) + (positive) = positive and (negative) + (negative) = negative
1. Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs.
2. Keep the sign of the number with the larger positive value.
Example
–2 + 3 =
1. Subtract the positive values of the numbers: 3 – 2 = 1.
2. The number 3 is the larger of the two positive values. Its sign in the original example was positive, so the sign of the answer is positive. The answer is positive 1.
Example
8 + –11 =
1. Subtract the positive values of the numbers: 11 – 8 = 3.
2. The number 11 is the larger of the two positive values. Its sign in the original example was negative, so the sign of the answer is negative. The answer is negative 3.

#### Subtracting

When subtracting integers, change the subtraction sign to addition and change the sign of the number being subtracted to its opposite. Then follow the rules for addition.

Example
(+10) – (+12) = (+10) + (–12) = –2
(–5) – (–7) = (–5) + (+7) = +2

#### Multiplying and Dividing

A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when multiplying or dividing two quantities, the answer will be positive. If the signs are different, the answer will be negative.

Example
(10)(–12) = –120
–5 × –7 = 35

#### Sequence of Mathematical Operations

There is an order in which a sequence of mathematical operations must be performed:

P: Parentheses/Grouping Symbols. Perform all operations within parentheses first. If there is more than one set of parentheses, begin to work with the innermost set and work toward the outside. If more than one operation is present within the parentheses, use the remaining rules of order to determine which operation to perform first.
E: Exponents. Evaluate exponents.
M/D: Multiply/Divide. Work from left to right in the expression.
A/S: Add/Subtract. Work from left to right in the expression.

This order is illustrated by the acronym PEMDAS, which can be remembered by using the first letter of each of the words in the phrase: Please Excuse My Dear Aunt Sally.

Example

### Properties of Arithmetic

There are several properties of mathematics:

• 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.
Example
5 × 2 = 2 × 5
5a = a5
b + 3 = 3 + b
• However, neither subtraction nor division is commutative, because reversing the order of the numbers does not yield the same result.

Example
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.
Example
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.
Example
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)
= 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 of the sets of numbers defined previously (counting numbers, integers, rational numbers, etc.), the additive identity is 0.
Example
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 of the sets of numbers defined previously (counting numbers, integers, rational numbers, etc.) the multiplicative identity is 1.
Example
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 .

Here are some numbers and their reciprocals:

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

### Factors and Multiples

#### Factors

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.

Example
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

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

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

The most important thing to remember about decimals is that the first place value to the right of the decimal point is the tenths place. The place values 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 × .1) + (4 × .01) + (5 × .001) + (7 × .0001)

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

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

### Multiplication of Decimals

Multiplication of decimals is exactly the same as multiplication of integers, except one must make note of the total number of decimal places in the factors.

Example
What is the product of 0.14 and 4.3?

First, multiply as usual (do not line up the decimal points):

Now, to figure out the answer, 4.3 has one decimal place and .14 has two decimal places. Add 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. When finished multiplying, start from the right side of the answer, and move to the left the number of decimal places previously calculated.

In this example, 602 turns into .602, because there have to be three decimal places in the answer. 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 problem; therefore, the answer must contain three decimal places. Starting to the right of 6, move left three places. The answer becomes 0.006.

### Dividing Decimals

Dividing decimals is a little different from integers for the setup, and then the regular rules of division apply. It is easier to divide if the divisor does not have any decimals. In order to accomplish that, simply move the decimal place to the right as many places as necessary to make the divisor a whole number. If the decimal point is moved in the divisor, it must also be moved in the dividend in order to keep the answer the same as the original problem; 4 ÷ 2 has the same solution as its multiples 8 ÷ 4 and 28 ÷ 14. 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 answer to accommodate the required move, simply add zeros until the desired placement is achieved. 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

First, to make .425 a whole number, move the decimal point three places to the right: 425. Now move the decimal point three places to the right for 1.53: 1,530. The problem is now a simple long division problem.

### Comparing Decimals

Comparing decimals is actually quite simple. Just line up the decimal points and then fill in zeros at the ends of the numbers until each one has an equal number of digits.

Example
Compare .5 and .005.

Line up decimal points.         .5

.005

.005
Now, ignore the decimal point and consider, which is bigger: 500 or 5?
500 is definitely bigger than 5, so .5 is larger than .005.

### Rounding Decimals

It is often inconvenient to work with very long decimals. Often it is much more convenient to have an approximation for a decimal that contains fewer digits than the entire decimal. In this case, we round decimals to a certain number of decimal places. There are numerous options for rounding:

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, we look at two digits of the decimal: the digit we are rounding to, and the digit to the immediate right. If the digit to the immediate right is less than 5, we leave the digit we are rounding to alone, and omit all the digits to the right of it. If the digit to the immediate right is 5 or greater, we increase the digit we are rounding by one, and omit all the digits to the right of it.

Example
Round to the nearest tenth and the nearest hundredth.
Dividing 3 by 7 gives us the repeating decimal .428571428571. … If we are rounding to the nearest tenth, we need to look at the digit in the tenths position (4) and the digit to the immediate right (2). Because 2 is less than 5, we leave the digit in the tenths position alone, and drop everything to the right of it. So, to the nearest tenth is .4.
To round to the nearest hundredth, we need to look at the digit in the hundredths position (2) and the digit to the immediate right (8). Because 8 is more than 5, we increase the digit in the hundredths position by 1, giving us 3, and drop everything to the right of it. So, to the nearest hundredth is .43.

### Fractions

To work well with fractions, it is necessary to understand some basic concepts.

### Simplifying Fractions

Rule:

• 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
The GCF of 63 and 72 is 9, so divide 63 and 72 each by 9 to simplify the fraction:

• ### Adding and Subtracting Fractions

Rules:

To add or subtract fractions with the same denominator:

To add or subtract fractions with different denominators:

• 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

### Multiplication of Fractions

Rule:

• Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators.
Example
• If any numerator and denominator have common factors, these may be simplified before multiplying. Divide the common multiples by a common factor. In the following example, 3 and 6 are both divided by 3 before multiplying.

Example

### Dividing Fractions

Rule:

• Dividing fractions is equivalent to multiplying the dividend by the reciprocal of the divisor. When dividing fractions, simply multiply the dividend by the divisor's reciprocal to get the answer.
Example
(dividend) ÷ (divisor)
•

Determine the reciprocal of the divisor:

Multiply the dividend by the reciprocal of the divisor and simplify if necessary.

### Comparing Fractions

Rules:

Sometimes it is necessary to compare the sizes of fractions. This is very simple when the fractions are familiar or when they have a common denominator.

Example
• If the fractions are not familiar and/or do not have a common denominator, there is a simple trick to remember. 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.
Example
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,

### Converting Decimals to Fractions

• To convert a nonrepeating decimal to a fraction, the digits of the decimal become the numerator of the fraction, and the denominator of the fraction is a power of 10 that contains that number of digits as zeros.
Example
Convert .125 to a fraction.
The decimal .125 means 125 thousandths, so it is 125 parts of 1,000. An easy way to do this is to make 125 the numerator, and since there are three digits in the number 125, the denominator is 1 with three zeros, or 1,000.
•

Then we just need to reduce the fraction.
• When converting a repeating decimal to a fraction, the digits of the repeating pattern of the decimal become the numerator of the fraction, and the denominator of the fraction is the same number of 9s as digits.
Example
Convert to a fraction.
•

You may already recognize as . The repeating pattern, in this case 3, becomes our numerator. There is one digit in the pattern, so 9 is our denominator.

Example

Convert to a fraction.
The repeating pattern, in this case 36, becomes our numerator. There are two digits in the pattern, so 99 is our denominator.