Arithmetic for Praxis II ParaPro Test Prep Study Guide (page 2)
The practice quiz for this study guide can be found at:
This section covers the basics of mathematical operations and their sequence. It also reviews whole numbers, integers, fractions, decimals, percents, and estimation.
The Number Line
The number line is a graphical representation of the order of numbers. As you move to the right, the value increases. As you move to the left, the value decreases.
A number line can show the approximate values of certain numbers. For example, the following number line shows that –4.5 is halfway between –5 and –4 and that 2.2 is closer to 2 than it is to 3.
The following table will illustrate some comparison symbols:
In addition, the numbers being added are called addends. The result is called a sum. The symbol for addition is called a plus sign. In the following example, 4 and 5 are addends and 9 is the sum:
- 4 + 5 = 9
In subtraction, the number being subtracted is called the subtrahend. The number being subtracted from is called the minuend. The answer to a subtraction problem is called a difference. The symbol for subtraction is called a minus sign. In the following example, 15 is the minuend, 4 is the subtrahend, and 11 is the difference:
- 15 – 4 = 11
When two or more numbers are being multiplied, they are called factors. The answer that results is called the product. In the following example, 5 and 6 are factors and 30 is their product:
5 ×6 = 30
There are several ways to represent multiplication in this mathematical statement.
A dot between factors indicates multiplication:
5 · 6 = 30
Parentheses around any one or more factors indicate multiplication:
(5)6 = 30, 5(6) = 30, and (5)(6) = 30.
Multiplication is also indicated when a number is placed next to a variable: 5a = 30. In this equation, 5 is being multiplied by a.
In division, the number being divided by is called the divisor. The number being divided into is called the dividend. The answer to a division problem is called the quotient.
There are a few different ways to represent division with symbols. In each of the following equivalent expressions, 3 is the divisor and 8 is the dividend:
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. For example, 2, 3, 5, 7, 11, 13, 17, 19, and 23 are all prime numbers.
- A composite number is a number that has more than two factors. For example, 4, 6, 8, 9, 10, 12, 14, 15, and 16 are all composite numbers.
- The number 1 is neither prime nor composite since it has only one factor.
Whole numbers are the set of nonnegative numbers that are not expressed as a fraction or a decimal. For example, 0, 4, 39, and 3,318 are all whole numbers. For the ParaPro Assessment, you will be expected to know how to add, subtract, multiply, divide, compare, and order them.
Comparing and Ordering Whole Numbers
To compare and order whole numbers, it is essential that you are familiar with the place value system. The following table shows the place values for a very large number: 3,294,107.
- To compare or order whole numbers, you need to look at the digits in the largest place value of a number first.
- Compare 3,419 and 3,491.
Begin by comparing the two numbers in their largest place value. They both have the digit 3 in the thousands place. Therefore, you do not know which number is larger. Move to the smaller place values (to the right) of each number and continue comparing. The digit in the hundreds place for each number is 4.You still do not know which number is larger. However, when you compare the digits in the tens places, you see that the 9 is greater than the 1. That means 3,491 is greater than 3,419. This can be represented with the greater than symbol: 3,491 > 4,419.
- Put the following numbers in order from greatest to least: 307, 319, 139, 301.
To order these numbers, the digits in their place values must be compared. Three of the numbers have a 3 in the hundreds place, but one number has a 1 in the hundreds place. Therefore, 139 is the smallest number. Next the digits in the tens places must be compared with the remaining numbers. The tens digit in 319 is 1, and the tens digit in 307 and 301 is 0. Therefore, 319 is the largest number. To order 307 and 301, compare the digits in the ones place: 7 is greater than 1, so 307 is greater than 301.
The correct order of the numbers, from greatest to least, is 319, 307, 301, and 139.
Adding Whole Numbers
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.
- Add 40 + 129 + 24.
- 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 that it gets added with the other tens:
- Add the tens column, including the regrouped 1.
- Then add the hundreds column. Because there is only one value, write the 1 in the answer.
Subtracting Whole Numbers
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.
- If Becky has 52 clients and Claire has 36, how many more clients does Becky have?
- 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.
- 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.
Multiplying Whole Numbers
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.
- A school auditorium contains 54 rows, each containing 34 seats. How many seats are there in total?
- 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.
- 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.
- 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.
- 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.
- 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.
Dividing Whole Numbers
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.
- 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?
- 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 into 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).
- Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide into 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.
If you get a number other than zero after your last subtraction, this number is your remainder.
- What is 9 divided by 4?
- 1 is the remainder.
The answer is 2 R1. This answer can also be written as 2 , because there was one part left over out of the four parts needed to make a whole.
Estimating with Whole Numbers
Some questions on the ParaPro Assessment will ask you for an estimate. That means you will not need to find the actual answer, but should instead find an answer that is close to the actual answer. One way to solve estimation problems with whole numbers is to use numbers that are easy to work with, and that are close to the actual numbers.
- A television set weighs 21 pounds. About how much will a case weigh if it carries 46 television sets?
- The number 21 is close to 20, and 20 is much easier to work with than 21. The number 46 is close to 50, and 50 is much easier to work with than 46. To find the approximate weight of the 46 television sets, you can just multiply 20 by 50. A proper estimate would be 1,000 pounds.
An integer is a whole number or its opposite. Here are some rules for performing operations with integers:
Adding numbers with the same sign results in a sum of the same sign:
- (positive) + (positive) = positive
- (negative) + (negative) = negative
- When adding numbers of different signs, follow this two-step process:
- Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs.
- Keep the sign of the number with the larger positive value.
- –2 + 3 =
- Subtract the positive values of the numbers: 3 – 2 = 1.
- 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.
- 8 + –11 =
- Subtract the positive values of the numbers: 11 – 8 = 3.
- 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.
When subtracting integers, change the subtraction sign to an addition sign, and change the sign of the number being subtracted to its opposite. Then follow the rules for addition.
- (+10) – (+12) = (+10) + (–12) = –2
- (–5) – (–7) = (–5) + (+7) = +2
Multiplying and Dividing Integers
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.
(positive) × (positive) = positive
(positive) × (negative) = negative
(negative) × (negative) = positive
- (10)(–12) = –120
- –5 × –7 = 35
- 12 ÷ –3 = –4
- 15 ÷ 3 = 5
An exponent indicates the number of times a base is used as a factor to attain a product.
- 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
Any nonzero number raised to the zero power equals 1.
- 50 = 1 700 = 1 29,8740 = 1
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 … Therefore, the perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …
The number 53 is read as "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 … Therefore, the perfect cubes are 0, 1, 8, 27, 64, 125 …
The Order of Operations
There is an order in which a sequence of mathematical operations must be performed, known as PEMDAS:
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 and 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.
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 answer is 3.
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.
- 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.
- 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
- The answer is .
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.
- 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.
- 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.
- 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.
- 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.
- 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)
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.
- 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.
- 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.
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.
- 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.
- 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
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.
- What is the sum of 0.45, 0.8, and 1.36?
- Take away 0.35 from 1.06.
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.
- 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.
- 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.
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.
- 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.
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.
To simplify fractions, identify the greatest common factor (GCF) of the numerator and denominator and divide both the numerator and denominator by this number.
- The GCF of 16 and 24 is 8, so divide 16 and 24 each by 8 to simplify the fraction:
Adding and Subtracting Fractions
To add or subtract fractions with like denominators, just add or subtract the numerators and keep the denominator.
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.
To multiply fractions, simply multiply the numerators and the denominators.
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.
- 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.
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.
- 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.
- 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 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:
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