CliffsNotes CBEST - BTPS TESTING Ph.D., Jerry Bobrow Ph.D. & 8 more 2021

Arithmetic review
Arithmetic review

Common Math Operation Symbols

Math symbols represent a quick method to identify math operations of numbers. Below is a reference list of commonly used math symbols.

The Number System

The structure for representing and expressing math is called the number system. It is the foundation for all mathematics problems. In completing arithmetic problems on the CBEST, you will work with several basic sets of numbers and the number line.

Sets of Numbers

Numbers can be represented in a variety of ways and have special rules to express how they relate to other numbers. This section introduces the sets of numbers and their definitions. Note: The terms listed in the table are “basic” sets of numbers and do not include other types of numbers (e.g., irrational numbers) that do not relate to the CBEST.

Examples:

1. Which of the following are integers?

For this set, the integers are .

2. List the prime numbers between 0 and 50.

A prime number is an integer greater than 1 that can be divided only by itself or 1. Only the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 satisfy this definition for integers between 0 and 50.

Number Line

On a number line, the numbers to the right of 0 are positive. Numbers to the left of 0 are negative as follows:

Given any two integers on a number line, the integer located the farthest to the right is always larger, regardless of its sign (positive or negative). Note that fractions may also be placed on a number line and can be similarly compared.

Examples:

For each pair of values, select the one with the greater value.

1. −8, −3

−3 > −8 since −3 is farther to the right on the number line.

2. 

since 0 is farther to the right on the number line.

Basic Math Properties and Operations

The “Sets of Numbers” section described the different types of numbers. The numerical value of every real number fits between the numerical values of two other real numbers, and the result of adding or multiplying real numbers is always another real number. This fact is described as a closure property. For example, if you add two even numbers, the answer will always be an even number: 8 + 6 = 14. Therefore, the set of even numbers is called “closed for addition.” This section helps you understand these basic properties of mathematical operations, makes it easier for you to work with real numbers, and helps you conceptually understand how sets of numbers fit together.

Properties of Addition and Subtraction

There are four mathematical properties of addition that are fundamental mathematical building blocks: commutative, associative, identity, and inverse.

Properties of Multiplication and Division

The four mathematical properties of multiplication (commutative, associative, identity, and inverse) make it easier to solve problems. Before we discuss these properties, notice the different ways to show multiplication.

Multiplication Notation

The operation of multiplication may be represented in a number of ways. For example, the product of two numbers, a and b, may be expressed as

Distributive Properties

The distributive property is one of the most used properties in math and is considered a basis for understanding mental operations of math. To distribute means to “spread out,” and this basic operation makes it easier for you to work with numbers as you separate them into component parts. In the distributive property, it is possible to take a number and separately distribute it across the sum of two or more other numbers before it is either added to or subtracted. The following examples show two important distributive properties: multiplication over addition and multiplication over subtraction.

Math Words and Phrases

The CBEST math questions appear as word problems. Before we discuss the operations of addition, subtraction, multiplication, and division, this section introduces words that signal an operation. Use these examples as a reference to help you deconstruct and translate word problems into numeric equations.

Words That Signal an Operation

Math Operations and Number Rules

Math fluency begins with understanding the basic rules for real number operations: addition, subtraction, multiplication, and division.

Addition

Steps to Adding Integers

1. If the two integers have the same sign (either both positive or both negative), add the integers and keep the same sign.

2. If the two integers have different signs (one positive and one negative), subtract the integers and keep the sign of the integer with the greater value.

3. If the two integers are opposites, their sum is zero.

Examples:

Two Integers with the Same Sign

1. (−15) + (−8) = −23

2. 

Two Integers with Different Signs

3. −38 + 25 = −13

4. 

Two Integers That Are Opposites

5. 14 + (−14) = 0

Subtraction

Steps to Subtracting Integers

1. To subtract two integers (positive and/or negative), first change the sign of the number being subtracted.

2. Then add.

Examples:

1. −12 − 6 = −12 + −6 = −18

2. 

3. 

Note: When number values are positive, the “+” is dropped: +5 = 5.

Steps to Subtracting Integers If the Minus Sign Precedes Parentheses

1. If a minus sign precedes parentheses, it means everything within the parentheses should be subtracted. Therefore, using the same rule as in the subtraction of integers, change every sign within the parentheses to its opposite.

2. Then add.

As a formula, a − b = a + (the opposite of b).

Examples:

1. 16 − (−7) = 16 + 7 = 23

2. (−22) − 23 = (−22) + (−23) = −45

Multiplication

Steps to Multiplying Integers

1. If the two integers have the same sign, multiply the integers, and their product will be positive.

2. If the two integers have different signs, multiply the integers, and their product will be negative.

3. If the number of negative signs in the problem is even, the product is positive.

4. If the number of negative signs in the problem is odd, the product is negative.

Note: Zero times any integer always equals zero.

Examples:

1. (−3)(8)(−5)(−1)(−2) = 240

2. (−3)(8)(−1)(−2) = −48

3. (0)(5) = 0

4. (8)(9)(0)(3)(−4) = 0

Division

Steps to Dividing Integers

1. If the two integers have the same sign, divide the integers, and their quotient will be positive.

2. If the two integers have different signs, divide the integers, and their quotient will be negative.

3. If the first integer is zero and the divisor is any nonzero integer, the quotient will always be zero.

Important note: Dividing by zero is “undefined” and is not permitted. and are not permitted because there are no values for these expressions. The answer is not zero.

Examples:

1. 

2. 

3. 0 ÷ 5, also written as , = 0

Divisibility Rules

Divisibility rules are shortcuts that help you to quickly determine whether a number can be divided with no remainder. Memorizing the rules on p. 92 can help you immediately evaluate and rule out incorrect answer choices.

Examples:

1. The number 2,730 is divisible by which integers between 1 and 10?

2 — 2,730 ends in a 0.

3 — The sum of the digits is 12, which is divisible by 3.

5 — 2,730 ends in a 0.

6 — The rules for 2 and 3 both work.

7 — 2,730 ÷ 7 = 390.

Even though 2,730 is divisible by 10, 10 is not between 1 and 10.

2. The number 2,648 is divisible by which integers between 1 and 10?

2 — 2,648 ends in 8.

4 — 48, the number formed by the last two digits, is divisible by 4.

8 — 648, the number formed by the last three digits, is divisible by 8.

Grouping Symbols

Now that we have discussed the basic operating steps of adding, subtracting, multiplying, and dividing, let’s review the grouping symbols and order of operations. Note: Although order of operations is not a common question on the CBEST, it’s important to be familiar with how to group numbers mathematically.

Parentheses ( ), brackets [ ], and braces { } are frequently needed to group numbers in mathematics. Generally, parentheses are the only symbols you will see on the CBEST. Parentheses are used first, followed by brackets and then braces. Operations inside grouping symbols must be performed before any operations outside of the grouping symbols.

Parentheses are used to group numbers or variables. Calculations inside parentheses take precedence and should be performed before any other operations.

Order of Operations

There is an order in which the operations on numbers must be done so that everyone doing a problem involving several operations and parentheses will get the same results. The order of operations is:

Order of Operations

Tip: An easy way to remember the order of operations is PEMDAS: Please Excuse My Dear Aunt Sally (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).

Example:

Fractions

All rules for the arithmetic operations involving integers also apply to fractions. Fractions consist of two numbers: a numerator (above the line) and a denominator (below the line).

The fraction bar indicates division, or 1 ÷ 2.

The denominator indicates the number of equal parts into which something is divided. The denominator cannot be zero since dividing by zero is undefined.

The numerator indicates how many of these equal parts are contained in the fraction.

Thus, if the fraction is of a pie, then the denominator 5 indicates that the pie is divided into 5 equal parts, of which 3 (numerator) are in the fraction. Sometimes it helps to think of the dividing line (in the middle of a fraction) as meaning “out of.” In other words, also means 3 “out of” 5 equal pieces from the whole pie.

Negative Fractions

Fractions may be negative as well as positive. However, negative fractions are typically written with the negative sign next to the fraction bar. For example, .

You may sometimes see negative fractions expressed as .

Proper Fractions, Common Fractions, and Improper Fractions

Proper Fraction—a fraction where the numerator is smaller than the denominator. All proper fractions have a value that is less than one. For example, .

Common Fraction—a fraction in which the numerator and denominator are both integers. For example, .

Improper Fraction—there are two forms of improper fractions.

The first is a fraction where the numerator is greater than the denominator. For example, .

The second is a fraction where the numerator and the denominator are equal. Note: The result is always equal to 1. For example,

Mixed Numbers

When a fraction contains both a whole number and a fraction, it is called a mixed number. For example, and are both mixed numbers.

To change an improper fraction to a mixed number, divide the denominator into the numerator and write the remainder as a fraction. For example:

To change a mixed number to an improper fraction, multiply the denominator times the whole number, add in the numerator, and put the total over the original denominator. For example:

Examples:

1. Change to an improper fraction.

2. Change to a mixed number.

Reducing/Simplifying Fractions

On the CBEST, a fraction must be reduced to its lowest terms. To reduce to the lowest terms, divide both the numerator and denominator by the largest number that will divide them both evenly.

Examples:

1. 

2. 

3. 

Least Common Denominator

When adding or subtracting fractions, the denominators must be the same. If the denominators are not the same, you must change all denominators to their least common denominator (LCD). The LCD is also known as the least common multiple (LCM) of the denominators. After all the denominators are the same, add (or subtract) the fractions by adding (or subtracting) the numerators; notice the denominator remains the same.

One way to find this value is to make a list of the multiples for the values involved and then find the least common one.

Examples:

1. Find the LCM for 24 and 36.

Notice that 72 and 144 are both common multiples, but that 72 is the least common multiple.

Now apply this to the adding of fractions.

2. 

As we saw above, the LCD for 24 and 36 is 72. Since 24 is multiplied by 3 to get 72, the 5 is also multiplied by 3:

Since 36 is multiplied by 2 to get 72, the 7 is also multiplied by 2:

Now that the denominators are the same, add the numerators and keep the denominator:

Of course, if the denominators are already the same, keep that denominator, and simply add the numerators.

3. 

Adding and Subtracting Positive and Negative Fractions

he rules for integers apply to adding or subtracting positive and negative fractions.

Examples:

1. 

2. 

3. 

Adding Mixed Numbers

The rules for adding and subtracting integers also apply to mixed numbers. To add mixed numbers, add the fraction portions together, add the whole numbers, and then combine the two results.

Example:

1. 

Subtracting Mixed Numbers

When you subtract mixed numbers, sometimes you may have to “borrow” from the whole number, just as you sometimes borrow from the next column when subtracting ordinary numbers.

Examples:

1. 

2. 

3. 

4. 

5. 

Multiplying Fractions

The rules for multiplying and dividing integers also apply to multiplying and dividing fractions when working with positive and negative terms. To multiply fractions, multiply the numerators and then multiply the denominators. Reduce to lowest terms if necessary.

Examples:

1. 

2. 

3. 

Whole numbers can be written as fractions: .

Example:

4. 

When multiplying fractions, it is often possible to simplify the problem by cross-canceling. To cross-cancel, find a number that divides into one numerator and one denominator. In Example 5 that follows, 2 in the numerator and 12 in the denominator are both divisible by 2.

Examples:

5. 

6. 

Remember: You can cross-cancel only when multiplying fractions.

Multiplying Mixed Numbers

To multiply mixed numbers, first change any mixed number to an improper fraction. Then multiply as shown in the preceding section. To change mixed numbers to improper fractions:

1. Multiply the whole number by the denominator of the fraction.

2. Add this to the numerator of the fraction.

3. This is now your numerator.

4. The denominator remains the same.

Examples:

1. 

2. 

Dividing Fractions

To divide fractions or mixed numbers, invert (turn upside down) the second fraction (the one “divided by”) and multiply. Then reduce if necessary.

Examples:

1. 

2. 

3. 

Complex Fractions

Sometimes a division of fractions problem may appear in the following form, called complex fractions.

The line separating the two fractions means “divided by.” This problem may be rewritten as . Now follow the same procedure as previously shown.

Decimals

As you work with real numbers, it is also important to understand that each position in any decimal number has place value.

Place Value

For example, in the number 485.03, the 4 is in the hundreds place, the 8 is in the tens place, the 5 is in the ones place, the 0 is in the tenths place, and the 3 is in the hundredths place. The following chart will help you identify place value and will help you visually identify the positions of decimal points.

You can write fractions in decimal form by using a decimal point. All numbers to the left of the decimal point are whole numbers. All numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, and so on.

Examples:

1. 

2.

3.

4.

5.

6.

7. 

Rounding Off (Approximation)

To round off any positive number:

1. Underline the place value to which you’re rounding off.

2. Look to the immediate right (one place) of the underlined place value.

3. Identify the number (the one to the right). If it is 5 or higher, round the underlined place value up by 1. If the number (the one to the right) is 4 or less, leave your underlined place value as it is and change all the other numbers to the right of it to zeros or drop them if the place value is to the right of the decimal point.

Examples:

1. Round 4.4584 to the nearest thousandth.

The 8 is in the thousandth place. To its right is a 4. Thus, the 8 is left unchanged and the digits to the right are dropped. The rounded-off answer becomes 4.458.

2. Round 3,456.12 to the nearest ten.

The 5 is in the tens place. To its right is a 6. Thus, the 5 is increased by 1, and the digit in the ones place becomes zero. Then, the remaining digits, to the right of the decimal point, are dropped. The rounded-off answer is 3,460.

Estimating

Estimating Sums

Use rounded numbers to estimate sums.

Example:

Give an estimate for the sum 3,741 + 5,021 rounded to the nearest thousand.

Therefore, the estimate of 3,741 + 5,021 ≈ 9,000. (Note: The symbol ≈ means approximately equal to.)

Estimating Differences

Use rounded numbers to estimate differences.

Example:

Give an estimate for the difference 317,753 − 115,522 rounded to the nearest hundred thousand.

Therefore, 317,753 − 115,522 ≈ 200,000.

Estimating Products

Use rounded numbers to estimate products.

Example:

Estimate the product of 722 × 489 by rounding to the nearest hundred.

Therefore, 722 × 489 ≈ 350,000.

If both multipliers end in 50, or are halfway numbers, then rounding one number up and one number down gives you a better estimate of the product.

Example:

Estimate the product of 650 × 350 by rounding to the nearest hundred. Round one number up and one down.

Therefore, 650 × 350 ≈ 210,000.

Now, round the first number down and the second number up to get the estimate.

Therefore, 650 × 350 ≈ 240,000.

Rounding one number up and one number down gives you a closer approximation than rounding both numbers up, which is the standard rule.

Estimating Quotients

Use rounded numbers to estimate quotients.

Example:

Estimate the quotient of 891 ÷ 288 by rounding to the nearest hundred.

Therefore, 891 ÷ 288 ≈ 3.

Adding and Subtracting Decimals

To add or subtract decimals, line up the decimal points (and place values), then add or subtract in the same manner that you would add or subtract other numbers. Note: It is often helpful to place zeros to the right of the decimal point before adding or subtracting to make the problem more readable.

Examples:

1. 

2. 

A whole number has an understood decimal point to its right.

3.

Multiplying Decimals

To multiply decimal numbers, perform multiplication as usual as if there were no decimal points in the numbers. As you perform the multiplication calculations, you will notice that the decimal points are not necessarily aligned. Now that you have a numeric answer, it’s time to insert the decimal point. To accomplish this, count the total number of digits to the right of the decimal point in all the numbers being multiplied.

Place the decimal point in your answer so that there are the same number of digits to the right of it as there are above the line.

Examples:

1.

In Example 2, notice that it is sometimes necessary to insert zeros immediately to the right of the decimal point in the answer to have the correct number of digits

2.

Dividing Decimals

Dividing decimals is the same as dividing other numbers. Note that the divisor (the number you’re dividing by) should always be a whole number. If the divisor is not a whole number and has a decimal, move the decimal point to the right as many places as necessary until it’s a whole number. Then move the decimal point to the right the same number of places in the dividend (the number being divided into).

Note: Sometimes you may have to add zeros in the dividend (the number inside the division bracket).

Examples:

1.

The decimal point was moved to the right one place in each number.

2.

The decimal point was moved three places to the right in each number. This required inserting three zeros in the dividend.

Changing Decimals to Fractions

To change a decimal to a fraction:

1. Move the decimal point two places to the right.

2. Put that number over 100.

3. Reduce if necessary.

Move the decimal point two places to the right, place that number over 100, and reduce if necessary.

Examples:

1. 

2. 

3. 

Changing Fractions to Decimals

To change a fraction to a decimal, divide the numerator by the denominator.

For example, means 13 divided by 20, or .

Zeros may be written to the right of the decimal point in the numerator without changing its value. Every fraction, when changed to a decimal, either terminates (ends) or has a number or block of numbers that repeat indefinitely. To indicate a repeating decimal, a bar is used over only the number or block of numbers that repeats.

Examples:

Change each fraction into its decimal name.

1. 

2. 

3. 

Ratios and Proportions

Ratios and proportional relationships are important concepts on the CBEST.

Ratio

A ratio is a comparison of two quantities and is usually written as a fraction. The ratio of 3 to 5 can be expressed as 3:5 or .

Examples:

1. The ratio of 9 to 20 is or 9:20.

2. The ratio of 32 to 40 is or 4:5.

3. An SAT preparatory program advertises a teacher-student ratio of no more than one teacher for every five students. Which of the following numbers of teachers and students would exceed the program’s advertised ratio?

1. 10 teachers and 5 students

2. 14 teachers and 3 students

3. 4 teachers and 18 students

4. 5 teachers and 26 students

5. 6 teachers and 29 students

The ratio in choice A simplifies to 2 teachers to every 1 student, and choices B, C, and E all contain ratios of fewer than five students per teacher. The ratio in choice D can be simplified to 1 teacher for every 5.2 students, which exceeds the stated maximum ratio. The correct choice is D.

Proportion

A proportion is an equation that states that two ratios are equal. Because and both have values of , it can be stated that , or is proportional to .

Proportions and the Cross-Multiplication Rule

To prove that two ratios are equal, using the cross-multiplication rule (multiplying diagonally across the equal sign) should always produce equal answers.

You can use this cross-multiplication rule to solve any proportion problems. For example, you can test to see if by multiplying across the equal sign.

Therefore, is a true proportion.

Examples:

1.  is a true proportion since 32 × 5 = 40 × 4.

2.  is not a true proportion since 12 × 4 ≠ 18 × 3.

On the CBEST, you will be asked to analyze proportional relationships to solve real-world problems.

Examples:

1. The chart below shows Katie’s scores on her weekly problem sets through the first 6 weeks of her junior year. In which of the following weeks did she achieve a score proportionate to her score in Week 6?

1. Week 1

2. Week 2

3. Week 3

4. Week 4

5. Week 5

Katie scored 20 correct out of 24 total questions in Week 6, which can be simplified to . In Week 4, Katie scored 15 correct out of 18 total questions, which can also be simplified to . The correct choice is D.

2. In researching her genealogy for a class project, Ashley discovers that one of her great-grandparents was Paraguayan. Assuming Ashley has no other Paraguayan lineage, which of the following percentages is proportionate to Ashley’s Paraguayan ancestry?

1. 8%

2. 12.5%

3. 16.7%

4. 18%

5. 25%

The first step to solving this problem is to figure out how many total biological great-grandparents Ashley would have had. Assuming each biological parent had two biological parents, and so forth, Ashley would have had 2 × 2 × 2 = 8 biological great-grandparents. Therefore, the proportion of Ashley’s Paraguayan ancestry is . Converted into percentage form, this value is 12.5%. The correct choice is B.

Percents

A percent is a fraction whose denominator is 100. The word “percent” means hundredths (per hundred). The symbol for percent is %. For example, the expression 23% is read as 23 hundredths and can be expressed either as a fraction or decimal: .

Changing Percents

Fraction-Decimal-Percent Equivalents

A time-saving tip is to try to memorize some of the following equivalents before you take the test to eliminate unnecessary computations on the day of the exam.

Solving Percent Problems

Percentage problems appear frequently on the CBEST. To determine the percent of a number, change the percent to a fraction or decimal (whichever is easier for you) and multiply. Keep in mind that the word “of” means to multiply.

Examples:

1. What is 20% of 80?

2. What is 79% of 64?

3. What is 15% of 50?

4. What is of 36?

The fraction method works best in this case.

5. 24 is 30% of what number?

Solving Percent Problems Using Proportion

Turn the question into an equation—word for word. For “what,” substitute x; for “is,” substitute an equal sign (=); for “of,” substitute a multiplication sign (×). Change percents to decimals or fractions, then solve the equation.

The proportion will look like this:

Examples:

1. 18 is what percent of 90?

Therefore, 18 is 20% of 90.

2. 40% of what is 20?

Therefore, 40% of 50 is 20.

3. What percent of 45 is 30?

Therefore, of 45 is 30.

Finding Percent Increase or Percent Decrease

Percentage word problems are especially common on the CBEST and will test your ability to analyze problems as you translate English words into numeric equations. Make sure that you are familiar with solving this type of problem. Note: The terms percentage increase (rise), percentage decrease (fall), and percentage change are the same as percent change.

To find percent change (percent increase or decrease), use this formula:

Examples:

1. What is the percentage decrease of a $500 item on sale for $400?

The amount of change from 500 to 400 equals 100, therefore

2. What is the percent increase of a rise in temperature from 80° to 100°?

The amount of change is the difference between 100 and 80, or 20.

3. What is the percent decrease of Jordan’s salary if it went from $150 per hour to $100 per hour?

4. What is the percent change in the monthly sales from 2,100 to 1,890?