CHAPTER 8
LESSON 1
Find a percent of numbers
Multiply the number by the percent (e.g. 87 * 68 = 5916)Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)
Find a percent of numbers
Multiply the number by the percent (e.g. 87 * 68 = 5916)Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)
LESSON 2
Find a percent
Convert the problem to an equation using the percentage formula: Y/P% = X.
Y is 25, P% is 20, so the equation is 25/20% = X.
Convert the percentage to a decimal by dividing by 100.
Converting 20% to a decimal: 20/100 = 0.20.
Find a percent
Convert the problem to an equation using the percentage formula: Y/P% = X.
Y is 25, P% is 20, so the equation is 25/20% = X.
Convert the percentage to a decimal by dividing by 100.
Converting 20% to a decimal: 20/100 = 0.20.
LESSON 3
Find a number when a percent is known
Multiply the number by the percent (e.g. 87 * 68 = 5916)Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)
Find a number when a percent is known
Multiply the number by the percent (e.g. 87 * 68 = 5916)Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)
CHAPTER 7
LESSON 1
Ratios
the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.
Ratios
the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.
LESSON 2
Rates
The numbers or measurements being compared are called the terms of the ratio. A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces.
Rates
The numbers or measurements being compared are called the terms of the ratio. A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces.
LESSON 3
Equivalent ratios
Equivalent ratios (which are, in effect, equivalent fractions) are two ratios that express the same relationship between numbers. We can create equivalent ratios by multiplying or dividing both the numerator and denominator of a given ratio by the same number.
Equivalent ratios
Equivalent ratios (which are, in effect, equivalent fractions) are two ratios that express the same relationship between numbers. We can create equivalent ratios by multiplying or dividing both the numerator and denominator of a given ratio by the same number.
LESSON 4
Solve proportions
Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation.
Solve proportions
Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation.
LESSON 5
Distance, speed, and time
Distance Speed Time Formula. Speed is a measure of how quickly an object moves from one place to another. It is equal to the distance traveled divided by the time. It is possible to find any of these three values using the other two.
Distance, speed, and time
Distance Speed Time Formula. Speed is a measure of how quickly an object moves from one place to another. It is equal to the distance traveled divided by the time. It is possible to find any of these three values using the other two.
LESSON 6
Scale drawings
A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale). The scale is shown as the length in the drawing, then a colon (":"), then the matching length on the real thing.
Scale drawings
A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale). The scale is shown as the length in the drawing, then a colon (":"), then the matching length on the real thing.
LESSON 7
Decimals and percents
To convert from decimal to percent: multiply by 100, and add a "%" sign. move the decimal point 2 places to the right, and add the "%" sign.
Decimals and percents
To convert from decimal to percent: multiply by 100, and add a "%" sign. move the decimal point 2 places to the right, and add the "%" sign.
LESSON 10
Fractions and percents
Convert the fraction to a decimal number. The fraction bar between the top number (numerator) and the bottom number (denominator) means "divided by." ...
Multiply by 100 to convert decimal number to percent. 0.25 × 100 = 25%
Fractions and percents
Convert the fraction to a decimal number. The fraction bar between the top number (numerator) and the bottom number (denominator) means "divided by." ...
Multiply by 100 to convert decimal number to percent. 0.25 × 100 = 25%
LESSON 11
Fractions, decimals, and percents
Decimals and Percents. "Percent" is actually "per cent", meaning"out of a hundred". (It comes from the Latin per cent um for "thoroughly hundred".) You can use this "out of a hundred"meaning, along with the fact that fractions are division, to convert between fractions, percents, and decimals.
Fractions, decimals, and percents
Decimals and Percents. "Percent" is actually "per cent", meaning"out of a hundred". (It comes from the Latin per cent um for "thoroughly hundred".) You can use this "out of a hundred"meaning, along with the fact that fractions are division, to convert between fractions, percents, and decimals.
CHAPTER 6
LESSON 1
Equation & Expression
An equation is two expressions that are equal to each other. In other words, expressions are just numbers, variables (like x and y), constants (like pi and e), and operators (addition, subtraction, division, exponentiation, etc.). That's why 4/0 is an expression.
Equation & Expression
An equation is two expressions that are equal to each other. In other words, expressions are just numbers, variables (like x and y), constants (like pi and e), and operators (addition, subtraction, division, exponentiation, etc.). That's why 4/0 is an expression.
LESSON 2
Addition properties to evaluate expressions
Addition properties to evaluate expressions
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. |
Associative property
In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. |
Identity V property
The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity propertyfor multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity." |
LESSON 3
Distributive property use the distributive property to evaluate expressions
apply the distributive property to the expression 3 (2 + 𝘹) to produce the equivalent expression 6 + 3𝘹; apply the distributive property to theexpression 24𝘹 + 18𝘺 to produce the equivalent expression 6 (4𝘹 + 3𝘺); applyproperties of operations to 𝘺 + 𝘺 + 𝘺 to produce the equivalent expression 3𝘺.
apply the distributive property to the expression 3 (2 + 𝘹) to produce the equivalent expression 6 + 3𝘹; apply the distributive property to theexpression 24𝘹 + 18𝘺 to produce the equivalent expression 6 (4𝘹 + 3𝘺); applyproperties of operations to 𝘺 + 𝘺 + 𝘺 to produce the equivalent expression 3𝘺.
LESSON 4
Evaluate express with fractions
Multiply a fraction with another fraction by multiplying the numerators together and the denominators together. For example, 3/8 x 2/5 = 6/40 = 3/20. Follow the same procedure when you divide, except first flip the fraction you are dividing by. For example: 3/8 ÷ 2/5 = 3/8 x 5/2 = 15/16.
Evaluate express with fractions
Multiply a fraction with another fraction by multiplying the numerators together and the denominators together. For example, 3/8 x 2/5 = 6/40 = 3/20. Follow the same procedure when you divide, except first flip the fraction you are dividing by. For example: 3/8 ÷ 2/5 = 3/8 x 5/2 = 15/16.
LESSON 5
Write addition and subtraction expressions
To add or subtract rational expressions with unlike denominators, first find the LCM of the denominator. ...
Write each expression using the LCD. ...
Add or subtract the numerators.
Simplify as needed.
Write addition and subtraction expressions
To add or subtract rational expressions with unlike denominators, first find the LCM of the denominator. ...
Write each expression using the LCD. ...
Add or subtract the numerators.
Simplify as needed.
LESSON 6
Order of operations
This phrase stands for, and helps one remember the orderof, "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". ... Exponents. Multiplication and Division (from left to right) Addition and Subtraction (from left to right).
Order of operations
This phrase stands for, and helps one remember the orderof, "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". ... Exponents. Multiplication and Division (from left to right) Addition and Subtraction (from left to right).
LESSON 7
Write multiplication and division expression
A mathematical operation performed on a pair of numbers in order to derive a third number called a product. For positive integers, multiplication consists of adding a number (the multiplicand) to itself a specified number of times. Thus multiplying 6 by 3 means adding 6 to itself three times.
A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further.
Write multiplication and division expression
A mathematical operation performed on a pair of numbers in order to derive a third number called a product. For positive integers, multiplication consists of adding a number (the multiplicand) to itself a specified number of times. Thus multiplying 6 by 3 means adding 6 to itself three times.
A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further.
LESSON 8
Equations with addition and subtraction
In mathematics, an equation is a statement of an equality containing one or more variables. Solving theequation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the equation.
the action or process of adding something to something else.
Subtraction in mathematics means you are taking something away from a group or number of things. When you subtract, what is left in the group becomes less. An example of a subtraction problem is the following: 5 - 3 = 2. Notice that there are three parts to the subtraction problem shown.
the action or process of adding something to something else.the action or process of adding something to something else.
Equations with addition and subtraction
In mathematics, an equation is a statement of an equality containing one or more variables. Solving theequation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the equation.
the action or process of adding something to something else.
Subtraction in mathematics means you are taking something away from a group or number of things. When you subtract, what is left in the group becomes less. An example of a subtraction problem is the following: 5 - 3 = 2. Notice that there are three parts to the subtraction problem shown.
the action or process of adding something to something else.the action or process of adding something to something else.
LESSON 9
Equations with multiplication and division
To solve a multiplication equation, use the inverse operation of division. Divide both sides by the same non-zero number. Click the equation to see how to solve it.
Equations with multiplication and division
To solve a multiplication equation, use the inverse operation of division. Divide both sides by the same non-zero number. Click the equation to see how to solve it.
LESSON 10
Write expression with fractions
A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions.
Write expression with fractions
A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions.
LESSON 11
Equations with fractions
Clear of fractions as follows: Multiply both sides of the equation -- every term -- by the LCM of denominators. Each denominator will then divide into its multiple. We will then have an equation without fractions.
Equations with fractions
Clear of fractions as follows: Multiply both sides of the equation -- every term -- by the LCM of denominators. Each denominator will then divide into its multiple. We will then have an equation without fractions.
CHAPTER 5
LESSON 1
Multiply integer
To multiply or divide signed integers, alwaysmultiply or divide the absolute values and use these rules to determine the sign of the answer. When youmultiply two integers with the same signs, the result is always positive. Just multiply the absolute values and make the answer positive. |
Divide integer
Just multiply the absolute values and make the answer positive. When you multiply two integers with different signs, the result is always negative. Just multiply the absolute values and make the answer negative. When you divide two integers with the same sign, the result is always positive. |
LESSON 2
Integer
Step 1: Take the absolute value of each number. Step 2: Subtract the number with a smaller absolute value from the number with bigger or larger absolute value. Step 3: Copy the sign of the number with the bigger or larger absolute value. |
Compare and order integers
If we compare numbers with different signs, then negative number is less than positive. If numbers are both positive then this is the case when we compare whole numbers. If numbers are both negative then we compare numbers without signs. The bigger positive number, the smaller negative. |
LESSON 3
Add integer
Step 1: Take the absolute value of each number. Step 2: Subtract the number with a smaller absolute value from the number with bigger or larger absolute value. Step 3: Copy the sign of the number with the bigger or larger absolute value. |
Subtract integer
To subtract integers, change the sign on the integer that is to be subtracted. If both signs are positive, the answer will be positive. If both signs are negative, the answer will be negative. If the signs are different subtract the smaller absolute value from the larger absolute value. |
LESSON 4
RATIONAL NUMBERS
Step 1: Factor both the numerator and the denominator. ...
Step 2: Write as one fraction. ...
Step 3: Simplify the rational expression. ...
Step 4: Multiply any remaining factors in the numerator and/or denominator. ...
Step 1: Factor both the numerator and the denominator.
Step 2: Write as one fraction.
Step 1: Factor both the numerator and the denominator. ...
Step 2: Write as one fraction. ...
Step 3: Simplify the rational expression. ...
Step 4: Multiply any remaining factors in the numerator and/or denominator. ...
Step 1: Factor both the numerator and the denominator.
Step 2: Write as one fraction.
LESSON 5
Compare and rational numbers
Rational numbers are numbers that can be written as the division of two integers. To compare rational numbers, we must first divide them to get a decimal number. After getting the equivalent decimal numbers, we can them compare them to othernumbers to see which one is greater or lesser.
Rational numbers are numbers that can be written as the division of two integers. To compare rational numbers, we must first divide them to get a decimal number. After getting the equivalent decimal numbers, we can them compare them to othernumbers to see which one is greater or lesser.
CHAPTER 4
LESSON 1
Add fractions and mixed numbers with unlike denominators.
Add the numerators of the two fractions.
Place that sum over the common denominator.
If this fraction is improper (numerator larger than or equal to the denominator) then convert it to a mixed number.
Add the integer portions of the two mixed numbers.
Add fractions and mixed numbers with unlike denominators.
Add the numerators of the two fractions.
Place that sum over the common denominator.
If this fraction is improper (numerator larger than or equal to the denominator) then convert it to a mixed number.
Add the integer portions of the two mixed numbers.
LESSON 2
Subtract fractions and mixed numbers with like denominator
Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators.
Step 1: Find the Lowest Common Multiple (LCM) between the denominators.
Step 1: Convert all mixed numbers into improper fractions.
In this second method, we will break the mixed number into wholes and parts. Thus, is equivalent to.
Subtract fractions and mixed numbers with like denominator
Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators.
Step 1: Find the Lowest Common Multiple (LCM) between the denominators.
Step 1: Convert all mixed numbers into improper fractions.
In this second method, we will break the mixed number into wholes and parts. Thus, is equivalent to.
LESSON 3
Add fractions and mixed number like denominator
Add the numerators of the two fractions.
Place that sum over the common denominator.
If this fraction is improper (numerator larger than or equal to the denominator) then convert it to a mixed number.
Add the integer portions of the two mixed numbers.
Add fractions and mixed number like denominator
Add the numerators of the two fractions.
Place that sum over the common denominator.
If this fraction is improper (numerator larger than or equal to the denominator) then convert it to a mixed number.
Add the integer portions of the two mixed numbers.
LESSON 4
Subtract fractions and mixed numbers with unlike denominators
Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators. ...
Step 1: Find the Lowest Common Multiple (LCM) between the denominators. ...
Step 1: Convert all mixed numbers into improper fractions. ...
In this second method, we will break the mixed number into wholes and parts. ...
Subtract fractions and mixed numbers with unlike denominators
Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators. ...
Step 1: Find the Lowest Common Multiple (LCM) between the denominators. ...
Step 1: Convert all mixed numbers into improper fractions. ...
In this second method, we will break the mixed number into wholes and parts. ...
LESSON 5
Multiply fractions
Simplify the fractions if not in lowest terms.
Multiply the numerators of the fractions to get the new numerator.
Multiply the denominators of the fractions to get the new denominator.
Multiply fractions
Simplify the fractions if not in lowest terms.
Multiply the numerators of the fractions to get the new numerator.
Multiply the denominators of the fractions to get the new denominator.
LESSON 6
Multiply mixed numbers
Consider multiplying 41/2 with 62/5
Convert your first mixed number to an improper fraction. ...
Convert your second mixed number to an improper fraction. ...
Multiply the two improper fractions. ...
Reduce your answer to the lowest terms. ...
Convert your answer to a mixed number.
Multiply mixed numbers
Consider multiplying 41/2 with 62/5
Convert your first mixed number to an improper fraction. ...
Convert your second mixed number to an improper fraction. ...
Multiply the two improper fractions. ...
Reduce your answer to the lowest terms. ...
Convert your answer to a mixed number.
LESSON 7
Divide fractions
Leave the first fraction in the equation alone.
Turn the division sign into a multiplication sign.
Flip the second fraction over (find its reciprocal).
Multiply the numerators (top numbers) of the two fractions together. ...
Multiply the denominators (bottom numbers) of the two fractions together.
Divide fractions
Leave the first fraction in the equation alone.
Turn the division sign into a multiplication sign.
Flip the second fraction over (find its reciprocal).
Multiply the numerators (top numbers) of the two fractions together. ...
Multiply the denominators (bottom numbers) of the two fractions together.
LESSON 8
Divide whole numbers and fraction
To divide a whole number with a fraction, make the whole number into a fraction by putting it over a denominator of 1. Next, reverse the numerator and denominator of the fraction you're dividing the whole number with. Multiply this new fraction and the whole number
Divide whole numbers and fraction
To divide a whole number with a fraction, make the whole number into a fraction by putting it over a denominator of 1. Next, reverse the numerator and denominator of the fraction you're dividing the whole number with. Multiply this new fraction and the whole number
LESSON 9
Divide mixed numbers
Step 1: Write the mixed numbers as improper fractions.
Step 2: Rewrite the division problem using the improper fractions. So to divide mixed numbers you should change the fractions to improper fraction and the convert the problem to a multiplication question.
Divide mixed numbers
Step 1: Write the mixed numbers as improper fractions.
Step 2: Rewrite the division problem using the improper fractions. So to divide mixed numbers you should change the fractions to improper fraction and the convert the problem to a multiplication question.
LESSON 10
Metric system of measurement
The metric system is an internationally adopted decimal system of measurement. It is in widespread use, and where it is used, it is the only or most common system of weights and measures. It is now known as the International System of Units.
Metric system of measurement
The metric system is an internationally adopted decimal system of measurement. It is in widespread use, and where it is used, it is the only or most common system of weights and measures. It is now known as the International System of Units.
Chapter 3
LESSON 1
Greatest Common Factor (GCF)
greatest common factor the greatest whole number that is a common factor of two or more numbers. It is also called the greatest common divisor.
USE PRIME FACTORS
step 1 write the prime factorization of both numbers. Circle the factors common to both.
step 2 find the product of the common factors.
Greatest Common Factor (GCF)
greatest common factor the greatest whole number that is a common factor of two or more numbers. It is also called the greatest common divisor.
USE PRIME FACTORS
step 1 write the prime factorization of both numbers. Circle the factors common to both.
step 2 find the product of the common factors.
LESSON 6
DIVISIBILITY RULE divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. |
LESSON 7
FACTORS AND PRIME NUMBERS factors and prime factorization of the natural numbers from 1 to 1000. When n is a prime number, the prime factorization is just n itself, written in bold below. The number 1 is called a unit. |
LESSON 12
TERMINATING DECIMAL
A terminating decimal is a decimal that ends.It's a decimal with a finite number of digits.
REPEATING DECIMAL
A repeating or recurring decimal is decimal representation of a number whose digits are periodic and the infinitely-repeated portion is not zero.