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Pre-Algebra

Math Review of Applications of Percents

150 150 Deborah

Overview

Percents are based on the decimal system and have many useful applications in business, sales, and finance. Other applications of percents include percent increase and decrease.

Estimating Percentages

Many of the most useful applications of percents involve estimating and calculating percentages for sales tax and tips at a restaurant. Suppose the sales tax is 8.7% in a particular town. That means that for any taxable item, the cost to buy it is the cost C plus .087C, calculated automatically at the cash register. It is always better to estimate high to make sure there is enough money for a purchase, and it is easier to use a value such as 10%. Then, the price of that item is the cost C plus .10C. If the cost of a CD is $15.99, estimate $16.00, and add 10% of $16.00 or $1.60, so the estimated cost will be a little less than $16.00 + $1.60 or $17.60. If the cost is higher, the estimated tax will also be higher. Suppose a DVD is $24.00. Its estimated price with tax will be $24.00 + $2.40 or about $26.40.

Tips

Estimating tips works on the same principle, except the percentage of tip to leave is up to the diner, and depends on the quality of service and the type of service. Suppose the waitperson brings the check and it is $25.36, including tax. The quality of service is good, and the diners decide to leave a 20% tip. The cost of the meal will then be the price P plus 0.20C. It is easier to estimate the tip by adding 0.10C + 0.10C, or $2.54 + 2.54, so the cost of the meal is $25.36 + $5.08 or $30.44.

Commissions and Fees

Commissions are typically a percentage of total sales and are either added to a base salary or paid as commission only. Suppose that Lori works in a salon, and her employer pays her $9.50 an hour plus a 1.5% commission on all products she sells. Her gross wages after a 40-hour work week are $9.50 times 40, or $380.00. Suppose she sells $125.00 in products during that week. Her commission will be $125.00 times 0.015, or $1.88. (When using percents in a calculation, change the percent to a decimal by multiplying by 1/100 and using decimal points.) $380.00 +$1.88 = $381.88.

Calculating Simple Interest

Simple interest follows a formula of I=Prt, where I is the amount of interest, P is the principal amount of money invested or loaned, r is the interest rate, given as a percentage, and t is the amount of time in years. It is only paid on the principal. In order to find the amount of simple interest paid on a loan of $1000.00 at a rate of 15% over 2 years, first substitute the known values in the formula such that I = $1000.00 (0.15)(2). Again, the 15% is changed to a decimal 0.15. Solving the equation, I = $300.00.

Percent Increase and Decrease

Percent change refers to the amount of increase or decrease given as a percentage. Percent increase is often referred to as a markup, and percent decrease is a discount. Suppose the regular price of an item is $12.00 and it is being sold at 30% off. In this case, the sale price is equal to the regular price P minus the discount of 30% P, or the sale price is equal to $12.00 – (0.30) P or $12.00 – 3.60 = $8.40.

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Math Review of Inequalities with More than One Step

150 150 Deborah

Overview

If an inequality contains more than one operation, it will take more than one step to solve it. Suppose that in order to solve an inequality, both multiplication and addition statements need to be undone by using the multiplicative and additive inverses. As with other inequalities that take one step, the variable will have many correct solutions that can be graphed.

Undoing Multiple Steps

Study the Inequality

Study the inequality to see what operations it contains. Then, the inverses of those operations are used in order to solve it. Suppose the inequality is 260+ 4x≤ 500. It contains addition (260 + 4x) and multiplication (4x), so inverse operations of subtraction and division will be needed.

Solve Each Step

Before the variable can be isolated on one side of the inequality, the 260 must be moved away by subtracting its opposite, or inverse. 260 – 260 +4x ≤500 – 260, or 4x ≤ 240. Since x is still multiplied by 4, both sides of the inequality must be divided by 4, or 4x/4 ≤240/4 or x ≤60.

Reverse the Inequality Sign

Suppose the inequality is 6 -3y≤ 21. It contains subtraction (6-3y) and multiplication (3y), so the inverses addition and division must be used to solve it. Solving each step, 6-6-3y ≤21 -6 or -3y ≤21. Since y is multiplied by   -3, both sides must be divided by -3. Dividing by a negative number adds an additional step, since the direction of the inequality sign must be reversed. Therefore -3y/-3 ≥21/-3, or y ≥-7.

Simplifying before Solving

Some inequalities must be simplified before they can be solved. Suppose (3 +9) > 7c -2. Using order of operations, the addition within parentheses must be solved first, and like terms are combined. The inequality 12> 7c -2 contains subtraction (7c-2) and multiplication (7c), so the inverses addition and division must be used. Isolating the variable, 12 +2 >7c -2 +2 or 14 >7c. Using the inverse, 14/7 >7c/7 or 2>c.

Distributive Property

Sometimes other properties must be used to simplify the inequality. Suppose the problem were 3 +2(x +4) ≥3. The distributive property can be used to expand 2(x +4) as 2x +8. Then 3 + 2x + 8 can be further combined as 11 +2x ≥3, or 2x ≥ 3- 11 , or 2x≥-8 or x ≥-4. Since both sides are divided by a positive number, the direction of the inequality sign is not reversed.

Graphing with a Number Line

Similar to the solutions of inequalities with one step, those with more than one step can be graphed on a number line. Remember the direction of the sign, so that the direction of the graph is correct. Also, if the sign is either < or >, the circle at the endpoint is open, while if the sign is ≤ or ≥, the circle at the endpoint is closed, showing the equals portion of the solution.

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Math Review of Inequalities

150 150 Deborah

Overview

While equations are statements that two quantities are equal, inequalities state that quantities are not equal. One side of the expression might be greater than the other side, or one side of the expression may be less. Correct solutions for variables encompass more than one number.

Inequalities

Inequalities include symbols such as greater than >, less than <, greater than or equal to ≥, less than or equal to ≤, or not equal to ≠. They can be expressed in words such as the real numbers that are greater than 5, in symbols such as x >5, or in set builder notation {x| x>5}. The values that solve the inequality are those that make it true. If x>5, then 4 cannot be a solution, but 5.01 can be. Similarly, if y +3 >5, then 3 can be a solution, because 3 +3 is 6 and 6 is greater than 5.

Graphing Inequalities

Writing a solution set in set-builder notation and listing enough members to give the pattern can consume both time and space. For example, the solution to {z | z -3 <12} can include {15.1, 15.2, 15.3, 15.4 …16, 16.1, 16.2 …} and so on. It can also be graphed on the number line. If the inequality is stated as either greater than or less than, than the endpoint of the ray is not a solution and it can be left as an open circle. If the variable is isolated on the left side of the inequality symbol, then the graph on the number line can point in the same direction as the inequality symbol. For example, 3 > x can be rewritten as x <3, and all numbers on the number line are less than 3 can be shaded in the same direction.

Solving Inequalities by Adding or Subtracting

Solving inequalities uses the same identity properties for addition and subtraction as solving equations. For example, to solve an inequality such as   3 + x <12, use the additive inverse so that 3-3 +x <12 -3. That isolates the variable, so x <9. Suppose that q – ½ >6. Then q- ½ + ½ > 6 + ½.

Solving Inequalities by Multiplying or Dividing

Solving inequalities by multiplying or dividing uses the same identity properties as equations with one important difference. If the number to be multiplied or divided is positive, then the direction of the inequality stays the same. However, if the number to be multiplied or divided is negative, the direction of the inequality is reversed.

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Math Review of Percents

150 150 Deborah

Overview

Percents are special types of ratios that compare the number to 100. Similar to any ratio, a percent can be expressed as a fraction. Since percents compare to 100, they can also be expressed in decimal form.

Fractions to Percents

Percent or the percent sign (%) means “per 100,” so 5% means the same as 5/100, 31% means 31/100 and so on. Many common fractions can be expressed as percents by converting the fraction in terms of 1/100, so ½ is 50/100 or 50%, ¾ is 75/100 or 75%, 4/5 is 80/100 or 80%. Percentages, like fractions, can be greater than 1, so that something that is 200% is two times as much, just as 200 is twice as much as 100.

Decimals to Percents

Any decimal can be expressed as a percentage, and any percentage as a decimal. (This can be a handy shortcut if the calculator does not have a percent key). For example, 20% is 0.20, and 7% is 0.07. Similarly, 0.003 is .03%, and 3.12 is 312%. A percentage such as 120% can be expressed in decimal form as 1.20.

Percents as Proportions

Percents can be expressed by the formula part/whole =percent/100, so that they can be compared as 2 equivalent ratios, the definition of proportion. Crossproducts can be used to then solve the equation. Suppose the problem were to find 30 is 20% of what number. 30/x = 20/100 is the equation for the proportion, or 20x = 30·100, or 3000. Dividing both sides by 20 to isolate the variable, x = 3000/20, or 150. Similarly, find 50% of 50. In this case, 50/100 = x/50 gives the proportion, so 100x =2500, or x= 2500/100, or x = 25. Also, 30 is what percentage of 150? This is also a proportion, as 30/150 = x/100, or 150x =3000, or x=3000/150, x = 20%.

Applications of Percents

Percents are used in many different ways. For example, if a ring is 24-karat gold, it is nearly 100% pure gold. It is too soft for most jewelry, so most gold is in the 10-karat (41.7% pure gold) to 18-karat (75% pure gold) range. Suppose a ring weighs about 10 g. If it were 10 karat, 4.17 g would be gold and the rest of the ring would be other metals, but if it were 18 karat, 7.5 g would be gold. Percents are often used in sale prices, as items are often on sale for 30% off or 40% off.

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Math Review of Rates, Ratios, and Proportions

150 150 Deborah

Overview

A rate compares 2 quantities that have different types of units, such as kilometers per hour or miles per gallon. A ratio compares quantities with the same type of units by dividing the quantity in the numerator by the quantity in the denominator, as in a fraction. A proportion states that 2 ratios are equivalent, such as 3/6 = ½.

Rates

Rates are often used when calculating a relationship between 2 different types of measurements. For example, a mile is a measure of length or distance, and a gallon is a measure of volume. A question such as “How many miles per gallon does the new car get?” really asks “How long of a distance per gallon can the new car travel?” That is important, because it is an indirect estimate of how much it will cost the owner to operate the car. Suppose the car gets 30 miles to a gallon, and its tank has a capacity of 11 gallons. That means that it can travel about 330 miles on a tank of gas. The mileage per gallon is often an important selling point for new cars.

Ratios

While a rate compares different types of units, a ratio compares the same types of units in a fraction. In symbol language, a ratio can be written as a: b or a/b, when b is ≠ 0. Suppose that the sale price is 40% off the regular price, and the regular price is 20.00. 40% of $20.00 is $8.00, so 20-8 = $12.00.

Proportions

Ratios and proportions are closely related, because a proportion is simply an equation of two ratios. One of the ways to estimate a ratio is by using proportions. In finding a value for the sale price, several different properties were used. If the regular price were $20.00, 40% of that price would also be equal to 4(10%), and 10% of $20.00 is $2.00. So 4($2.00) = $8.00, and $20.00 -8.00 = 12.00. Similarly, $20.00/1 times 40/100 = 800.00/100, or $8.00. In that example, all the values are known. If one value is unknown, a proportion can be solved by using cross products, so that 8/20 = 40/100. In this case, 20·40 =800, and 8·100 =800. In algebraic language, If a/b =c/d, and b≠0;   d ≠0, then ad =bc.

Scale

Scale drawings and scale models use ratios and proportions in 2 and 3 dimensional applications. Suppose a map has the scale ¼ in = 10 miles. It is about 2 ¾ in between 2 cities in a state. In other words, ¼ /11/4 or 1/11 = 10x or x = 110. Similarly, many scale model vehicles are made on a scale of 1:18, so that anything that is 2 inches long on the model will be 2(18) or 36 inches long on the real thing.

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Math Review of Two-Step and Multi-Step Equations

150 150 Deborah

Overview

Not all equations can be solved in one step. Many take several steps to solve, but the principles of solving one-step equations can be applied. Use the order of operations to simplify the equation; math properties to combine like terms; then apply inverses to isolate the variable on one side of the equation in order to solve it.

PEMDAS to the Rescue

A messy equation can almost always be cleaned by applying the order of operations. Start simplifying the equation by solving within parentheses, clearing exponents, multiplication, division, addition and subtraction. Sometimes the order of operations will suggest shortcuts. Suppose a problem were (3x –x) +2 =8. Solving within the parentheses, 3x –x =2x + 2 = 8.

Review Math Properties

The commutative properties of addition and multiplication; associative properties of addition and multiplication, and the distributive property of multiplication over addition or subtraction are good strategies when solving equations. Students are able to apply them to tame an equation.

Combine Like Terms

Whenever possible, combine like terms. They become obvious when they are not hiding behind parentheses, standing next to one another in the equation. For example, an expression such as 5x +2 + 3x +4 +6y +3-y can be simplified to 5x + 3x + 6y –y +2 +4 +3 or 8x +5y +9. After like terms are combined, the variable can be isolated on one side of the equation and solved. If 2x +2 = 8, then 2x +2 -2 =8-2 or 2x =6. Both sides can then be divided by 2 so that x = 6/2 or 3.

 

Multi-Step Equations Represent the Real World

Many problems in the everyday world are modelled by equations that represent more than one step. Suppose that the regular price of a gallon of gasoline is $3.00 a gallon, but a discount card gives 10 cents less per gallon. If a car has an 11 gallon tank, and it is empty, how much will it cost to fill the tank? The first step in solving the problem is to find the cost of the gasoline with the discount card. $3.00 – 0.10 = 2.90 per gallon. The second step is to multiply $2.90 ·11, the number of gallons, or $31.90. In this problem, the x was already isolated on one side of the equation.

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Math Review of One-Step Equations

150 150 Deborah

Overview

Some algebra equations can be solved with one step, using operations such as addition, subtraction, multiplication, and division. In order to solve them, use inverse operations so that the variable is on one side and real numbers are on the other side. Then, solve the problem with arithmetic.

Balanced Equations

An equation is solved when both the left side of the equation and the right side of the equation mean the same thing. Take a math fact such as 2 + 3 = 5. It is true for real numbers because 2 + 3 equals 5, and 5 equals 5. Suppose the equation were 3 +7 =x. In order for the equation to be balanced on both sides, x = 10, because 3 +7 equals 10 and 10 equals 10.

Inverse Operations

Addition and subtraction are inverse operations; and multiplication and division are inverse operations. That is because adding a number and its inverse equal the additive identity, or zero; and multiplying a number by its inverse equal the multiplicative identity or 1. In math symbol language a – a =0, and a/a, as long as a doesn’t equal zero, is 1. Adding or subtracting zero doesn’t change anything, and multiplying or dividing by 1 doesn’t change anything. The equation stays balanced by using the inverse on both sides.

Solving Problems by Adding or Subtracting

An equation such as 12.5 +7.2 = x is easy to solve because the variable x is already on one side and the real numbers (also called constants) are on the other. By adding up the constants, 12.5 + 7.2 = 19.7, so x equals 19.7. Suppose the equation were 2 + x = 5. Keep the equation balanced by using the same inverse on both sides. Since it is an addition problem, the inverse of addition is subtraction. In this example, 2 -2 +x = 5 -2, or 0 + x = 5-2, or x =3. Similarly, y -3 = 10 remains balanced when y – 3 +3=10 +3, or y +0 = 10 +3, or y = 13. The equation is a subtraction problem, and the inverse of subtraction is addition.

Solving Problems by Multiplying or Dividing

In an equation such as 4·11 = z, z equals 44. The constants are already on the left side and the variable is already on the right side of the equation. Suppose the equation were 9x =72. In that case, it is a multiplication problem, and the inverse of multiplication is division. 9 ·1/9·x = 72 ·1/9, because 9 ·1/9 ·x = 1x, or x, and 72/9 = 8, so x = 8. Similarly, x/3 = 12 becomes 3 ·1/3 ·x = 12·3 or x = 36.

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Math Review of Simplifying Expressions

150 150 Deborah

Overview

Students can use the commutative, associative, and distributive properties of addition and multiplication to simplify algebraic expressions. Problems are more easily solved if compatible terms can be combined first. Some problems can then be solved mentally.

Compatible Terms

Real numbers, whether whole or mixed, are constants that can be added, subtracted, multiplied or divided. The only exception is dividing by zero, which is undefined. Compatible terms can also be simplified, as long as each term has the same variable or variables at the same degree. The terms 2x + 4x can be combined to equal 6x, and the terms 3y2– 8y2 equal -5y2. However, 2x + 3z cannot be added, and 3x3 + 2x -4x is in its simplest form as 3x3 –2x.

Commutative Property of Addition and Multiplication

According to the commutative property of addition or multiplication, the order of adding or multiplying numbers does not matter, as long as all of them are added or multiplied. For example, 30 + 40 = 70, and 40 +30 = 70. The law in math symbol language is a + b = b + a for addition, and ab=ba for multiplication. Similarly, 2x + 3y +3x – 2y is the same as 2x +3x +3y -2y. The commutative property is a useful tool in mental math. Suppose the column of figures is 51 + 25 + 25 +49 + 50. It can be rearranged as 51 +49 +25 + 25 +50 to equal 200.

 

Associative Property of Addition and Multiplication

According to the associative property of addition or multiplication, the way numbers are grouped doesn’t matter, as long as all the numbers are added or multiplied. In math symbol language, (a + b) + c = a + (b +c).   The associative property is also a tool that can be used in mental math. When the mental math problem 51 +49 +25 + 25 +50 was rearranged, it was also grouped, as (51 + 49) + (25 +25 + 50). Solving within parentheses, 100 +100 =200. Both the commutative and associative problems were used together to solve the problem.   An expression such as 4x + 3y – .5y +7x can be rearranged as 4x +7x +3y -.5y and then regrouped as (4x +7x) + (3y – .5y) to equal 11x + 2.5y.

 

Distributive Property

The long formal name of the Distributive Property is the Distributive Property of Multiplication over Addition, or the Distributive Property of Multiplication over Subtraction. In math symbol language, it means that (a +b)c = ac + bc, or (a-b)c = ac – bc. This property is very useful, because it allows monomials to be combined that use the same variables. For example, 3y + 2y = 5y because (3 +2)y equals 5y. Similarly, 17a -11a =6a because (17-11)a = 6a.

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Math Review of Order of Operations

150 150 Deborah

Overview

Order of operations within math problems gives students a set of rules to follow in order to get the same answer. In the standard order of operations, expressions within parentheses are worked first, then exponents, then multiplication, division, addition, and subtraction.

Conventional Order

Suppose the problem is as simple as 85 +9 X 2. Without an order of operations, one person might calculate the problem as (85 +9) X 2 or 94 X 2 = 188, while another might calculate it as 85 + (9 X 2) or 85 + 18 =103. Similarly, does 2 X 34 mean 64, or 1296, or does it mean to work the 34 and then multiply by 2, or 81 X 2 = 162?

 

Ensuring Solutions

By convention, exponents are solved first, then multiplications or divisions, from left to right, then additions or subtractions, from left to right. Therefore, the correct answer to 85 + 9 X 2 is not 188, but 103. There are no exponents in the problem, but there is a multiplication, 9 X 2, equaling 18. Working from left to right, the last step is adding 85 + 18, to equal 103. Similarly, in 2 X 34, the exponent is worked first, then the multiplication.

Parentheses

Parentheses in mathematical sentences take priority over all other operations, so they are solved first, from left to right. Suppose the problem really were (85 +9) X 2. In that case, the correct answer would be to solve within the parentheses first, then the multiplication. Similarly, (4x)3 does not mean the same as 4x3. Within the order of operations, if x equals 2, (4x)3 would equal 83 or 512. If x equals 2, 4x3 would equal 4 X 23 or 32.

Types of Calculators and Order of Operations

When they are buying school supplies, students may wonder why their teachers specify certain types of calculators. That is because scientific calculators follow the algebraic order of operations, while other calculators do not. A scientific calculator will calculate the problem 85 +9 X 2 by doing the multiplication first, coming up with the answer 103, while a regular calculator will do the operations from left to right only, answering 188.

 

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Math Review of Operations with Real Numbers

150 150 Deborah

Overview

Operations with real numbers are the most familiar types in mathematics. Those numerical expressions with constants and operations include such actions as addition, subtraction, multiplication, and division.   Many algebraic expressions are usually operations with real numbers.

Real Numbers

The set of real numbers include the entire set of rational numbers, such as positive and negative integers, whole numbers, and natural numbers. Positive and negative fractions are rational numbers, because the numerator and denominator of the fraction represents a ratio. Whole numbers are rational numbers, because they are ratios in simplified form. For example, the fraction 4/1 is usually represented as 4. Irrational numbers, such as pi π, the constant e, √7, or √2 are also real numbers, but imaginary numbers and the infinities are not.

Addition of Real Numbers

Addition of real numbers can be represented on the number line by moving right on the number line. For example, 4 +3 =7 can be modeled on the number line by starting at 4 spaces from 0 and then moving right 3 more spaces. Fractions can be added by finding their common denominator, then adding the numerators. For example, ½ +1/3 = 3/6 +2/6 = 5/6.

Subtraction of Real Numbers

Real numbers can be subtracted on the number line by moving to the left. Subtraction is the same as adding the additive inverse, so that 7-5 means the same thing as adding 7 + (-5). The additive inverse of a number is the opposite of that number, so that the additive inverse of 3 is -3, the additive inverse of 1/5 is -1/5, and the additive inverse of -12 is 12. Fractions can also be subtracted by adding the additive inverse, as long as the common denominators are found.

 

Multiplication and Division of Real Numbers

In the most common model of multiplication and division, multiplication is repeated addition and division is repeated subtraction. When multiplying two positive or two negative numbers, the product is positive, but if one number is positive and the other is negative, the product will be negative. Division by zero is undefined, the product of any number and 0 is zero, and the quotient of 0 and any nonzero number is 0.

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