MATH 08 Diagnostic Review Materials PART Chapters 5 to 7 YOU WILL NOT BE GIVEN A DIAGNOSTIC TEST UNTIL THIS MATERIAL IS RETURNED DO NOT WRITE IN THIS MATERIAL Revised Winter 0
PRACTICE TEST: Complete as many problems as you can, then check your solutions with the answers that follow Keep in mind that you are not expected to know how to solve every problem Simplify: x 5x 6 y y Simplify: 5a 7a b b Evaluate: x 8y if x and Evaluate: 5a b if a and 5 Complete: x? 6 Complete:? 5a 5a 7 Simplify: 0 x(x ) 8 Simplify: x 5 9 Simplify: (8x ) 0 Simplify: ( x 5) Simplify: a 9 Simplify: 6 9a 8 y b Is x = 8 a solution of the equation x? Is x = a solution of the equationx 7? 5 Solve: x 7 5 6 Solve: 8 x 0 7 Solve: 5 5 6 x 8 Solve: 7 x 9 Solve: x 0 Solve: x 8 Solve: (a ) 5 9 Solve: ( x ) 5 9 Translate the following into a math equation and solve: The difference of five times a number and three is twelve Translate the following into a math equation and solve: Five less than two times a number is the same as seven 5 A committee consists of 0 teachers, students, and administrators What is the ratio of teachers to students on the committee? 6 A wheelchair ramp rises feet for every feet it runs What is the ratio of rise to run? 7 An office assistant can type 0 words in minutes Determine the rate of words to minutes 8 A runner can complete 8 laps around the track in 56 minutes What is the rate of laps to minutes for this runner? 9 An international call costs $98 for minutes What is the cost per minute? 0 At a warehouse store, 0 cans of soup cost $8 What is the price per can of soup? Is 5 a true statement? 6 7 Is 5 a true statement? 0 5 Find the value of x that makes statement x a true 7 5
Find the value of x that makes statement x a true 9 0 5 A college has a ratio of male students for every female students If there are 5 male students, how many female students attend the college? 6 There are 5 mg of cholesterol in oz of egg substitute How many mg of cholesterol are there in oz of egg substitute? 7 Write 56% as a fraction in simplest form 8 Write % as a fraction in simplest form 9 Write % as a decimal 0 Write % as a decimal Write 8 Write 5 6 as a percent as a percent Write as a percent Write 0 as a percent 5 What is 89% of 0? 6 What is 68% of 80? 7 What percent of 50 is 0? 8 What percent of 80 is 50? 9 5 is 5% of what number? 50 60 is 0% of what number? 5 A survey asked 60 mathematics students what they thought were the most important factors that contributed to their success in mathematics education Among the factors considered were the role of the instructor, taking responsibility for their homework assignments, study skills and time management, being tutored on a one-on-one basis, and study groups The results of the survey are presented in the circle graph below Study Groups, 0% Tutoring, 5% Doing Homework 5% Math Students Study Skills, 5% Instructor, 5% A What factor was considered to be the most important factor for achieving success in math? B How many students thought instructors play a major role in getting them to be successful in mathematics? C What percent of the students thought that study skills and tutoring were important factors for success in their math education? D How many students thought that study skills and tutoring were important factors for success in their math education? 5 A bank charges 5% simple interest How much interest must you pay on a loan of $700 for years? 5 What is the interest on a $650 simple interest loan with a 0% annual rate for 8 months? 5 What is the commission made on a sale if the commission rate is % and the amount of sales is $5,5? 55 The price tag on a dress is $0 but Crystal has a coupon for 5% off any item How much will Crystal save if she uses her coupon on the dress?
PRACTICE TEST ANSWERS x 9y Part A Page 5 a b Part A Page 5 0 Part B Page 5 Part B Page 5 5 x Part C Page 5 6 0 Part C Page 5 7 x Part C Page 5 8 x 8 Part C Page 5 9 x 6 Part D Page 6 0 8x 0 Part D Page 6 9a Part D Page 6 6 a Part D Page 6 Yes Part E Page 6 No Part E Page 6 5 x 7 Part F Page 6 6 x 9 Part F Page 6 7 x 8 Part G Page 7 8 77 x 8 8 5 Part G Page 7 9 x 8 Part H Page 8 0 x 0 Part H Page 8 a Part H Page 8 x 9 Part H Page 8 Part I Page 8 6 Part I Page 8 5 6 7 8 5 teachers students ft 6 ft 60 words minutes lap min Part J Page 9 Part J Page 9 Part K Page 9 Part K Page 9 9 $09 per min Part L Page 0 0 $08 per can Part L Page 0 No Part M Page 0 Yes Part M Page 0 8 Part M Page 0 5 5 7 7 7 Part M Page 0 0 0 5 7,98 Part N Page 6 675 mg Part N Page 7 8 5 6 5 Part O Page Part O Page 9 0 Part P Page 0 00 Part P Page 5% Part Q Page 5% Part Q Page 0% Part R Page % Part R Page 5 089 Part S Page 6 Part S Page 7 0% Part S Page 8 65% Part S Page 9 60 Part S Page 50 50 Part S Page 5 A Doing homework Part T Page B 0 students C 0% D students 5 $70 Part U Page 5 5 $9750 Part U Page 5 5 $5705 Part V Page 6 55 $650 Part V Page 6
Part A LIKE TERMS: terms that have the same variables raised to the same power or two constants Like Terms: x and 5x Like Terms: x and 5x Like Terms: and 5 Unlike Terms: x and 5y Unlike Terms: x and Unlike Terms: x and 5 5x Part B COMBINING LIKE TERMS: add the coefficients and keep the variable part the same Combine like terms x 5x y 8y 7 9x 5y 6 8x 5x + 7 x + x 6x 5 0 -x 5 x 5x + x + + 8 -x + 9 x x + 5y + y x + 9y EVALUATING AN EXPRESSION: Evaluate 7x y if x 5 and y replace the variables with the given values and perform the arithmetic using the proper order of operations 7(5) () 5 6 9 Part C 5 Evaluate 6 + 5x if x = Evaluate x + 8y if x = - and y = - - Evaluate x + y if x = and y = 9 Evaluate x 6y if x and y ALGEBRAIC PROPERTIES PROPERTY DESCRIPTION ADDITION MULTIPLICATION Commutative Changing the order does not change the answer a b b a ab b a Associative Changing the grouping does not change the ( a b) c a ( b c) ( ab) c a( b c) answer (5 8) 5 (8 ) (5) 5 ( ) Identity a0 a 6 0 6 a a The sum of a term and 0 is that same term and and and and The product of a factor and is that same factor 0a a 0 6 6 a a Inverse The sum of a number and its opposite is 0 a ( a) 0 a The product of a number and its reciprocal is 7 ( 7) 0 a Zero The product of a number and 0 is 0 a 0 0 0 0 and and 0 a 0 0 0
Complete the equation using the appropriate Algebraic Property 8a? 0? 6x? 5? 0 8a 6x 5 Simplify using the appropriate Algebraic Property (x ) 0 7y x 8y x x 5( a ) x 5y x x 5a Part D DISTRIBUTIVE PROPERTY: Multiply each term inside the parentheses by the term outside the parentheses Distribute 5( x 7) ( k 5) 5x 5 k 5 7x 5 9 5 a 7 8x 5 a Part E DETERMINING IF A GIVEN VALUE IS A SOLUTION OF AN EQUATION Is x 5 a solution of x 7? 5 7? 7 7 Replace the variable with the given value Perform the arithmetic See if both sides are equal Yes, x 5 is a solution Determine whether the given value is a solution to the equation Is w = 9 a solution of the equation 5 w? No Is x = 8 a solution of the equation x 8 0? Yes Is z = 7 a solution of the equation z? No Part F SOLVING EQUATIONS: Determine the value of the variable that makes the equation true How? Correct Answer? Use inverse operations to undo the operations applied to the variable Check your answer by substituting it in the original problem 6
ADDITION PROPERTY OF EQUALITY: If a b, then a+c b+c If the same number is added to (or subtracted from) both sides of an equation, the two sides remain equal SOLVING EQUATIONS USING THE ADDITION PROPERTY OF EQUALITY Solve x 8 x 8+8 +8 x 0 x Solve x 5 x 5-5 - 5 x 0 6 x 6 Determine the operation: Subtraction by 8 Perform the inverse operation: Addition (on both sides) by 8 Do the arithmetic The solution is x Determine the operation: Addition by 5 Perform the inverse operation: Subtraction (on both sides) by 5 Do the arithmetic The solution is x 6 Part G Solve n 6 y 8 p 5 6 n 9 y 9 7 b p b MULTIPLICATION PROPERTY OF EQUALITY: If a b, then a c b c If both sides of an equation are multiplied (or divided) by the same number, the two sides remain equal SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY OF EQUALITY Solve 8x 96 8x 96 8 8 x x Solve 7 x x 7 x x x 9 7 8 Determine the operation: Multiplication by 8 Perform the inverse operation: Division (on both sides) by 8 Perform the arithmetic The solution is x Determine the operation: Division Perform the inverse operation: Multiplication (on both sides) Perform the arithmetic The solution is x 8 Solve x 6 x 0 0 x 8 x 8 6 x 8 x x 900 x 8 7
Part H SOLVING EQUATIONS USING BOTH THE ADDITION AND MULTIPLICATION PROPERTIES OF EQUALITY Solve 5x x 8 0 7x 7x - - 7x 7 7 x Combine like terms on each side of the equation Get the variable term alone on one side of the equation Perform the inverse operation (Inverse of Addition is Subtraction) Get the variable alone on one side of the equation Perform the inverse operation (Inverse of Multiplication is Division) Simplify x 6 5 y 5 (x ) x 0 (x 5) 6 x 0 y 0 x x Part I KEY WORDS AND PHRASES Addition Subtraction Multiplication Division Equals sum increased by more than plus total combined added to in all difference decreased by less than minus fewer than reduced by take away product times of factor of doubled tripled etc quotient divided by ratio of per is/are was/were will be gives yields TRANSLATING ENGLISH SENTENCES INTO MATHEMATICAL EQUATIONS The sum of a number and 5 is Find the number Let x = the number x 5 x 5-5 -5 x 6 8 Read carefully What are you asked to find? Select a variable to represent the unknown Translate the words into an algebraic equation Solve the equation using inverse operations 5 Simplify (Remember to check your answer) Three times a number decreased by six is 7 Determine the number x 6 7 x Five less than times a number is 7 Determine the number x 5 7 x The sum of twice a number and 5 is eleven Determine the number x 5 x
Part J RATIO: a comparison of two quantities Write the ratio of cm to 8 cm in ways () to 8 () :8 () 8 Always reduce to lowest terms Never write as a whole number or mixed number There are 8 blue cars and 0 green cars on the lot a Write the ratio of blue cars to green cars b Write the ratio of green cars to the total number of cars Blue Cars 8 8 Green Cars 0 0 5 Green Cars 0 0 5 Total Cars 8 8 7 There are 8 black marbles, 5 blue marbles, and 6 red marbles in a bag What is the ratio of black marbles to red marbles? There are 6 men and 8 women that work in the department What is the ratio of men to the total number of employees in the department? A tree is 0 feet high and 6 feet wide What is the ratio of the height of the tree to the width of the tree? 8 7 5 Part K RATE: a ratio that compares two quantities that have different units Always reduce to lowest terms Never write as a whole number or mixed number The patient weighs 50 pounds and receives 5 mg of medicine What is the rate of mg to pounds for this medication? 5 mg 5 mg 5 mg 50 pounds 50 pounds 5 0 pounds An office assistant can type 60 words in minutes Determine the rate of words to minutes A patient requires 5 milligrams of medicine for every 5 pounds of body weight What is the rate of milligrams to pounds for this patient? A car traveled 00 miles in 6 hours What is the rate of miles to hours? 55 words minutes 5 mg 9 pounds 00 miles hours 9
Part L UNIT RATE: a rate in which the denominator is What is the unit rate for driving 6 miles on gallons of gas? * Divide numerator and denominator by the denominator: 6 miles 6 miles 5 miles gallons gallons gallon 5 miles per gallon A ounce can of beans costs $08 What is the unit price per ounce? A secretary can type 876 words in 5 minutes What is the secretary s unit rate in words per minute? An international call costs $8 for minutes What is the cost per minute? $00 per ounce 58 words per minute $0 per minute Part M PROPORTION: a statement that two ratios (fractions) are equal Is 6 a true statement? 8 9 5 Determine cross products:? ( 6)(9 5) (8)() The proportion is not true: a c only if ad bc b d Solve for n: 7 5 n Determine cross products: n 5 7 Solve for variable: n 05 n 05 n 6 Determine if the following are true statements 8 5 5 6 0 5 Solve for n: n 7 n 8 0 5 7 No Yes n n 0
Part N APPLICATIONS OF PROPORTIONS: On a map, cm represents 9 km How many cm would represent 6 km? Assign Variable: Write Ratio in Words: Write Ratio of Given Values: Write Ratio with Variable: 5 Write Proportion: 6 Solve Proportion: n = number of centimeters Centimeters Kilometers cm 9 km n cm 6 km cm n cm 9 km 6 km n 9 6 6 9n 7 9n 7 9n 9 9 8 n On the map, 8 cm will represent 6 km For every 0 kg of body weight mg of a drug are to be given Find the weight of a person requiring 5 mg of the drug A baker can make 7 cookies using cups of flour How many cups of flour are needed to make 88 cookies? Harry gets miles per gallon of gasoline in his truck If he has gallons of gasoline in his truck, how far can he go? 65 kg 6 cups 9 miles
Part O PERCENT: the number of parts out of 00 parts CONVERTING PERCENTS TO FRACTIONS: P% P Divide the percent by 00 00 Example: Write 5% as a fraction 5 00 55 00 5 7 0 Convert the following percents to fractions: 5% 90% 6% 00% 5 00 90 9 00 0 6 00 00 50 00 Part P CONVERTING PERCENTS TO DECIMALS: P% P P00 00 Divide percent by 00 Example: Write 6% as a decimal 6 6 00 00 0 06 Divide percent by 00 OR Move decimal point places to the left Convert the following percents to decimals: 5% % 5% 75% 05 00 5 075
Part Q CONVERTING FRACTIONS TO PERCENTS: Example: Write as a percent 8 n % d 00% P Multiply the fraction by 00% 00% 8 00% 8 5% Multiply fraction by 00% To simplify, divide numerator and denominator by 75 % 7 % 7 5% Multiplying gives the answer as an improper fraction percent OR Convert answer to a mixed number or decimal percent Convert the following fractions to percents: 5 7 8 8 8% 50% 875% 75% Part R CONVERTING DECIMALS TO PERCENTS : Decimal 00% P% Multiply decimal by 00% Example: Write 0 as a percent 0 00% % Multiply decimal by 00% OR Move decimal point places to the right Convert the following decimals to percents: 065 00 7 05 65% % 70% 5%
Part S PERCENT PROBLEMS: What percent of 5 is 7? Percent Whole Part Could Also Be Written As 7 is what percent of 5? Part Percent Whole % of 50 is what number? Percent Whole ParT Could Also Be Written As What is % of 50? ParT Percent Whole 5% of what number is 6? Percent Whole Part Could Also Be Written As 6 is 5% of what number? Part Percent Whole Percent Part 00 Whole P 7 00 5 5P 700 5 5 P 0% Percent ParT 00 Whole T 00 50 600 00T 00 00 T 6 Percent 00 5 6 00 W Part Whole 5W 600 5 5 W 6 8 What percent of 5 is 5? is what percent of 60? What number is 65% of? 0% 0% 078 5% of 90 is what number? 5 is 5% of what? 6 80% of what number is? 5 5 60 6 60 Part T CIRCLE GRAPH: useful for showing parts out of a whole each sector (pie-sliced wedge) is part of the whole the total of all sectors must equal 00% Example: How many total students are in this class? 8 5 0 What percent of students earned A s? P 8 00 0 0P 800 0 0 P 0%
Use the circle graph below to answer the following questions Ms Smith is a school teacher She earns $,800 per month for her hard work as a teacher Her monthly expenses are represented by the circle graph below Budget Savings, 0% Rent, 0% Utilities, 5% Transportation, 0% Food, 5% How much does Ms Smith spend on rent? How much does she save every month? What percent of her monthly income does Ms Smith spend on utilities and transportation combined? What is the amount that Ms Smith spends on utilities and transportation combined? 5 What is Ms Smith s total expenditure other than rent? $,50 $80 5% $950 5 $,80 Part U SIMPLE INTEREST: Principal: (a) amount of money borrowed OR (b) amount of money deposited in an account Interest: (a) fee charged to borrow money OR (b) money earned on a deposit Interest Rate: (a) percent charged for money borrowed OR (b) percent earned on money deposited Simple Interest: interest calculated on original principal only Simple Interest = Principal Rate Time in Years OR I=Prt Example: What is the simple interest on a loan of $9000 for years if the interest rate is 75%? I = Interest =? P = Principal = $9000 r = Rate = 75% t = Time (in years) = I = P r t I ($9000)(7 5%)() I ($9000)(0 075)() I $, 700 5
A bank charges 6% simple interest How much must you pay on a loan of $000 for three years? What is the total amount in interest made on a $000 simple interest investment if the rate is 8% and the term is years? How much would Jerry pay in total if he borrows $000 at 5% simple interest for 5 years? $60 $0 $,500 Part V APPLICATIONS: Sales Tax, Commission, Rates, Tips Example: Jake is buying a shirt for $7 How much sales tax will he pay if the tax rate is 6%? Sales Tax = Rate Purchase T 6% $7 T 0 06 $7 T $ 0 Example: What is the commission made on a sale if the commission rate is % and the amount of sales is $68,5? Commision = Rate Sales C % $68,5 C 0 0 $68,5 T $,75 0 Alberto wants to buy a remote control car with a sales price of $50 If the tax rate is 6%, how much will he pay in tax? Dave sells $56,000 worth of advertising in one month If his commission rate is 8%, how much is his commission check? Edward finds a refrigerator listed for $,900 that is marked 0% off What is the sale price of the refrigerator? $087 $,80 $,0790 6