GMAT exam success Episode 2 Part 8 pps

– ARITHMETIC – If you are subtracting, change the subtraction sign to addition, and change the sign of the number following to its opposite Then follow the rules for addition: a –5 + –6 = –11 b –12 + (+7) = –5 Remember: When you subtract, you add the opposite M ULTIPLYING AND D IVIDING I NTEGERS If an even number of negatives is used, multiply or divide as usual, and the answer is positive a –3 × –4 = 12 b (–12 –6) × = If an odd number of negatives is used, multiply or divide as usual, and the answer is negative a –15 = –3 b (–2 × –4) × –5 = –40 This is helpful to remember when working with powers of a negative number If the power is even, the answer is positive If the power is odd, the answer is negative Fractions A fraction is a ratio of two numbers, where the top number is the numerator and the bottom number is the denominator R EDUCING F RACTIONS To reduce fractions to their lowest terms, or simplest form, find the GCF of both numerator and denominator Divide each part of the fraction by this common factor and the result is a reduced fraction When a fraction is in reduced form, the two remaining numbers in the fraction are relatively prime a  23 b 32x 4xy  y8 When performing operations with fractions, the important thing to remember is when you need a common denominator and when one is not necessary A DDING AND S UBTRACTING F RACTIONS It is very important to remember to find the least common denominator (LCD) when adding or subtracting fractions After this is done, you will be only adding or subtracting the numerators and keeping the common denominator as the bottom number in your answer a  23 b 3y  xy4 LCD  15 LCD  xy 5  10 16 15  15  15 x y x  xy4  3x xy 329 – ARITHMETIC – M ULTIPLYING F RACTIONS It is not necessary to get a common denominator when multiplying fractions To perform this operation, you can simply multiply across the numerators and then the denominators If possible, you can also cross-cancel common factors if they are present, as in example b a 23  b 12 25 53  12 255  D IVIDING F RACTIONS A common denominator is also not needed when dividing fractions, and the procedure is similar to multiplying Since dividing by a fraction is the same as multiplying by its reciprocal, leave the first fraction alone, change the division to multiplication, and change the number being divided by to its reciprocal a 43  45 431  35 b 3x y 1 3x 5xy 5x 12x 5xy  y1 124x1  Decimals The following chart reviews the place value names used with decimals Here are the decimal place names for the number 6384.2957 T H O U S A N D S O N E S D E C I M A L T E N T H S H U N D R E D S T E N S P O I N T H U N D R E D T H S T H O U S A N D T H S T T EH NO U S A N D T H S It is also helpful to know of the fractional equivalents to some commonly used decimals and percents, especially because you will not be able to use a calculator 0.1  10%  101 0.3  3313 %  13 0.4  40%  25 330 – ARITHMETIC – 0.5  50%  12 0.6  66 23 %  23 0.75  75%  34 A DDING AND S UBTRACTING D ECIMALS The important thing to remember about adding and subtracting decimals is that the decimal places must be lined up a 3.6 +5.61 b 9.21 5.984 –2.34 3.644 M ULTIPLYING D ECIMALS Multiply as usual, and count the total number of decimal places in the original numbers That total will be the amount of decimal places to count over from the right in the final answer 34.5 × 5.4 1,380 + 17,250 18,630 Since the original numbers have two decimal places, the final answer is 186.30 or 186.3 by counting over two places from the right in the answer D IVIDING D ECIMALS Start by moving any decimal in the number being divided by to change the number into a whole number Then move the decimal in the number being divided into the same number of places Divide as usual and keep track of the decimal place 1.53 5.1 5.1 1.53    ⇒ 51 15.3    15.3 331 – ARITHMETIC – Ratios A ratio is a comparison of two or more numbers with the same unit label A ratio can be written in three ways: a: b a to b or a b A rate is similar to a ratio except that the unit labels are different For example, the expression 50 miles per hour is a rate—50 miles/1 hour Proportion Two ratios set equal to each other is called a proportion To solve a proportion, cross-multiply  10x Cross multiply to get: 4x  50 4x  504 x  12 12 Percent A ratio that compares a number to 100 is called a percent To change a decimal to a percent, move the decimal two places to the right .25 = 25% 105 = 10.5% = 30% To change a percent to a decimal, move the decimal two places to the left 36% = 36 125% = 1.25 8% = 08 Some word problems that use percents are commission and rate-of-change problems, which include sales and interest problems The general proportion that can be set up to solve this type of word problem is Part Whole % , although more specific proportions will also be shown  100 332 – ARITHMETIC – C OMMISSION John earns 4.5% commission on all of his sales What is his commission if his sales total $235.12? To find the part of the sales John earns, set up a proportion: change part %   whole original cost 100 x 235.12 4.5  100 Cross multiply 100x  1058.04 100x 100  1058.04 100 x  10.5804  $10.58 R ATE OF C HANGE If a pair of shoes is marked down from $24 to $18, what is the percent of decrease? To solve the percent, set up the following proportion: change part %   whole original cost 100 24 18 24 24 x  100 x  100 Cross multiply 24x  600 24x 24  600 24 x  25% decrease in price Note that the number in the proportion setup represents the discount, not the sale price S IMPLE I NTEREST Pat deposited $650 into her bank account If the interest rate is 3% annually, how much money will she have in the bank after 10 years? 333 – ARITHMETIC – Interest = Principal (amount invested) × Interest rate (as a decimal) × Time (years) or I = PRT Substitute the values from the problem into the formula I = (650)(.03)(10) Multiply I = 195 Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her account Exponents The exponent of a number tells how many times to use that number as a factor For example, in the expression 43, is the base number and is the exponent, or power Four should be used as a factor three times: 43 = × × = 64 Any number raised to a negative exponent is the reciprocal of that number raised to the positive exponent:  113 22  19 Any number to a fractional exponent is the root of the number: 25  25  273  23 27  2564  24 256  Any nonzero number with zero as the exponent is equal to one: 140° = Square Roots and Perfect Squares Any number that is the product of two of the same factors is a perfect square × = 1, × = 4, × = 9, × = 16, × = 25, Knowing the first 20 perfect squares by heart may be helpful You probably already know at least the first ten 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 334 – ARITHMETIC – Radicals A square root symbol is also known as a radical sign The number inside the radical is the radicand To simplify a radical, find the largest perfect square factor of the radicand  32 =  16 ×  Take the square root of that number and leave any remaining numbers under the radical  32 = 4 To add or subtract square roots, you must have like terms In other words, the radicand must be the same If you have like terms, simply add or subtract the coefficients and keep the radicand the same Examples 3 + 2 = 5 2 4 –  = 3 6 + 3 cannot be combined because they are not like terms Here are some rules to remember when multiplying and dividing radicals: Multiplying:  x ×  y =  xy  ×  =  Dividing: x x  B y y 25 25   B 16 16 Counting Problems and Probability The probability of an event is the number of ways the event can occur, divided by the total possible outcomes P1E2 Number of ways the event can occur Total possible outcomes The probability that an event will NOT occur is equal to – P(E) 335 – ARITHMETIC – The counting principle says that the product of the number of choices equals the total number of possibilities For example, if you have two choices for an appetizer, four choices for a main course, and five choices for dessert, you can choose from a total of × × = 40 possible meals The symbol n! represents n factorial and is often used in probability and counting problems n! = (n) × (n – 1) × (n – 2) × × For example, 5! = × × × × = 120 Permutations and Combinations Permutations are the total number of arrangements or orders of objects when the order matters The formula is nPr  n! 1n r 2! , where n is the total number of things to choose from and r is the number of things to arrange at a time Some examples where permutations are used would be calculating the total number of different arrangements of letters and numbers on a license plate or the total number of ways three different people can finish first, second, and third in a race Combinations are the total number of arrangements or orders of objects when the order does not matter The formula is nCr  r!1n n! r 2! , where n is the total number of objects to choose from and r is the size of the group to choose An example where a combination is used would be selecting people for a committee Statistics Mean is the average of a set of numbers To calculate the mean, add all the numbers in the set and divide by the number of numbers in the set Find the mean of 2, 3, 5, 10, and 15    10  15  355 The mean is Median is the middle number in a set To find the median, first arrange the numbers in order and then find the middle number If two numbers share the middle, find the average of those two numbers Find the median of 12, 10, 2, 3, 15, and 12 First put the numbers in order: 2, 3, 10, 12, 12, and 15 Since an even number of numbers is given, two numbers share the middle (10 and 12) Find the average of 10 and 12 to find the median 10  12  222 The median is 11 336 – ARITHMETIC – Mode is the number that appears the most in a set of numbers and is usually the easiest to find Find the mode of 33, 32, 34, 99, 66, 34, 12, 33, and 34 Since 34 appears the most (three times), it is the mode of the set NOTE: It is possible for there to be no mode or several modes in a set Range is the difference between the largest and the smallest numbers in the set Find the range of the set 14, –12, 13, 10, 22, 23, –3, 10 Since –12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them 23 – (–12) = 23 + (+12) = 35 The range is 35 337 C H A P T E R 21  Algebra Translating Expressions and Equations Translating sentences and word problems into mathematical expressions and equations is similar to translating two different languages The key words are the vocabulary that tells what operations should be done and in what order Use the following chart to help you with some of the key words used on the GMAT® quantitative section   SUM DIFFERENCE PRODUCT QUOTIENT EQUAL TO MORE THAN LESS THAN TIMES DIVIDED BY TOTAL ADDED TO SUBTRACTED FROM MULTIPLIED BY PLUS MINUS INCREASED BY DECREASED BY FEWER THAN 339 – ALGEBRA – The following is an example of a problem where knowing the key words is necessary: Fifteen less than five times a number is equal to the product of ten and the number What is the number? Translate the sentence piece by piece: Fifteen less than five times the number equals 5x – 15 = the product of 10 and x 10x The equation is Subtract 5x from both sides: 5x – 15 = 10x 5x – 5x – 15 = 10x – 5x Divide both sides by 5: –15  5x5 –3 = x It is important to realize that the key words less than tell you to subtract from the number and the key word product reminds you to multiply  Combining Like Terms and Polynomials In algebra, you use a letter to represent an unknown quantity This letter is called the variable The number preceding the variable is called the coefficient If a number is not written in front of the variable, the coefficient is understood to be one If any coefficient or variable is raised to a power, this number is the exponent 3x xy –2x 3y Three is the coefficient and x is the variable One is the coefficient, and both x and y are the variables Negative two is the coefficient, x and y are the variables, and three is the exponent of x Another important concept to recognize is like terms In algebra, like terms are expressions that have exactly the same variable(s) to the same power and can be combined easily by adding or subtracting the coefficients Examples 3x + 5x 4x 2y + –10x 2y 2xy + 9x 2y These terms are like terms, and the sum is 8x These terms are also like terms, and the sum is –6x2y These terms are not like terms because the variables, taken with their powers, are not exactly the same They cannot be combined 340 – ALGEBRA – A polynomial is the sum or difference of many terms and some have specific names: 8x2 3x + 2y 4x + 2x –  This is a monomial because there is one term This is a binomial because there are two terms This is a trinomial because there are three terms Laws of Exponents ■ When multiplying like bases, add the exponents: x × x = x + = x ■ When dividing like bases, subtract the exponents: ■ When raising a power to another power, multiply the exponents: 1x2 23  x 23  x ■ Remember that a fractional exponent means the root:  x = x 2 and  x = x 3 x5 x 2= x5 – 2= x3 The following is an example of a question involving exponents: Solve for x: 2x + = 83 a b c d e The correct answer is d To solve this type of equation, each side must have the same base Since can be expressed as 23, then 83 = (23)3 = 29 Both sides of the equation have a common base of 2, so set the exponents equal to each other to solve for x x + = So, x =  Solving Linear Equations of One Variable When solving this type of equation, it is important to remember two basic properties: ■ ■ If a number is added to or subtracted from one side of an equation, it must be added to or subtracted from the other side If a number is multiplied or divided on one side of an equation, it must also be multiplied or divided on the other side 341 – ALGEBRA – Linear equations can be solved in four basic steps: Remove parentheses by using distributive property Combine like terms on the same side of the equal sign Move the variables to one side of the equation Solve the one- or two-step equation that remains, remembering the two previous properties Examples Solve for x in each of the following equations: a 3x – = 10 Add to both sides of the equation: 3x – + = 10 + Divide both sides by 3: 3x  153 x=5 b (x – 1) + x = Use distributive property to remove parentheses: 3x – + x = Combine like terms: 4x – = Add to both sides of the equation: 4x – + = + Divide both sides by 4: 4x  44 x=1 c 8x – = + 3x Subtract 3x from both sides of the equation to move the variables to one side: 8x – 3x – = + 3x – 3x Add to both sides of the equation: 5x – + = + Divide both sides by 5: 5x  105 x=2  Solving Literal Equations A literal equation is an equation that contains two or more variables It may be in the form of a formula You may be asked to solve a literal equation for one variable in terms of the other variables Use the same steps that you used to solve linear equations 342 – ALGEBRA – Example Solve for x in terms of a and b: Subtract b from both sides of the equation: 2x + b = a 2x + b – b = a – b Divide both sides of the equation by 2: 2x  a b x  a b  Solving Inequalities Solving inequalities is very similar to solving equations The four symbols used when solving inequalities are as follows: ■ ■ ■ ■  is less than  is greater than  is less than or equal to  is greater than or equal to When solving inequalities, there is one catch: If you are multiplying or dividing each side by a negative number, you must reverse the direction of the inequality symbol For example, solve the inequality –3x +  18: First subtract from both sides: –3x  6  18 Then divide both sides by –3: –3x –3 The inequality symbol now changes: x  12  –3 Solving Compound Inequalities A compound inequality is a combination of two inequalities For example, take the compound inequality –3  x +  To solve this, subtract from all parts of the inequality –3 –  x + –  – Simplify –4  x  Therefore, the solution set is all numbers between –4 and 3, not including –4 and 343 – ALGEBRA –  Multiplying and Factoring Polynomials When multiplying by a monomial, use the distributive property to simplify Examples Multiply each of the following: (6x 3)(5xy 2) = 30x 4y (Remember that x = x 1.) 2x (x – 3) = 2x – 6x x (3x + 4x – 2) = 3x + 4x – 2x When multiplying two binomials, use an acronym called FOIL F O I L Multiply the first terms in each set of parentheses Multiply the outer terms in the parentheses Multiply the inner terms in the parentheses Multiply the last terms in the parentheses Examples (x – 1)(x + 2) = x + 2x – 1x – = x + x – F O I L (a – b)2 = (a – b)(a – b) = a2 – ab – ab – b2 F O I L Factoring Polynomials Factoring polynomials is the reverse of multiplying them together Examples Factor the following:  2x + = (x + 1) Take out the common factor of x – = (x + 3)(x – 3) Factor the difference between two perfect squares 2x + 5x – = (2x – 1)(x + 3) Factor using FOIL backwards 2x – 50 = 2(x – 25) = 2(x + 5)(x – 5) First take out the common factor and then factor the difference between two squares Solving Quadratic Equations An equation in the form y = ax + bx + c, where a, b, and c are real numbers, is a quadratic equation In other words, the greatest exponent on x is two 344 – ALGEBRA – Quadratic equations can be solved in two ways: factoring, if it is possible for that equation, or using the quadratic formula By Factoring In order to factor the quadratic equation, it first needs to be in standard form This form is y = ax2+bx + c In most cases, the factors of the equations involve two numbers whose sum is b and product is c Examples Solve for x in the following equation: x – 25 = This equation is already in standard form This equation is a special case; it is the difference between two perfect squares To factor this, find the square root of both terms The square root of the first term x is x The square root of the second term 25 is Then two factors are x – and x + The equation x – 25 = then becomes (x – 5)(x + 5) = Set each factor equal to zero and solve x – = or x + = x = or x = –5 The solution is {5, –5} 2 x + 6x = –9 This equation needs to be put into standard form by adding to both sides of the equation x + 6x + = –9 + x + 6x + = The factors of this trinomial will be two numbers whose sum is and whose product is The factors are x + and x + because + = and × = The equation becomes (x + 3)(x + 3) = Set each factor equal to zero and solve x + = or x + = x = –3 or x = –3 Because both factors were the same, this was a perfect square trinomial The solution is {–3} x2 = 12 + x This equation needs to be put into standard form by subtracting 12 and x from both sides of the equation x – x – 12 = 12 – 12 + x – x x – x – 12 = 345 – ALGEBRA – Since the sum of and –4 is –1, and their product is –12, the equation factors to (x + 3) (x – 4) = Set each factor equal to zero and solve: x + = or x – = x = –3 or x = The solution is {–3, 4} By Quadratic Formula Solving by using the quadratic formula will work for any quadratic equation, especially those that are not factorable Solve for x: x + 4x = Put the equation in standard form x + 4x – = Since this equation is not factorable, use the quadratic formula by identifying the value of a, b, and c and then substituting it into the formula For this particular equation, a = 1, b = 4, and c = –1 x –b ; b – 4ac 2a x –4 ; 42 41121–12 2112 x –4 ; 16  x –4 ; 20 x –4 22 ; 2 x  –2 ; The solution is  –2  5, –2 –  The following is an example of a word problem incorporating quadratic equations: A rectangular pool has a width of 25 feet and a length of 30 feet A deck with a uniform width surrounds it If the area of the deck and the pool together is 1,254 square feet, what is the width of the deck? 346 – ALGEBRA – Begin by drawing a picture of the situation The picture could be similar to the following figure x 30 x x 25 x Since you know the area of the entire figure, write an equation that uses this information Since we are trying to find the width of the deck, let x = the width of the deck Therefore, x + x + 25 or 2x + 25 is the width of the entire figure In the same way, x + x + 30 or 2x + 30 is the length of the entire figure The area of a rectangle is length × width, so use A = l × w Substitute into the equation: Multiply using FOIL: Combine like terms: Subtract 1,254 from both sides: Divide each term by 2: Factor the trinomial: Set each factor equal to and solve 1,254 = (2x + 30)(2x + 25) 1,254 = 4x + 50x + 60x + 750 1,254 = 4x + 110x + 750 1,254 – 1,254 = 4x + 110x + 750 – 1,254 = 4x + 110x – 504 = 2x +55x – 252 = (2x + 63 )(x – 4) 2x + 63 = or x – = 2x = –63 x = x = –31.5 Since we are solving for a length, the solution of –31.5 must be rejected The width of the deck is feet  Rational Expressions and Equations Rational expressions and equations involve fractions Since dividing by zero is undefined, it is important to know when an expression is undefined The fraction x is undefined when the denominator x – = 0; therefore, x = 347 – ALGEBRA – You may be asked to perform various operations on rational expressions See the following examples Examples Simplify x b x3b2 Simplify x 3x Multiply 4x x – 16 Divide Add xy a  2a a2  3a  x  2x a – 3a 2a   y3 Subtract x  x – x – 3x Solve 23 x  16 x  14 Solve x  14  16 Answers x2b x3b2 1x  21x  1x 3 31x 3 4x1x  2 2x2 1x 21x   x1x 4 21a  2 a1a  2 1a  21a  2 a1a  a  3x xy 3x  18 x  3x  xb1  20  2x 3x 348 ... Solve x  14  16 Answers x2b x3b2 1x  21 x  1x 3 31x 3 4x1x  2 2x2 1x 21 x   x1x 4 21 a  2 a1a  2 1a  21 a  2 a1a  a  3x xy 3x  18 x  3x  xb1  20  2x 3x 3 48 ... particular equation, a = 1, b = 4, and c = –1 x –b ; b – 4ac 2a x –4 ; 42 41 121 – 12 21 12 x –4 ; 16  x –4 ; 20 x –4 22 ; 2 x  ? ?2 ; The solution is  ? ?2  5, ? ?2 –  The following is an example... range of the set 14, – 12, 13, 10, 22 , 23 , –3, 10 Since – 12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them 23 – (– 12) = 23 + (+ 12) = 35 The range is