GMAT Quant Common Mistakes

Is x < y ? (1) 2x < 3y (2) xy > 0 Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

(1) 2x < 3y (2x < 3y)/2 = x< 3/2x NS bc if x or y is positive then the formula won't work. It could work if negative but we don't know if it's positive or negative (2)xy>0 which means that x & y are either both negative or both positive

If x and y are integers, is x > y ? x + y > 0 y^x < 0

C If x and y are integers, is x > y? (1) x + y > 0. Given that the sum of two numbers is greater than zero, but we cannot determine which one is greater. Not sufficient. (2) y^x < 0. This statement implies that y is a negative number. Now, if y=-1 and x=1, then x>y BUT if y=-1 and x=-1, then x=y. Not sufficient. (1)+(2) Since from (2) we have that y is a negative number, then -y is a positive number. Therefore from (1) we have that x>-y=positive, which means that x is a positive number. So, we have that x=positive>y=negative. Sufficient. Answer: C.

Store X sold 1/6 more television sets in March than in February, and in February, it sold 3/8 fewer television sets than in January. If the store sold 480 television sets in January, how many sets were sold in March? 210 300 350 550 770

CAREFULLY READ THE QUESTION J=480, F= -(3/8)J, M= (1/6)+ F To find F, x/480 = (3/8) CROSS MULTIPLY 8x = 1440 DIVIDE x(f) = 180 J-F= 480-180= 300 To find M, x(m)/300 = (1/6) CROSS MULTIPLY 6x(m) = 300 DIVIDE x(m) = 50 300+50

Is x + y > z? (1) z > y (2) z > x

Can skip to C or E because we don't have both variables in either statements > However, because z > y & x while there are cases where they could work there are also cases where it wouldn't work so therefore is NOT SUFFICIENT > IF, one of the signs had been flipped and the other stayed the same, the flipped sign would've been sufficient

A certain square has sides of length x. If the length of each side is doubled, then which of the following represents the area of the resulting square, in terms of x? x^2 2x^2 4x 4x^2 x^2 + 4

DRAW THE SHAPE AND IT'LL MAKE THE QUESTION EASIER Square A: x * x = 2x Ex. 2 * 2 = 4 Square B 2x*2x = 4x^2 Ex. 2(2)*2(2)= 4(2)^2 = 16

A circle is drawn in the xy-coordinate plane. If there are n different points (x, y) on the circle such that xy = 0, then the possible values of n are 0, 1, or 2 0, 2, or 4 0 or 4 0, 2, 3, or 4 0, 1, 2, 3, or 4

E

All of the furniture for sale at Al's Discount Furniture is offered for less than the manufacturer's suggested retail price (MSRP). Once a year, Al's holds a clearance sale. If Jamie purchased a certain desk during the sale, did she get a discount of more than 50% of Al's regular price for the desk? (1) Al's regular price for the desk is less than 60% of the MSRP of $2,000. (2) The sale price was $600 less than Al's regular price for the desk.

E

In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through? (-14, 10) (-7, 5) (12, -4) (14, -5) (21, -9)

FIND THE SLOPE ( RISE/RUN) = (0-5)/(7-0) = -(5/7) **DIDN'T REALIZE THE Y-INTERCEPT WAS GIVEN** ** READ MORE CAREFULLY** y= -(5/7)x+5 Then just plug in the answers x's to get the correct answer (14,-5)

Which of the following is the lowest positive integer that is divisible by 8, 9, 10, 11, and 12? 7,920 5,940 3,960 2,970 890

Find the prime factors for each number and then multiply the highest exponent of the factors together to get final answer 8:2^3 9:3^2 10:2 x 5 11:11 12:2^2 x 3 2^3x3^2x5x11=8 x 9 x 5 x 11 = 72 x 55 = 3960

If n divided by 7 has a remainder of 2, what is the remainder when 3 times n is divided by 7? 1 2 3 5 6

First identify all possible values for n that would have a remainder of 2 n= 9,16,23... plug in n 3(9)/7 = 27/7 = 3 R 6

If 3|3 - x| = 7, what is the product of all the possible values of x? 1/9 1/3 2/3 16/9 32/9

First simplify the equation (3|3 - x| = 7)/3 = |3 - x| = 7/3 Now find the two sets of absolute values REMEMBER NEVER CHANGE THE LEFT, ONLY CHANGE THE RIGHT 3-x= 7/3 MOVE THE X AND 7/3 TO THE OTHER SIDES 3-(7/3)=x (9/3)-(7/3)=x 2/3 = x 3-x= -(7/3) MOVE THE X AND 7/3 TO THE OTHER SIDES 3+(7/3)=x (9/3)+(7/3)=x 16/3 = x Multiple the two products (2/3)x(16/3)= (32/9)

Is a/b < 0? (1) a2 / b3 > 0 (2) ab4 < 0

For their quotient to be less than zero, a and b must have opposite signs. In other words, if the answer to the question is "yes," EITHER a is positive and b is negative OR a is negative and b is positive. The question can be rephrased as the following: "Do a and b have opposite signs?" (1) INSUFFICIENT: a2 is always positive so for the quotient of a2 and b3 to be positive, b3 must be positive. That means that b is positive. This does not however tell us anything about the sign of a. (2) INSUFFICIENT: b4 is always positive so for the product of a and b4 to be negative, a must be negative. This does not however tell us anything about the sign of b. (1) AND (2) SUFFICIENT: Statement 1 tells us that b is positive and statement 2 tells us that a is negative. The yes/no question can be definitively answered with a "yes." The correct answer is C.

Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs? 2 5 6 8 9

Formula for 2 set overlapping sets: Total = Group 1 + Group 2 - (people in both) + (people in neither) Formula for 3 set overlapping sets: Total = Group 1 + Group 2 + Group 3 - (people in 2 groups) - 2(people in all 3 groups) + none. 59 = 22+27+28-6-2x+0 59= 71 - 2x -12 = -2x x = 6

At a constant rate of flow, it takes 20 minutes to fill a swimming pool if a large hose is used and 30 minutes if a small hose is used. At these constant rates, how many minutes will it take to fill the pool when both hoses are used simultaneously? 10 12 15 25 50

It's important to combine the rates together properly Large= 1/20 , Small 1/30 To combine, convert the denominator to LCM > Large (3*(1/20) = 3/60 >Small (2*(1/30) = 2/60 Combined: 3/60+2/60= 5/60 or 1/12 Rate combined is 12

2 2 a 3 +4 b= 9 0 If a and b represent positive single digits in the correctly worked computation above, what is the value of a + 2b ? 2 7 9 11 12

JUST BE CAREFUL OF READING PROPERLY WOULD'VE HELPED YOU ANSWER THE QUESTION PROPERLY MISTAKE WAS THAT YOU MISREAD THE VARIABLE A > Should of just put both in their own algebraic phrase 2+3+b = 10~ 5+b = 10 b= 5 6+1(carried over from rt side) + a = 9 ~ 7+a=9 ~ a=2

Working simultaneously at their respective constant working rates, Machine A & B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours, in terms of x and y, how many hours does it take Machine B, working alone at its constant rate to produce 800 nails x/(x+y) y/(x+y) xy/(x+y) xy(x-y) xy/(y-x)

MAKE SURE TO NOTE THE INDIVIDUAL RATES GIVEN Machine A & B Together = (800/x) Machine A alone = (800/y) To find Machine B alone = (800/x)-(800/y) (800/x)-(800/y) MAKE COMMON DENOM (XY) (800y-800x)/xy (800y-800x)/xy = (800/b) MOVE NUMBERS B= 800xy/ (800y-800x) --> xy/(y-x)

10 students took a chemistry exam that was graded on a scale of 0 to 100. Five of the students were in Dr. Adams' class and the other five students were in Dr. Brown's class. Is the median score for Dr. Adams' students greater than the median score for Dr. Brown's students? (1) The range of scores for students in Dr. Adams' class was 40 to 80, while the range of scores for students in Dr. Brown's class was 50 to 90. (2) If the students are paired in study teams such that each student from Dr. Adams' class has a partner from Dr. Brown's class, there is a way to pair the 10 students such that the higher scorer in each pair is one of Dr. Brown's students.

(1) Not Sufficient bc while we have the ranges we're not able to determine what the median score is despite have the range (2) States that when students are paired between the two classes B > A. Every pair of students from A < B so this will be sufficient

If x, y, and z are nonzero numbers, is (x)(y + z) > 0? (1) |x + y| = |x| + |y| (2) |z + y| = |y| + |z|

(1) Not sufficient because we don't know about z (2) Not sufficient because we don't know about x Combined Sufficient Because we know about all 3 variables and that they are > 0

If a and b are positive integers such that a < b, is b even? 1)(b/2)-(a/2) is an integer. (2) (3b/4)-(a/2)is an integer.

(1) Simplified the equation for Statement (1) can be translated to (b-a)/2. When you plug in b as a even number and use a as a odd number it does produce an integer. However, b could be a odd number and a could be a even and the result would still come out the same therefore NOT SUFFICIENT (2) Simplified the equation for Statement (2) can be translated to (3b-2a)/4. For these equations, b can't work with odd numbers because the result isn't a integer. However, when b is even and a is odd the result does work after b=4. SUFFICIENT

What was Bill's average speed on his trip of 250 miles from New York City to Boston? (1) The trip took Bill 5 hours. (2) At the midpoint of his trip, Bill was going exactly 50 miles per hour.

(1) Sufficient bc your are given the distance and the time which allows to calculate speed (2) Insufficient bc you are told that at the midpoint, he's going 50 mph but it's unclear whether or not his speed changed over time.

What is the value of |x|? (1) |x^2 + 16| - 5 = 27 (2) x^2 = 8x - 16

(1) Sufficient because the value of x^2 must be non-negative, the value will always be positive |x2 + 16| - 5 = 27 x2 + 16 - 5 = 27 x2 + 11 = 27 x2 = 16 x = 4 or x = -4 Since |-4| = |4| = 4, we know that |x| = 4; this statement is sufficient. (2) Sufficient because x^2-8x+16 is a perfect square and translate to (x-4)(x-4) or x = 4

Is x > y? (1) √x > y (2) x^3 >y

(1) √x > y works well if it is a certain whole number Ex. √2 > 1, but it doesn't work for fractions √(1/3) > 2 NOT SUFFICIENT (2) x^3 >y works well if it a whole number, but it doesn't work for fractions (1/2)^3> 1 NOT SUFFICIENT You have the two unique equations to answer yes or no. NEVER ASSUME IT'S AN INTEGER, ALWAYS USE FRACTIONS TOO UNLESS SPECIFIED OTHERWISE.

The charge for a single room at Hotel P is 25 percent less than the charge for a single room at Hotel R and 10 percent less than the charge for a single room at Hotel G. The charge for a single room at Hotel R is what percent greater than the charge for a single room at Hotel G ? 15% 20% 40% 50% 150%

20 Easiest to solve in fractions P = (3/4)R , P = (9/10)G (3/4)R = (9/10)G (4/3)*(3/4)R= (4/3)*(9/10)G R = (36/30)G R=6/5G or 1.2G 20%

If water is leaking from a certain tank at a constant rate of 1,200 milliliters per hour, how many seconds does it take for 1 milliliter of water to leak from the tank? 1/3 1/2 2 3 20

3

Two sides of a triangle have lengths x and y and meet at a right angle. If the perimeter of the triangle is 4 x, what is the ratio of x to y ? 2 : 3 3 : 4 4 : 3 3 : 2 2 : 1

3:4 Drawing the figure first helps assign the unknown sign variable and solve for the perimeter x + y+ z = 4x SIMPLIFY z=3x-y Since it's a right triangle, can use Pythagorean theorem x^2+y^2= (3x-y)^2 SIMPLIFY x^2+y^2=9x^2-6xy+y^2 (-8x^2=-6xy)/x -8x=-6y SWITCH RULE -8/-6=y/x or x/y = 6/8 = 3/4

At what point does the line ax + y + b = 0 intersect the y-axis? (1) b = 5 (2) a = 3

A Before jumping into the option, we are trying to find the y-axis which is when x=0. so the equation will be y+b = 0 or y=-b. Looking for B (1) B= 5 SUFFICIENT

If k is a positive integer, what is the remainder when (k+2)(k3-k) is divided by 6 0 1 2 3 4

ADVANCED DIVISIBILITY RULES TO REMEMBER HERE: > The product of 3 consecutive integers is always 3 > A product is divisible by 4 if it contains two 2s. (k+2)(k3-k) Factor k out of the second equation (k+2)(k)(k2-1) Simply 2nd equation into quadratic form (a2-b2) (k+2)(k)(k+1)(k-1) Put in sequential order (k-1)(k)(k+1)(k+2) ~ indicates that this is a sequential series of numbers Remembering the set of rules from above, this indicates that the sequence is divisible by 2 and 3 which would leave the remainder of this sequence at 0

If the length of side AB is 17, is triangle ABC a right triangle? (1) The length of side BC is 144. (2) The length of side AC is 145. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

According to the Pythagorean Theorem, in a right triangle a2 + b2 = c2. (1) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2. (2) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2. (1) AND (2) SUFFICIENT: With all three side lengths, we can determine if a2 + b2 = c2. It turns out that 172 + 1442 = 1452, so this is a right triangle. However, even if it were not a right triangle, this formula would still be sufficient, so it is unnecessary to finish the calculation. The correct answer is C. IF A^2+B^2 DIDN'T EQUAL C^2, THE ANSWER WOULD STILL BE C BECAUSE WE HAVE ENOUGH INFORMATION

A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible? A. 144 B. 152 C. 160 D. 168 E. 176

B Find the possible values if 0 is the middle term and if 1 is the middle terms If 0 is middle= (8*9) = 72 If 1 is middle = (8*10) = 80 72+80= 152

If a and b are negative integers, which of the following statements must be true? I. (-a)^b is positive. II. (-a)^-b is positive. III. a^-b is positive. None II only I and II only I and III only I, II and III

BREAKDOWN THE THREE STATEMENTS PUTTING NEG #S I. (-a)^b --> (-(-a))^-(b) --> a^(-b). TRUE because the values will always be positive but will be in fraction form a^-b = 1/(a^b) II. (-a)^-b--> (-(-a))^-(-b)-->a^b TRUE III.a^-b works if a if both are even

In the xy-plane, lines k and l intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of l (1) The slope of k is less than the slope of l (2)The slope of l is positive

C (1) Initially looks good k could go either way in terms of the y-intercept (2) Important to know but doesn't mean but doesn't solve the equation properly. Combined, the y-intercept for K is always higher than y-intercept for L

What is the value of the two-digit positive integer n ? (1) The sum of the digits in n is 12. (2) If the digits in n are reversed, the value of the number formed is 36 more than the value of n.

C (1) List all the possible pairs (6,6) (8,4) (4,8) (9,3) (3,9) Too many answer choices, need to be able to narrow it down to a single answer choice Skip to (1)&(2) because 2 alone doesn't make sense Only value that works switched is (4.8)

The third-place finisher of the Allen County hot dog eating contest, in which each contestant was given an equal amount of time to eat as many hot dogs as possible, required an average of 15 seconds to consume each hot dog. How many hot dogs did the winner eat? (1) The winner consumed 24 more hot dogs than did the third-place finisher. (2) The winner consumed hot dogs at double the rate of the third-place finisher.

C (1) The winner consumed 24 more hot dogs than did the third-place finisher --> W=T+24 --> one equation two unknowns. Not sufficient. (2) The winner consumed hot dogs at double the rate of the third-place finisher --> at double rate the winner would consume twice as many hot dogs, so W=2T --> one equation two unknowns. Not sufficient. (1)+(2) We have two distinct linear equations with two unknowns, so we can get the single numerical values of T and W: T+24=2T --> T=24. Sufficient.

Was the number of books sold at Bookstore X last week greater than the number of books sold at Bookstore Y last week? (1)Last week, more than 1,000 books were sold at Bookstore X on Saturday and fewer than 1,000 books were sold at Bookstore Y on Saturday. (2)Last week, less than 20 percent of the books sold at Bookstore X were sold on Saturday and more than 20 percent of the books sold at Bookstore Y were sold on Saturday.

C A ] Does not tell us anything about the sales in the rest of the week .INSUFFICIENT B] Does not tell us 20% of what , hence we cannot compare . INSUFFICIENT Together Let sales at bookstore X on saturday be 1001 . Therefore >5005 books were sold in the week Let sales at bookstore Y on saturday be 999. Therefore <4995 books were sold during the week Hence we can conclude that sales at X were greater than Y for the given week

Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than π/100 square feet in area and each more than 3 feet from the nearest wall. Can the rug be placed to cover both stains? (1)Jack's rug covers an area of 9π square feet. (2)The centers of the stains are less than 4 feet apart.

C BEFORE JUMPING INTO THE ANSWER CHOICES, HERE'S WHAT YOU CAN FIGURE OUT CIRCULAR STAINS ARE π/100 OR π(1/10)^2 STAINS ARE MORE THAN 3 FT APART but we need to know how big the rug is and the distance between stains (1) The area of the circular rug is 9π or 3^2π --> radius = 3 , diameter is 6 NOT SUFFICIENT (2) Says that the stains are less than 4ft away. BUT we don't know how big the rug is NOT SUFFICIENT (1)&(2) We know both things

A certain library charges a flat fee of $5 for any overdue book for the first x days the book is overdue, and an additional fee of m dollars per day for every day after x that the book is overdue. What is the fee for a book that is 12 days overdue? (1) The fee for a book that is 9 days overdue is $2 more than the fee for a book that is 8 days overdue. (2) The fee for a book that is 8 days overdue is the same as the fee for a book that is 7 days overdue. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

Combo Logic in Math

A certain taxi driver charges a rate of r cents per person per mile. How much money, in dollars, would it cost three people to travel x miles if he gives them a 50% discount? (100 cents = 1 dollar) 3xr/2 3x/200r 3r/200x 3xr/200 xr/600

D Read carefully, r-cents/person, miles- x rate per person ~ rx *asking for three people* 3rx* .5 = (3rx/2) <-Only rate per cent ((3rx/2)/100)= (3rx/200)

If mn ≠ 0 and 25 percent of n equals 37.5 percent of m, what is the value of 12n/m

DON'T KEEP AS PERCENTAGE, CHANGE TO FRACTION .25n =.375m CONVERTS TO N/4= 3M/8 CROSS MULTIPLY 8n = 12m MOVE M & 8 TO THE DENOMINATOR N/M = 12/8 SIMPLIFY N/M = 3/2 MULTIPLY 12*N/M (12*3)/2= (36)/2 = 18

At a refreshment stand, each can of soda sells for the same price and each sandwich sells for the same price. What is the total price for 2 sandwiches and 3 cans of soda at the stand? (1) At the stand the total price for 1 sandwich and 1 can of soda is $3. (2) At the stand the total price for 3 sandwiches and 2 cans of soda is $8. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

DON'T NEED TO SOLVE DATA SUFFICIENCY Just need to make sure there is enough information Need to find 2S+3C =?? (1) S+C = 3 NS bc it doesn't help us to determine the unique price of each (2)3S+2C= 8 NS bc it doesn't help us to determine the unique price of each C & E left --> Trap Answer is E due to confusion HOWEVER, when combined you have two unique formula and WHENEVER you have two unique formulas you can solve for the equation.

If x is the sum of six consecutive integers, then x is divisible by which of the following: I. 3 II. 4 III. 6 I only II only III only I and III I, II, and III

DON'T OVERTHINK. 6x+15 > 3 can come out of the equation > 4 doesn't work because there's not two 2s >6 doesn't work because there isn't both a 2 or 3 in equation

3) Kevin drove from A to B at a constant speed of 60 mph. Once he reached B, he turned right around with pause, and returned to A at a constant speed of 80 mph. Exactly 4 hours before the end of his trip, he was still approaching B, only 15 miles away from it. What is the distance between A and B? (A) 275 mi (B) 300 mi (C) 320 mi (D) 350 mi (E) 390 mi

First take all the relevant info from the question a>b = 60mph; b>a=80mph; 4 hrs P--15-->B->A *note* doesn't say 4hr total trip, but part of the trip Looking for D Find T for the first leg of the given trip T=D/R = 15/60= 1/4 Total Time for the Second Leg 4-(1/4) = 15/4 Plug in Second Leg Time Simplify b>a = 80*(15/4) = 20 * 15 = 300 (B)

Is n/14 an integer? (1) n is divisible by 28. (2) n is divisible by 70. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

INTEGER IS A WHOLE NUMBER For a number to be divisible by 14, it needs to be divisible by 2 & 7 (1) n is divisible by 28 --> Sufficient bc 28 is divisible by 2, 7. The factors for 28 are 1,2,4,7,14,28 (2) n is divisible by 70 --> Sufficient bc 70 is divisible by 2,7. The factors for 70 are 1, 2, 5,7,10,14,35,70 Therefore the answer is D: EACH statement ALONE is sufficient.

What is the value of x? (1) x^2 - 5 x + 6 = 0 (2) x > 0 Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

NEED ONE SOLUTION, NOT MULTIPLE POSSIBLE SOLUTIONS (1) (X-3)(X-2) ~ x = 2 or 3 Not sufficient bc 2 solutions (2) x > 0 ~ x is positive Not sufficient bc there's too many solutions (1 & 2) Combined not sufficient bc we only have one solution

The average weight of the women in a room is 120 lbs, and the average weight of the men in the room is 150 lbs. What is the average weight of the people in the room? (1) There are twice as many men as women in the room. (2) There are a total of 120 people in the room.

NEED TO KNOW THE RATIO BETWEEN MEN AND WOMEN (2) Not sufficient because it only tells us about the total distribution, and not the specifics we need to know regarding the ratio of men and women (1) Provided m = 2f (2f(150)+f(120))/(2f+f) = (300f+120f)/(3f)= (420f/3f) =140

If x/y = c/d and d/c = b/a, which of the following must be true? I. y/x = b/a II. x/a = y/b III. y/a = x/b I ONLY II ONLY III ONLY I & II ONLY I & III ONLY I, II, & IIII

NUMERATOR AND DENOMINATORS CAN BE SWAPPED y/x = d/c = a/b or x/y = c/d=b/a I. Easy conversion -> y/x = a/b Yes II. y/x= a/b --> x/a = y/b Yes III. y/x = a/b --> Numerators & denominators can't go together I & II only.

A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach? A. 35 B. 42 C. 35√3 D. 7+35√3 E. 7+42√3

Need to draw the figure. The figure is a right triangle with the 70 as the hypotenuse, The 7 isn't part of the triangle.

If x is an integer, is x^3 even? (1) 2x + 2 is even. (2) 3x + 1 is even.

ODD/EVEN RULE IS SUPER IMPORTANT HERE. (1)2x+2 Even? No because x can be even or odd (2)3x+1 even? Yes because the 3x must be, so x must be odd and two odds can get used together SUFFICIENT

If a = 1 and b = -2, then (2a^2 + b)(x + y) + (a + b)(x - y) = 0 2x y - x x - y x + y

PEMDAS >EXPONENT DOESN'T APPLY TO BOTH 2 and a^2, ONLY a^2 (2(1)^2 + (-2))(x + y) + (1 + (-2))(x - y) (2+(-2))( x+y)+(-1)(x-y) (0)(x+y)-(x-y) -(x-y) -x+y y-x (C)

A certain store sells only black shoes and brown shoes. In a certain week, the store sold x black shoes and y brown shoes. If 2/3 of all shoes sold that week were black, which of the following expressions represents the value of y, in terms of x? x/3 x/2 2x/3 3x/2 2x

PICK SMART NUMBERS, THEN PLUG-IN, USE SOME SET OF THREE SINCE 3 IS IN THE DENOMINATOR, THIS CASE USE 6 black shoes(x): (2/3)(6) = 4 brown shoes (y) 1-x = 1-4= 2 Plug-in X into the equations and see if you get 2 (A) 4/3 = 1.33 Incorrect (B) 4/2 = 2 CORRECT (C) 2(4)/3 = 8/3 = 2.33 Incorrect (D) 3(4)/2 = 12/2 = 6 Incorrect (E) 2(4) = 8 Incorrect OR CAN SOLVE ALGEBRAICALLY SINCE WE KNOW X IS 2/3 TOTAL x=(2/3)(x+y) x=(2x+2y)/3 MULTIPLY BOTH SIDES BY 3 3x=2x+2y SIMPLIFY x= 2y DIVIDE BOTH SIDE BY 2 x/2 = y

If 2x - 2x - 2 = 3(213), what is the value of x ? 9 11 13 15 17

PLUGGING & CHUGGING DOESN'T WORK HERE NEED TO SIMPLIFY THE FIRST HALF OF THE EQUATION FIRST 2x - 2x - 2 = 3(213) SIMPLIFY (2x/1) - (2x/22)=3(213) FIND COMMON DENOMINATOR ON 1ST SIDE(22) & SIMPLIFY (222x- 2x/22) = ((3(213))/1) CROSS MULTIPLY 222x- 2x =3(213) 22 SIMPLIFY 222x- 2x =3(215) FACTOR 2x FROM THE EQUATION 2x(22 - 1) = 3(215) SIMPLIFY 3(2x)= 3(215) X = 15

x, y, a, and b are positive integers. When x is divided by y, the remainder is 6. When a is divided by b, the remainder is 9. Which of the following is NOT a possible value for y + b? 24 21 20 17 15

PLUGGING AND CHUGGING DOESN'T HELP HERE Utilize Algebra & Inequalities to Solve (1) x/y ~ R6 (2)a/b ~ R9 Combined y+b> 6+9 y+b> 15 Therefore the answer is 15

Malik's recipe for 4 servings of a certain dish requires 1.5 cups of pasta. According to this recipe, what is the number of cups of pasta that Malik will use the next time he prepares this dish? (1) The next time he prepares this dish, Malik will make half as many servings as he did the last time he prepared the dish. (2) Malik used 6 cups of pasta the last time he prepared this dish. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

Pay attention to EVERY word in the sentence. Don't fall for the E trap just bc the question does look confusing (1) Next Time = 1/2(Last Time) Seems like it would answer the questions bc you would just take half of what they provided to get the answer BUT we have to think of the question in present tense. Question doesn't refer to half of what was provided, it refers to half of what was done the last time he made this recipe. NS (2) It states the last time he made the recipe he used 6-cups of pasta,HOWEVER, this statement alone doesn't answer what he is going to make the NEXT time, it only indicates LAST time. NS Left w/ C or E If combined, works bc we know that the he's going to make half of what he made the LAST time, and he made 6 cups of pasta LAST time.

If n is a nonzero integer, is xn < 1? (1) x > 1 (2) n > 0 Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

Plug & chug a few different cases

If -1 < x < 0, which of the following must be true? I. x^3 < x^2 II. x^5 < 1 - x III. x^4 < x^2 I only I and II only II and III only I and III only I, II and III

Plug and chug numbers to solve From this we're given that x is a negative fraction Use -(1/2) I. -(1/2)^3< -(1/2)^2 = -(1/8) < (1/4) YES II.-(1/2)^5< 1-(-(1/2)) = -(1/32) < 1.5 YES III. -(1/2)^4 < -(1/2)^2 = (1/4)< (1/2) YES

Advanced Divisibility Concepts

Product of 3 consecutive integers Is always 3 A product is divisible bt 4 if contains twos

If, 5 years ago, Jamie was half as old as he is now, how old will he be in x years? x + 10 x + 5 x + 2 x - 5 2x

READ CAREFULLY STUPID Since the question says that Jamie was half of his current age 5 years ago, the only value that will work is 5 a nd 10. 5 years ago - 5, Now- 10 , To find the age in x years, take x +10

Quentin's income is 60% less than Rex's income, and Sam's income is 25% less than Quentin's income. If Rex gave 60% of his income to Sam and 40% of his income to Quentin, Quentin's new income would be what fraction of Sam's new income? 8/9 11/12 8/13 11/13 12/13

READ EACH TERM CAREFULLY Q is 60% Less than R or Q is 40% of R ~ Q = .4R S is 25% Less than Q or S is 75% of Q ~ S = .75Q PLUG IN SMART NUMBERS , LET R = 100 Q = .4(100) = 40 S = .75(40) = 30 CURRENT RATIO = 40/30= 4/3 DISTRIBUTE ACCORDINGLY 60% of R to S = 60 , 40% of R to Q = 40 Q= 40+40 =80 S= 30+60 = 90 Q/S = 80/90 = 8/9

A basketball team composed of 12 players scored 100 points in a particular contest. If none of the individual players scored fewer than 7 points, what is the greatest number of points that an individual player might have scored? 7 13 16 21 23

READ THE QUESTION CAREFULLY BEFORE FREAKING OUT Based on what is proved, if all 12 players scored the same 7 points it would only add up to 84 which isn't enough based on what is provided, it's very safe to assume that 11 players scored the 7 points. So to find out how many the max player gets take 100-(11x7) = 100-77=23 DON'T PUT THINGS IN THE QUESTION THAT AREN'T PROVIDED

If R = 1 + 2xy + x2y2, what is the value of xy ? (1) R = 0 (2) x > 0 Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

REMEMBER THE FOLLOWING QUADRATIC FORMS >(a+b)2= a2+2ab+b2 >(a-b)2= a2-2ab+b2 >a2-b2 = (a+b)(a-b) (1) R = 0 0 = 1 +2xy + x2y2 >Disguised quadratic form (a+b)^2 a=1 b= xy (1+xy)(1-+xy) --> 1 + 2xy + x2y2 0 = (1+xy)^2 --> sqrt(0)= sqrt((1+xy)^2)--> 0= 1+xy --> -1 = xy SUFFICIENT (2) x > 0 Knowing that x is positive doesn't help us to solve the equation A is correct answer

At the beginning of the year, the ratio of juniors to seniors in high school X was 3 to 4. During the year, 10 juniors and twice as many seniors transferred to another high school, while no new students joined high school X. If, at the end of the year, the ratio of juniors to seniors was 4 to 5, how many seniors were there in high school X at the beginning of the year? 80 90 100 110 120

REMEMBER TO READ THE QUESTION CAREFULLY Beginning Ratio of Junior to Seniors : 3x/4x During the Yr, 10 Juniors Left, Twice as Many Seniors(2x10=20) New Ratio of Junior to Seniors: (3x-10)/(4x-20) Given New Ratio of Juniors to Seniors: 4/5 (3x-10)/(4x-20)= (4/5) Cross Multiply 15x-50 = 16x - 80 Simplify x= 80 80(4)=120

<--w--x--y--z--> On the number line above, is the product of w, x, y, and z negative? (1) z is positive. (2) The product of w and x is positive. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

REMEMBER, This is a sequence of numbers on the number-line. In order, to find the product of the 4- variables one of the variables needs to be negative & the rest positive, or the one positive and the rest negative (1) Z is positive. Doesn't tell us what the rest of the variables are to determine whether or not the product is negative NS (2) The product of w and x is positive. knowing that the product of w and x is positive, it can be seen that w & x are either both negative or both positive. However, we don't know what's going on with y & z therefore it cannot be determined NS Combined, We can account for w,x, & z BUT don't know what y is and bc we don't know it is therefore NOT Sufficient (E)

A foreign language club at Washington Middle School consists of n students, 2/5 of whom are boys. All of the students in the club study exactly one foreign language. 1/3 of the girls in the club study Spanish and 5/6 of the remaining girls study French. If the rest of the girls in the club study German, how many girls in the club, in terms of n, study German? 2n/5 n/3 n/5 2n/15 n/15

Remember everything in terms of N. As long as the term stay in n you'l be able to figure out the answer Boys: 2n/5 Girls: n-(2n/5) = (5n/5)-(2n/5) = (5n-2n)/5 = 3n/5 Girls studying Spanish: (3n/5) x (1/3) = 3n/15 = (n/5) Girls NOT learning Spanish: (3n/5)-(n/5) = (2n/5) Girls studying French: (2n/5) x (5/6) = (10n/30) = (n/3) Girls Studying German: (3n/5)-(n/5)-(n/3)=(9n/15)-(3n/15)-(5n/15) = n/15

If x and y are non-zero integers, and 9x^4 - 4y^4 = 3x^2 + 2y^2, which of the following could be the value of x^2 in terms of y? -4y^2/3 -2y^2 (2y^2+1)/3 2y^2 6y^2/3

Remember the important quadratic function a^2-b^2= (a+b)(a-b) 9x^4 - 4y^4 = 3x^2 + 2y^2 FACTOR LFT SIDE EXPONENT (3x^2)^2-(2y^2)^2 = 3x^2 + 2y^2 PUT LFT SIDE INTO QUAD (3x^2+2y^2)(3x^2-2y^2)= 3x^2 + 2y^2 DIVIDE BOTH SIDES BY RT SIDE ((3x^2+2y^2)(3x^2-2y^2)= 3x^2 + 2y^2 )/3x^2 + 2y^2 SIMPLIFY (3x^2-2y^2)=1 MOVE 2y^2 OVER TO THE RT 3x^2=2y^2+1 DIVIDE BOTH SIDE BY 3 x^2= (2y^2+1)/3

If a(a+2) = 24 and b(b+2)= 24, where a ≠ b, then a+b = -48 -2 2 46 48

SINCE THE EQUATIONS ARE THE SAME W/ DIFFERENT JUST FOCUS ON A a2 + 2a = 24 Distribute a2 + 2a-24=0 Move 24 to the other side (a+6)(a-4) Since equation quadratic, factor out the pairs a & b ( since formula same)= -6,4 since a ≠ b use both variables and when combined -6+4= -2

If X is the average of Y numbers, what is Y? (1) X = 54.5 (2) The sum of the Y numbers for which X is the average is 2,616.

SKIP DIRECTLY TO CE BECAUSE THEY'RE BOTH NOT SUFFICIENT ALONE (1) X = 54.5 (2) The sum of the Y numbers for which X is the average is 2,616. Together it tells us that 2,616= 54.5y

At 10:00 a.m., Peter begins traveling on a certain bike path from Riverdale at a constant rate of 10 mph. If, at 2:00 p.m., John begins traveling from Riverdale on the same path at a constant rate of 15 mph, at what time will he catch up to Peter? 6:00 p.m. 6:30 p.m. 8:00 p.m. 8:30 p.m. 10:00 p.m.

Solve Algebraically , Read Carefully, Peter = 10mph ~ 10t John= 4 hours later, 15 mph ~15(t-4) 10t = 15(t-4) DISTRIBUTE 10t = 15t-60 SIMPLIFY t= 12 After 12 hours, they'll meet so ~ 10pm

The three-digit positive integer x has the hundreds, tens, and units digits of a, b, and c, respectively. The three-digit positive integer y has the hundreds, tens, and units digits of k, l, and m, respectively. If (2^a)(3^b)(5^c) = 12(2^k)(3^l)(5^m), what is the value of x - y? 21 200 210 300 310

TO SOLVE, NEED TO UTILIZE EXPONENTS AND ALGEBRA (2^a)(3^b)(5^c) = 12(2^k)(3^l)(5^m) SIMPLIFY RT SIDE (2^a)(3^b)(5^c) = (3^1)(2^2)(2^k)(3^l)(5^m) SIMPLIFY FURTHER (2^a)(3^b)(5^c) =(2^k+2)(3^l+1)(5^m) NOW SUBTRACT SIMPLIFIED ALL THE ALGEBRAIC VALUES CANCEL AND YOU'RE LEFT WITH 210

If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}? 1,800 1,845 1,890 1,968 2,016

To answer first need to find the average of the set ((13*6)+(28*6))/2 = (78+168)/2 = 246/2= 123 Then find the number of terms in the set (28-13)+1= 16 123*16= 1968 D

In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings? 5/21 3/7 4/7 5/7 16/21

Utilize Combinatorics & Probability Formula To find the number of combinations utilize the following equation:7!/(7-2)!2! = 7!/5!2! ~ (7 x 6)/(2 x 1)=42/2 =21 Now determine the # of possible combinations >Utilize letters to distinguish the different people >4 ppl w/ 1 sibling {A,B,C,D} >3 ppl w/ 2 siblings {E,F,G} Now list the possible combinations: AB,CD, EF,EG, FG ( Since A-D can only have one sibling only one set can exist) Utilize the probability formula P(sibs) + P(not sibs) = 1 (5/21) + p(not sibs) = 1 p(not sibs)= 1-(5/21) p(not sibs)= (16/21)

5) Cars P & Q are approaching each other on the same highway. Car P is moving at 49 mph northbound and Car Q is moving at 61 mph southbound. At 2:00 pm, they are approaching each other and 120 mi apart. Eventually they pass each other. At what clock time are they moving away from each other and 45 miles apart? (A) 3:06 pm (B) 3:30 pm (C) 3:54 pm (D) 5:21 pm (E) 6:15 pm

While the questions is worded strangely need to understand whether the gap is expanding or shrinking. The question is not looking for an shrinkage but a expansion. To calculate the time lapse from 2pm First combine the two rates 69+49= 110 Then find the total distance covered: 120+45 = 165 T=165/110 = 3/2 or 1.5 hrs 1.5hrs + 2pm = 3:30pm (B)

a, b, c, and d are positive consecutive integers and a < b < c < d. If the product of b, c, and d is twice that of a, b, and c, then bc = 2 6 12 20 30

bcd = 2abc (bc)d=(bc)2a DIVIDE BC d=2a Only value that works for a 3, 6 = 2(3) a<b<c<d ~ 3<4<5<6 ~ (4x5)

What is the remainder when 25 is divided by positive integer j? (1) j is even. (2) j < 9

c (1) 25/2 = 12 R1, 25/4= 6 R1...25/14= 1 R 11 NOT SUFFICIENT (2) 25/5 = 5 R 0 NOT SUFFICIENT (1)&(2) 25/2, 25/4, 25/6, 25/8 ALL HAVE REMAINDER OF 1, SUFFICIENT COMBINED

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenue from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of Newspaper A, which of the following expresses r in terms of p ? A. 100p/125−p B. 150p/250−p C. 300p/375−p D. 400p/500−p E. 500p/625−p

d