Rework of Magoosh Math
[What is the GCF?] [CAREFUL with choices such as A] If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y? Cannot be determined 1 2 5 6
x=1, y=1 we have to check this for corner case We get 2 and 3 and the GCF is 1 -What happens if we pick 3 and 2 for c and y respectively -We get 6 and 6, and the GCF is 6 -So the answer is A
[Start with the most restrictive scenario first] In a certain sock drawer, there are 4 pairs of black socks, 3 pairs of gray socks and 2 pairs of orange socks. If socks are removed at random without replacement, what is the minimum number of socks that must be removed in order to ensure that two socks of the same color have been removed?
-We draw one sock it is one color -We draw another and it is another color -We draw another and it is yet another color -The fourth one MUST be a color that matches. This is the least we have to go to ENSURE that we get a matching pair.
If x is a positive integer and x+2 is divisible by 10, what is the remainder when x^2+4x+9 is divided by 10? 1 3 5 7 9
-Whenever you see a quadratic, attempt to break it and see if any more patterns emerge. -So... x^2+4x+9 = x^2+4x+4+5 = (x+2)(x+2)+5 -We already know that (x+2) is divisible by 10 so the remaineder must be 5
[Write down what you need to solve for OR Assign it Variables] For a certain event, 148 people attended. If all 148 had paid full admission price, the total revenue would be three times the cost of sponsoring the event. (Admission price was the only source of revenue.) As it happens, only 50 paid the full admission price, and the others paid nothing. Column A the total revenue Column B the cost of sponsoring the event
Column A: R=revenue Column B: C=cost of sponsoring the event p: p=price of a single ticket -So, 148p=3C -Actual Revenue: R=50p Now we have... A: 50p = R B: 148p/3 = C Which is bigger? Clearly A is
Set A: {1, 3, 4, 6, 9, 12, 15} If three numbers are randomly selected from set A without replacement, what is the probability that the sum of the three numbers is divisible by 3? 3/14 2/7 9/14 5/7 11/14
-First thing to ALWAYS ask is: how many stages are there? -Second thing to ALWAYS ask is: does order matter? -3 stages -No, so it is a combination -Remember also the integer properties of divisibility. 3 and any of its multiples added together will give a number still divisible by 3. 5c3/7c3
[The wording "possible" implies all that are UNIQUE] The diagonal of a polygon is a line segment from any vertex to any non-adjacent vertex. The diagram at right shows a regular decagon, a 10-sided polygon, with two diagonals drawn. How many possible diagonals does the regular decagon have?
-Here our equations looks like: (10*7)/2=35
[Work with variables if the comparison involves the same thing!] Column A The perimeter of a rectangle Column B Twice the length of the diagonal of the same rectangle
-Here we should denote x and y as the sides of the rectangle -That means that the perimeter is 2x+2y -The other comparison is twice the diagonal with will be 2sqrt[(x^2)+(y^2)] -We can factor a 2 and square both sides -Our comparison should be simple now
In a population of chickens, the average (arithmetic mean) weight is 6.3 pounds, and the standard deviation is 1.2 pounds. Which of the following weights (in pounds) are within 1.5 units of standard deviation of the mean? 4.4 4.6 5.1 5.2 6.9 7.6 7.7 8.2
-How std works: -1 std unit is 1.2lbs so how much would be 1.5 units? -1.5 = 1 + 0.5. 1= 1.2 so half is 0.6 so 1.5 would be 1.2+0.6=1.8 -Rest is easy to solve
[Combinations vs. Old-fashioned way] Sid intended to type a seven-digit number, but the two 3's he meant to type did not appear. What appeared instead was the five-digit number 52115. How many different seven-digit numbers could Sid have meant to type? 10 16 21 24 27
-I did this the old-fashioned way of if the 3's are next to each other then we have 6 options+5+4+3+2+1 -Another way is that we have to pick 2 slots out of 7. That is 7c2
[FOR SOME BIZARRE reason I want to believe that 3+3 is 9 which is EFFED UP. IT IS 6] The average (arithmetic mean) of 3, 3, 5, 6 and x is 2. Column A x Column B -8
-IF you do the math correctly the answer is A
If y - 3x > 12 and x - y > 38, which of the following are possible values of x? -60 -30 -6 4 20 40 80
-If the inequalities are in the same direction, you can ADD them. -If they are in opposite directions, you can SUBTRACT them.
[WITHOUT Replacement] If four numbers are randomly selected without replacement from set {1, 2, 3, 4}, what is the probability that the four numbers are selected in ascending order?
-Not tough to solve but keep in mind that without replacement and how it works: 1: 1/4 2: 1/3 (since we 1 is gone from the group) 3: 1/2 4: 1
[HALF-and-HALF with numbers] A number, x, is randomly selected from the integers from 42 to 92 inclusive. Column A The probability that x is odd. Column B The probability that x is even.
-Note that numbers are usually divided half and half between even and odd -Inclusively we have 51 numbers meaning that we have more of one -Since we start with an even and end with an even the answer must be more even
[Don't FORGET to check 1.] How many positive integers less than 10,000 are such that the product of their digits is 210? 24 30 48 54 72
-Notice here that 1__,__,__,__ we have 2 spots since the factors of 210 are 2,3,5,7 -So once scenario is that we have 4! choices to arrange these -Another scenario is the 6,5,7 which gives us 3! -BUT THERE IS ONE MORE SCENARIO since we have FOUR SPOTS: 1,6,5,7 which gives us another 4! -This equals 54
The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k? A) 1 B) 13 C) 14 D) 18 E) 21
-Notice here that 4/5=(a+k)/(b+k). -The total parts are 9. a+b+2k=117. -117/9=13. So each part if worth 13. -So... 4/5=52/65 -Then... (52-k)/(63-k)=3/4 -k=13 ---- -On the other hand, don't create more variable than you need to: -Say that x=a+k and 117-x=b+k... -Then... x/(117-x)=4/5
[ALWAYS start with the most restrictive scenario FIRST] In the below addition A, B, C, D, E, F, and G represent the digits 0, 1, 2, 3, 4, 5 and 6. If each variable has a different value, and E ≠ 0, then G equals? __AB +CD EFG
-Notice that E has to be 1 since this is a 3 digit number but the highest any two numbers can add to is 11 -Knowing that, A&C can either be 6&5 or 6&4. The latter has to be the case because the former gives 11 and that would give us two 1s which is not possible. Now the only numbers left are 2,3,5. So G has to be 5.
[OPTIONS for STAGES.] How many three-digit numbers are there such that all three digits are different and the first digit is not zero? 504 648 720 729 810
-Notice that the first stage CANNOT be 0, so we have 9 choices -The second choice can be anything and since we have 9 numbers left, it is 9 -The last stage can be the remaining 8 numbers (9)(9)(8)=(81)(8)=648
[Counting starting with 0 OR 1] [DO NOT assume. WRITE out equations.] Column A Average (arithmetic mean) of integers from -50 to -1 inclusive. Column B Average (arithmetic mean) of integers from -50 to 0 inclusive.
-Notice that we can either start counting from 0 or 1. The final result is the same. -Sum of 0-9 inclusive = (10)(9)/2 = 45 -Sum of 1-9 inclusive = (9)(10)/2 = 45
[ABSOLUTE VALUES CANNOT equal less than 0] What is the sum of all possible solutions of the equation |x + 4|^2 - 10|x + 4| = 24? -16 -14 -12 -8 -6
-Notice that when we get the quadratic we have: |x+4| = 12 |x+4| = -2 -The second CANNOT be a solution. Since an absolute value CANNOT equal a negative!
In a group of 45 children, 60 percent of the children are boys, and 60 percent of the children are left-handed. Column A Number of boys who are left-handed Column B 8
-Once you've made a matrix, start with the information you are given, which is 8. -Then see what the logic is and move to another number.
[DON'T forget to try out a few numbers.] Triangle ABC has sides x, sqrt(x), and x^2, where x is an integer. What is the area of ABC?
-One might conclude that this cannot be determined since x can be anything but before you jump to that conclusion at least try two values... 1 and 2 -With 1 we realize this has to be an equilateral since any number above that violates the "two sides have to be greater than the third side"
[How coordinate equations work] Which of the following lines intersects the vertical line x = 3 between (3, 1) and (3, 2)?
-Realize that lines take in x and give out y -A line that intersects at or between certain points means that a POINT LIES BETWEEN THAT REGION -For this the x is circumscribed to be x=3. We want y's that are between 1 & 2 when we plug in 3
Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove the same route back from Villageton to Townville at an average speed of 60 miles per hour. Column A 50 Column B The average speed of Andy's entire trip in miles per hour.
-Remember if the distance is the same then the total distance is 2D. -Also you can make up a number for D. Use that if nothing else works.
[Putting things in relations to the SMALLEST object.] Cam is 20 percent taller than Bea, and Bea is 20 percent taller than Ann. Column A Cam's height minus Bea's height. Column B Bea's height minus Ann's height.
-Since A is the thing that relates everything and is really the "LCD" let's leave A as is -Then B=1.2A -So, C=1.2B=(1.2)(1.2A)=1.44A -Now we can compare 1.44A-1.2A=.24A -1.2A-A=.2A
[Area of a Triangle] Point A (-4, 2) and Point B (2, 4) lie in the xy-coordinate plane. If point C lies in the first quadrant and contains the coordinates (p, q), where p < 2 and q < 4, which of the following could be the area of triangle ABC? 1.1 3.9 11.9
-The lowest the area of the triangle regardless of the size can be 0+. -In this case the max can be 12 and the min can be 0+. Using that what's possible?
[DISTANCE and TIME] David drove to work at an average (arithmetic mean) speed of 45 miles per hour. After work, David drove home at an average speed of 60 miles per hour. If David spent a total of 2 hours commuting to and from work, how many miles does David drive to work? 48 256/5 360/7 105/2 160/3
-There are may ways to do this but realize that we are given the time of 2hours -And we can manipulate D=rt to get t since we are given the rate and D is the same -So... D/45+D/60=2
[How to write equations] For the first 5 hours of a trip, a plane averaged 120 kilometers per hour. For the remainder of the trip, the plane travelled an average speed of 180 kilometers per hour. If the average speed for the entire trip was 170 kilometers per hour, how many hours long was the entire trip? 15 20 25 30 35
-This is interesting -- the top is always going to be (time)(rate) -The bottom is always just going to be time -The way to write this is easy: (120)(5)(180)(x)/(5+x)=170
[PROPORTIONAL reasoning with the formulas you know] If the length of each side of an equilateral triangle were increased by 50 percent, what would be the percent increase in the area? 75% 100% 125% 150% 225%
-This is really simple if you know the formula of an equilateral and just use proportional reasoning
[Double MATRIX vs. Venn-Diagram] In a group of 50 students, 31 are taking French, 17 are taking Spanish, and 10 are taking neither French nor Spanish. How many students are taking both French and Spanish? 4 8 12 14 16
-We are dealing with two categories French and Not French [make's up one side of the box] AND Spanish and Not Spanish -Create a matrix to solve now
If x and y are integers, and w=(x^2)y+x+3y, which of the following statements must be true? If w is even, then x must be even. If x is odd, then w must be odd. If y is odd, then w must be odd. If w is odd, then y must be odd.
-We are essentially testing four scenarios: two stages composed of x and y either of which can be even or odd. -EE -EO -OE -OO -For even pick 0 -For odd pick 1 -Only the first three are correct
[QUESTIONS that work with % that don't need complicated solutions.] At the moment there are 54,210 tagged birds in a certain wildlife refuge. If exactly 20 percent of all birds in the refuge are tagged, what percent of the untagged birds must be tagged so that half of all birds in the refuge are tagged? 25 30 33 1/3 37 1/2 50
-We are going to get a percent, which is a ratio. This means we don't need to work with big numbers. As long as we maintain the ratio the answer will be correct regardless so let's work with 10 -20% or 2 of the birds are tagged -To get 50% or 5 we need 3 extra birds -These 3 birds are 3/8 = .375 = 37.5% of the population that isn't tagged
[I DIDN'T miss this again and paid attention to what was being asked!] Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d - b = 3, then what is the greatest possible range of the five numbers? 12 17 18 19 20
-We are looking for the GREATEST range. Good idea is to make an equation and circle it and come back to it -Lowest value is 1 -Highest value has to be 19 -Range is 18
[DIVISIBILITY] The Sargon Corporation, which employs both men and women, offers an optional stock-option buy-in program to its employees. If 85% of the men and 77% of the women choose to participate in this plan, then which of the following could be the total number of employees? Indicate all possible values for the number of employees. 100 200 350 460 525 640 750 880
-We can conclude that the number of women has to be divisible by 100. Since we have 77/100=77%. -We can conclude that the number of men has to be divisible by 85/100=17/20 "20". -Notice that 100 cannot work. Because we can't have a multiple of 100 and add a multiple of 20 to get something within the range of 0-100 unless one of the groups is 0, which is not the case since we are told exact percentages. -So 200, 460, 640, 880 work since we can add multiples of 100 and 20 together to get these numbers.
[USING Logic with GEOMETRY] Column A The perimeter of a rectangle Column B Twice the length of the diagonal of the same rectangle
-We can do this mathematically -But think about this. The diagonal is going to be shorter than the sum of the sides. It has to be, due to the triangle rules! -This means that the double is diagonal is going to be shorter than the four sides combined
[OUR Q's are Different] When positive integer N is divided by 167, the remainder is 35, and when positive integer K is divided by 167, the remainder is 17. What is the remainder when 2N+K is divided by 167?
-We can't use "creating the dividend" since our Q's are different -So let's just use proxy's N could be 35 and K could be 17 -Solve the problem now
[REMEMBER the equation R = n! - Q] In how many ways can Ann, Bob, Chuck, Don and Ed be seated in a row such that Ann and Bob are not seated next to each other? 24 48 56 72 96
-We know that n! is equal to 5! which is 120 -Q would be: 4*2=8 ways in which AB are next to each other (2 is for the four spots but we can reverse their order); multiple is by the rest of the group 3! = 48 120-48=72 ways is equal to R
[Be VERY careful reading the wording of the question] [What we know for certainty in probability is equal to 1] In basketball, when Monica takes her first free throw, she has a 50% chance of scoring. If she takes an additional throw, she has a 75% chance of scoring on this throw if she scored on the immediately previous throw, but only a 25% chance of scoring on this throw if she didn't score on the immediately previous throw. Suppose she has to take a set of three free throws in a row, and [she doesn't score on the first one]. What is the probability that she scores at least once on one of the subsequent two throws?
-We know that the best way to solve this is by the complement rule: P(at least 1) = 1 - P(0) -The thing to note is that we are told that she misses the first one. So there is no way that she can make the first one so she only has two chances to make it now. So we can disregard the first probability since we know it for certainty to be a miss. -The equation then becomes 1-[(3/4)(3/4)]=7/16
Half of w is x Half of y is w [w + x + y = 28] Column A w Column B 7
1) Always write down the equations neatly but more importantly write down the [ultimate equation] and circle it. 2) Since we are looking for w, try to put everything in terms of w. That's really what the test is looking for. Ans. A (8)
pqr<0 [(pq)^2]/r<0 Column A p*q Column B 0
1) Look at the equations one at a time closely and see what the possibilities can be: -If pqr<0 then either all the numbers are negative or only one is negative -Based off the second equation go back to the first to figure out our answer which is A
In Ophiuchus Corporation, 60% of the total revenue R is devoted to the advertising budget. Five-sixths of this advertising budget was spent on television advertising. Which of the following represents the dollar amount spent on television advertising? A) R/2 B) R/3 C) 2*R/3 D) 2*R/5 E) 4*R/5
1) Write down equations: 3/5R=A 5/6A=T 2) Can these equations be manipulated to be put into one equation? Notice that we have the entire value of A, which we need since 5/6 of that value is the Television advertising. (5/6)(3/5R)=T 3) If finding equation is taking too long use a number. I picked 300 for R.
[Tough Question] For how many unique coordinate points (P, Q), such that P and Q are integers, is it true that P^2 - Q^2 = 1155? 16 24 32 48 64
1155 = 11*105 = 3*5*7*11 P2 - Q2 = (P - Q)*(P + Q) The first factor (P + Q) could equal any one of those factors, or any combination, and the other factor (P - Q) would equal the remaining factors. Each pairing would result in different values for P & Q. First treat positive values. (P + Q) could equal a) 1 b) any of the four individual factors (4) c) any pair of factors (4C2 = 6) d) any three factors (4 possibilities) e) all four factors That's 1 + 4 + 6 + 4 + 1 = 16 different combinations. Thus, there are 16 different positive values for (P + Q) that would lead to different possibilities for P & Q. For every positive value of both (P + Q) and {P - Q), there would also be a possibility of two negative values. That's 16 more possibilities for P & Q. A total of 32 values.
[We start with 5c0] Car X can come with any of these 5 additional features: sunroof, stereo, tinted windows, leather seats and cruise control. Column A Number of different combinations possible Column B 25
5c0=1 5c1=5 5c2=10 5c3=10 5c4=5 5c5=1 -This equals 32 (we add the different scenarios together) -Another way to do it is via the __,__,__,__,__. Where each stage has 2 options (yes or no)
[MULTIPLE Scenario Question] A box contains 4 red chips and 2 blue chips. If two chips are selected at random without replacement, what is the probability that the chips are different colors? 1/2 8/15 7/12 2/3 7/10
Scenario Blue Chips: (2/6)(4/5)=4/15 Scenario Red Chips: (4/6)(2/5)=4/15 8/15
[How to write equations.] A helicopter company charges $85 for the first kilometer of a trip and $5 for every kilometer after that. If the total cost of a trip was $365, how many kilometers were flown?
T = 85 + 5(x-1)
[Setting things equal to TIME and adding things that have the SAME denominator -- including variables.] Every day at noon, a bus leaves for Townville and travels at a speed of x kilometers per hour. Today, the bus left 30 minutes late. If the driver drives 7/6 times as fast as usual, she will arrive in Townville at the regular time. If the distance to Townville is 280 kilometers, what is the value of x?
D=xt D=xT => T = 280/x Also we should get T=[280/(7/6x)]+1/2 This works out to be: 280/x=(240/x) + 1/2 -Notice that the denominators are the same (we can't cross multiply since x could be 0): 280/x - 240/x = 40/x = 1/2 -Taking the reciprocal we get x/40=2 -So x=80
Let N = 60! + 55! + 50! The unit digit of N and a number of digits to the left of the units digits are consecutive zeros before we come to the first non-zero digit. How many such consecutive zeros are there until the first non-zero digit?
We are concerned with the consecutive zeros between the decimal point and the first non-zero digit to the right in a number. If we add three numbers with three different strings of consecutive zeros, the one with the fewest consecutive zeroes would be the limiting factor. Here, this would be the smallest factorial. However many consecutive zeros 50! has, that will be the number of consecutive zeros that N has. 50! has factors of 5 and 2. Each pair, one factor of 5 and one factor of 2, equals a factor of 10 that would product one zero in the sequence. It has many more factors of 2 than factors of 5, so the latter will be the limiting agent in determining the number of factors of 10, and hence the number of consecutive zeros. Each multiple of 5 has a factor of 5, so there are ten of these from 5 to 50. In addition, both 25 and 50 have an extra factor of 5, because they are both divisible by 52 = 25. altogether, that's twelve factors of 5. This means that 50! has 12 consecutive zeros before the first non-zero decimal. In consequence, N also has 12 consecutive zeros before the first non-zero decimal. Answer = 12
[Treating a group as one stage] In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?
[__,__,__][,__,__,__] -There are 2 ways that these can be arranges -Each of the stages within can be arranged in (3)(2)(1)=6 ways -So we get (2)(6)(6)=72
[Think about this LOGICALLY and SCENARIOS.] In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?
-How many scenarios can we have with the Red shirts? R,R,R,__,__,__ = 4 when they are next to each other. So we can wind this down to 3, 2, 1. For a total of 10. -Since we have to do the same for Green we have 20
[Upper vs. lower LIMITS] [Get to a conclusion from the second, then go to first, then to go second] x^2 - y^2 < 8 x + y > 3 If x and y are integers in the above inequalities and 0 < y < x, what is the greatest possible value of x?
-Notice that the second equation tells us the lower limit of 3 -The second equation tells us the upper limit of 8 -Since we are dealing with integers, the largest the number can be is 7. How many ways can we get 7? 1+6, 2+5, 3+4 -Using these combinations what's possible? -Only the last one works
[How RATIOS work.] In 2006, the ratio of the number of widgets sold by Company C, Company E (not shown) and Company D was 5 to 8 to 2, respectively. How many widgets did Company E sell in 2006? D=1.2 300,000 600,000 2,400,000 4,800,000 6,000,000
-We know the given information and need to find an unknown E/D=4/1=x/1.2 -We get 4.8
[When the FIRST PICK is 10/10] A box contains 10 balls numbered from 1 to 10 inclusive. If Ann removes a ball at random and replaces it, and then Jane removes a ball at random, what is the probability that both women removed the same ball?
-Careful about this, although I got it right, it is a bit tricky -The first pick can be any of the 10 balls so it is 10/10 -The second pick is now dependent on the first pick and must be that first pick, so it is 1/10
[Follow the LOGIC and DO NOT doubt it.] Three friends are buying a gift for a friend. Declan contributes 4 dollars more than 1/4 the cost of the gift, Ed contributes 1 dollar less than 1/3 the cost of the gift, and Frank contributes the remaining 22 dollars. What is the cost of the gift? 48 54 60 66 72
-Declan: (G/4)+4 = D -Ed: (G/3)-1 = E -If Frank contributed the rest (22), then G-D-E=22 -Another way to see this is D+E+22=G -Solve for this to get 60
[Don't make silly mistakes. FOR problems like these WORK OUT slowly and METHODICALLY. It should be straight in the bag.] a + (b + c) = (a + b) + c a - (b - c) = (a - b) - c a*(b*c) = (a*b)* c a/(b/c) = (a/b)/c
-Here B doesn't work -The last one also doesn't work
[When you can't immediately make sense. Write down the equations neatly.] Anne pays 150 percent more for a wholesale widget than Bart pays. Anne's retail price per widget is 15 percent greater than the wholesale price she paid. Bart's retail price per widget is 185 percent greater than the wholesale price he paid. Column A Anne's retail price per widget. Column B Bart's retail price per widget.
A=2.5B a=1.15(2.5B) b=2.85B
[These kinds of things require you to look at patterns that repeat beforehand.] In a certain sequence of all positive terms, {a1, a2, a3, ...} each term equals the previous term times a constant factor. If (a1)(a5) = 900, what is the value of a3?
a1 a2=a1c a3=a1cc a4=a1ccc a5=a1cccc -So... (a1)(a1cccc)=(a1^2)(c^4)=900 -If we square both sides we get (a1)(c^2)=30=a3
