GRE math
What is the largest 4 digit multiple of 71?
The largest 4-digit number is 9,999 When 9,999 is divided by 71, the result is 140 with a remainder of 59. That means 9,999 is 59 greater then the nearest multiples of 71. Another way to approach this, after doing the division is to recognize that 71 x 141 would produce a number larger than 9,999 so the largest 4-digit multiple of 71 would be 71 x 140.
How many different appearing arrangements can be created using all the letters AAABBC
This question represents a pattern of a counting problem with certain conditions or restrictions added. There are 6! ways to arrange 6 different items. However, in this case many of those arrangements will appear identical. Consider the configuration in the question AAABBC. If the A's were not identical there would be 3!= 6 different appearing ways to arrange them. However since the A's are identical
What is relative frequency?
is the number of values with the characteristic interest expressed as a percent of the total number of values.
How many positive facts of 54 are odd?
the positive factors of 54 are 1,2,3,6,9,18,27 and 54. Of those four (1, 3, 9 and 27) are odd
If A:B is 3:7, C:D is 15:11 and B:C is 14:5, what is A:D?
Alternatively "chain" the ratios to find the answer. Looking at just the variables rather than the numbers, note that (A/B) x (B/C) x (C/D) = (A/D) because the Bs and Cs cancel out. Plug the values of the variables into this equation: (3/7) x (14/5) x (15/11) = (18/11)
The mean of a group of 6 numbers is 15. If the values of one-third of the numbers are increased by 12 each, what is the new mean?
An average is the sum of all the values in a group divided by the number of values. In this question, the average is given but the sum is not. Rearrange the formula for averages: sum= number of values x average, which 6 x 15= 90. Since there are 6 numbers in the group, one- third of that is 2. If those 2 numbers are increased by 12 each then the sum is increased by 24. Divide the new sum of 90 + 24= 114 by 6 to obtain the new average of 19
What is the largest prime factor of 46,000
Any multiple of 10 will have prime factors of 5 x 2. In this case, 46000= 46 x 10x 10 x 10= 23 x 2 x 5 x 2 x 5 x 2 x 5 x 2. The number 23 is prime and cannot be broken down further
The ratio of red to blue to black pens in a box 3:5:7. If all 75 pens in the box are one of these colors, how many are red?
Convert the part-to-part ratio to the part-to-part whole ratio for red pens: (3/ 3+5+7)= (3/15)=(1/5) set up the proportion (R/75)=(1/5) and cross multiply to get 5R=75. R=15.
What is the range of {-3, 0, 2, x, 6, y, 3} if x^2 + 8x= -16 and y^2 - y= 6?
In order to determine the range of the group of numbers, the values of x and y are needed. Rearrange the equation for x to: x^2 +8x + 16 =0. The factors to (x+4)^2= 0, so the only value for x is -4. At this point the smallest known value in the group is -4, and the greatest is 6. Set up the equation for y in standard quadratic format: y^2-y -6 =0. This factors out to (y-3)(y+2), so y can either be 3 or -2. Since both of these values are inside the range of values already known, they have no effect on the overall range of the group which is 6- (-4)= 10
Rafael has three more bus tokens than Tax. Chan has twice as many as Rafael. If Chan had 1 more token, he would have 3 times as many as Taz. How many totals tokens does the group have?
Represent the three peoples number of tokens by the first letters of their names and translate the information given into algebraic equations: R=T +3, C=2R and C= 3t -1 Since there are three distinct linear equations and three variables, you can solve this system of equations. Substitute 2R for C in the third equation: 2R=3T -1. Double the first equation to get 2R= 2T+6. This gives two different equations for 2R, so set them equal to each other: 2T + 6=3T -1. Subtract 2T from both sides and add 1 to both sides to get T=7. This means that R has 10 tokens, 3more than T and C has twice as many as R, which is 20. The total number of tokens among the three people is 7+ 10+ 20= 37
Paula has 10 books that she'd like to read on vacation, but she only has space for 3 books in her suitcase. How many different groups of 3 books can Paula pack?
Since the books are just being put in a suitcase, order doesn't matter, and the combinations formula can be used C= (10!/ 3!(10-3))
Working together, Kendra, Latasha and Melanie can complete a certain task in 4 hours. If Kendra alone could complete the task in 8 hours and Latasha could complete the task in half the time it would take Melanie, how long would it take Latasha to complete the task by herself?
Since there are 3 workers in this question, use the formula for adding rates of multiple workers: (1/T)= (1/K)+(1/L)+(1/M). the question states that the three women could compete the task together in 4 hours, and also provides the information that K=8. It then states that Latasha can complete the task in half the time Melanie can. Set M=2L and plug the values into the equation to get (1/4)= (1/8)+(1/L)+(1/2L). Multiple each term by the least common multiple of the denominators, 8L, to clear the fractions: which simplifies to 2L=L+8+4, so L=12
If the average of 6, 3, -2, 5,11 and x is 5, what is the value of x?
The formula for computing averages is average= (sum of values/ number of values). Rearrange this to: sum of values= number of values x average. Including x, there are 6 values so 6+3-2+5+11+x=6 x . Thus 23+x=3- and x=7.
A and B are overlapping sets. If |A| has 7 elements, |B| has 5 and | A^ B | has 3 elements, how many elements are in | A u B |?
The formula is based on the inclusion-exclusion principle for sets states that |A u B|= |A| + |B| - |A ^ B|. Substitute the numbers given in the question: |A u B|= 7+ 5 -3 =9
Pablo is allowed to choose 1 of 3 different fruit beverages and 2 of 4 different healthy grain bars for his afternoon snack. How many different combinations does he have from which to choose?
The number of options Pablo has for the beverage is simply 3, because he can only select one item of the three that are available to him. To calculate the number of options for the grain bars, use the combination formula because the order does not matter and you get 6. Since pablo gets to choose a beverage and two grain bars, and for each of the 3 beverages he can choose from 6 different options o grain bars, multiply the two numbers: 3 x 6 = 18
In a bag of coins, (2/5) are pennies, (1/3) are pickles, (1/4) are dimes, and there are 5 quarters. If there are no other coins in the bag, what is the total number of coins?
The question could also be approached using algebra, by setting up the equation C= (2/5)C+(1/3)C+(1/4)C+5 where C represents the total number of coins. Multiply through by the LCM of 60 to get 60C=24C+20C+15C+300. Combine like terms: 60C= 59C+300, so C=300
what is the probability of rolling a 6 on two consecutive rolls of a fair six-sided die?
There are 6 equally likely outcomes for one roll of a fair die. One of these outcomes is 6, so the probability of rolling a 6 is (1/6). The question asks for the probability of rolling a 6 on the first and the second roll. These events are independent, so multiple (1/6) x ( 1/6) = (1/36)
What is the probability of the result of 4 independent coin flips being exactly 1 head and 3 tails?
There are two possible outcomes for each flip of the coin: heads or tails. You might be tempted to think that the total number of possible outcomes for 4 consecutive flips would be 4 x 2= 8, but remember that the coin is flipped once and then a second time and then a third time and then a fourth time, so the total number of possible outcomes is actually 2^4=16. If only one head is the result that could occur on any one of the 4 flips, so there are 4 desired outcomes the probability is thus (4/16)= (1/4)
A bag contains only red and blue plastic chips. There were 10 chips in the bag and 1 blue chip was removed. The probability of drawing a blue chip was then (1/3). How many red chips were in the bag?
After 1 blue chip was removed, there were 9 chips left. If the probability of drawing another blue chip from the remaining 9 was (1/3), there must have been (1/3) x 9= 3 blue chips, and 9-3 = 6 red chips.
The ratio of x to y is 1:4. If the value of x were increased by 1 without changing the value of y, the ratio of x to y would become 1:3. If z= x+ 2y, what is the value of z?
Restate the ratios as fractions: Initially (x/y)=(1/4) so y=4x. If x were increased by 1, the new ratio would be (x+1/y)=(1/3).Cross multiply to get y=3x+3. Set the two values of y equal to each other: 4x=3x+3 which simplifies to x=3. Since y=4x, y=4(3)=12. Because the question states the increase of the x as a hypothetical statement, use the initial value for x to calculate z: 3+2(12)=27
Abdul recently made 200 mile trip. For the first 30 miles, he traveled at an average speed of 45 miles per hour. His average speed for the next 50 miles was 60 mph. Abdul averaged 50 mph for the final portion of his trip. How long did it take Abdul to complete his journey?
Use the Time-Speed-Distances formula in the format T=(D/S) to determine the time for each leg of abduls trip. For the first part T=(30/45)=(2/3). For the second leg, T=(50/60)=(5/6). The distance for the third part of the trip is not given, but can be calculated by subtracting the two known distances from the total: D=200-30-50= 120. Apply the equation for time: T= (120/50)= 2(2/5) Rather then trying to work with a common denominator to add the times, convert to minutes which gets you 40+50+144=234. Divide 234 minutes by 60 minutes in an hour to get 3 hours with 54 minutes remaining.
If A,B and C are positive integers and A and C are odd what is the smallest possible value of b given a x b x is even?
When multiplying integers at least one integer must be even to get an even result. If a and c are odd then b must be even for a x b x c to be even. That means the smallest value of b, which must be positive is 2