GRE math
The product of the greatest prime factor of 88 and the smallest prime factor of 117 is 6 33 99 104 858
33 Factor each number down to its component primes. For example, 88 = 2(44) = 2(4)(11) = 2(2)(2)(11) = 23(11). Similarly, 117 = 3(39) = 3(3)(13) = 32(13). So, 11(3) = 33.
In a certain geometric sequence, the first five terms are m, n, o, p, and q. If m = ½ and o = 18, the fifth term in the series is** 6 9 18 108 648
648 In a geometric sequence, each term is the previous one multiplied by a constant. The constant (let's use k) can also be expressed as any term divided by the previous term; in this case, k=n/m = 2n. So o = kn = 2n*n= 18, so n = √18/√2 =3. Now we can say k=6, so o = 18. So p = ko = 6(18) = 108, and q = kp = 6(108) = 648
y = -2x + 4 Column A: the area in square units bounded by the coordinate axes and the line Column B: 8
The quantity in Column B is greater. Sketch a quick graph of the line; make it easy by placing points where x = 0 and y = 0. This makes a triangle with base 2 units and height 4 units, so the area = 1/2bh = 4.
−7x − 2y = −13, and x − 2y = 11. The variable y = -3 -4 3 4 7
-4 This is a system of equations in two variables. Subtract the second equation from the first to get x = 3. Be sure to track the negatives! Then 4x = 12. Use the second equation to find y=(11-x)/-2 = (11-3)/-2 =-4. Again, it is easy to lose track of the negatives.
Point A on the coordinate plane is 24 units from point B and 48 units from point C. Set N= {all possible distances from B to C}. The range of set N =** 12 24 32 48 72
48 The range of a set is its maximum value minus its minimum value. The maximum and minimum values for the distances occur when points A, B, and C lie on a line. If A is the middle point, then the distance between B and C is 48+24=72. If B is in the middle, then the distance between B and C is 48-24=24. So the range is 72-24=48.
A 6-foot tall man observed that his shadow due to the sun is longer than his height. At that time of the day, the sun is up 45 degrees above the horizon. What is the length of the man's shadow? 12 √3 feet 6 √2 feet 3 √2 feet 3 feet 6 feet
6 feet This problem is equivalent to a right triangle with the height and length of shadow as legs and 30 degrees as one of the angles. This forms a 45-45-90 triangle which is a special triangle. In this triangle both legs have the same length. Thus the length of the shadow is also 6 feet.
Joe sells newspapers. On Monday he sold 15 newspapers, on Tuesday 20, and on Wednesday 25. To the nearest one percent, what was the percentage increase in his sales from Monday to Wednesday?*** 10% 17% 25% 67% 167%
67% The formula to use for percent of change is the 100(new value-old value)/(old value). In this case, the old value is Monday's 15 papers. The new value is Wednesday's 25. The formula is 100(25-15)/ (15) = 66 2/3% = 67%.
30 liters of a solution of water and acid contains 15% acid. If more water is added to change the solution to 5% acid, which of the following represents the number of liters of water in the final solution? 30 57 60 85.5 90
85.5 The equation of the mixture problem is 30(0.15) + x(0) = (30+x)(0.05). Multiplying the right hand side through and rearranging gives 0.05x + 1.5. Solve for x, the amount of water added, you get x=60, a number which awaits the unwary in choice C. But the question asks how much water is in the final solution, which is 95% of the 30 original liters plus the 60 liters just added.
John, David, and Andrew live in the same street. The distance from David's house to John's is twice that of David's to Andrew's. If the distance from John's to Andrew's is 300 feet, what is the distance between David's house to John's? Assume that their houses are aligned. 200 150 600 Cannot be determined from the given information. 100
Cannot be determined from the given information. We don't know the arrangement of their houses. There are three ways to arrange them which means there are virtually three solutions. Thus, we can't uniquely determine the distance between David's house and John's.
(s/t) < -1, which of the following CANNOT be true? s> 1 & t<-1 s>0 & t< 0 s>1 & t>-1 s<-1 & t<0 s=-2 & t= 1/2
s<-1 & t<0 This means both s and t are negative which results to s/t to be positive.
A certain triangle has angles such that the ratio of the angles A:B:C = 1:3:5. Four times the measure of angle B is 12 degrees 36 degrees 80 degrees 240 degrees 400 degrees
240 The angles of any triangle sum to 180. For this triangle, the 180 degrees are apportioned in the ratio 1:3:5, which means there are 9 "parts" total. Dividing 180 by 9 means that there are 20 degrees in each part. Angle A gets one "part", or 20 degrees, B 3 "parts" or 60 degrees, and C 5 "parts", or 100 degrees. Four times the measure of angle B is 4 * 60 =240.
12r2 - 51r + 45 = 3(4r -5)(r+3) 3(4r+5)(r-3) 3(4r+3)(r-5) 3(4r-5)(r-3) 3r2 - 15
3(4r-5)(r-3) This looks gnarly, but use your test-taking skills before your algebra skills - glance ahead at the answer choices before you factor. You should be able to eliminate choice E, since it has no "r" term. The other four choices all pull the factor 3 out, so that gives you a head start; rewrite the equation as 3(4r2 - 17r + 15). You will be using FOIL (Firsts, Outsides, Insides, Lasts) to factor further. If you think of it, a quick survey of the remaining choices shows that choice D is the only one whose "L" calculation could result in +15; pick it and you're done. If you miss this, continue by noting that the remaining choices all have 4r and r as the first, or "F", terms in the parentheses, and that the second choices are all +/- 3 and +/ 5, more useful clues. Set up your parentheses with what you already know: 3(4r __ __)(r___ ___). Now look for the combination of "O" and "I" pairs that add to -17r. This is only true for choice D: (4r)(-3) + (-5)(r) = -12r - 5r = -17r.
An electronics store had a 10% off sale. Items that did not sell were marked down another 10%, and then what was left was marked down another 15% for clearance. A certain item's original price before the sales was $150. What was its clearance price? $52.50 $97.50 $103.278 $103.27 $103.28
$103.28 Do not add up the percentages directly; each sale represented a markdown from the previous price. For three successive sales of 10%, 10%, and 15%, the final price of the item is (original price) (90%)(90%)(85%), or 150(0.9)(0.9)(0.85) = $103.278 or $103.28 (remember to round properly).
Ian regularly empties his pockets of coins into a jar. One day, he counts the money and finds that he has $22.50 in nickels (worth 5 cents each), dimes (each worth 10 cents), and quarters (each worth 25 cents.) He has 3 times as many nickels as dimes, and 6 more quarters than dimes. How much money does he have in quarters? $11.25 $12.00 $15.50 $19.75 $20.50
$12.00 Let n = # of nickels, d = # of dimes, and q = # of quarters. Then, n = 3d and q = d+6. Each nickel is worth 0.05 dollars, each dime 0.10, and each quarter 0.25, and it all has to add up to $22.50. So (0.05) n + (0.10)d + (0.25)q = 22.50. Multiply through by 100 to clear the decimals, to get 5n + 10d + 25q = 2250. Now, put everything in terms of d: 5(3d) + 10d + 25(d+6) = 2250. Multiply through: 15d + 10d + 25d +150 = 2250. Combine like terms and solve for d: 50d =2100, so d = 42. Then n = 3d = 126 and q = 48. The value of 48 quarters is 48(0.25) = $12.00.
A triangle ABC drawn on the coordinate plane has one side determined by points A = (1,3) and B = (5,6). The slope of the line BC is -3, while the slope of the line AC = 0. The coordinates of point C are (4,6) (1,18) (3,7) (6,3) (1,6)
(6,3) Draw yourself a sketch, including reasonably neat grid lines. Place and connect points A and B. Next, draw a line with slope -3 through point B (slope = rise over run; so for a slope of -3, for every one unit of x to the right, y goes down 3.) Now draw line AC with 0 slope (straight across). They intersect at point C, (6,3). If you thought a slope of 0 means a vertical line up from point A, erroneous choice B awaits you.
If y = 29/31 and x = 43, then (y0)(y1)(y2)(x) / xy3 = 0 1 x 1/y 29/31
1 Anything taken to the 0 power is 1. The y1 . y2 term cancels against the y3, and the x's cancel.
If the number of seconds in d days equals the number of minutes in 8 weeks, then d = 60 56 7 2/5 14/15
14/15 Set up the equality: (d days)(24 hrs/day)(60 min/hr)(60 sec/min) = (8 weeks)(7 days/week)(24 hours/day)(60 min/hr). Note that the 24 hr/day and the 60 min/hr appear on both sides of the equation, so they cancel. This leaves (ignoring the cleared units): d(60) = 56. Solve for d and reduce to simplest terms. By the way, 14/15 of a day isn't as weird as it sounds; it's 22 hours and 24 minutes.
In his last will, the millionaire willed 80% of his wealth to his only son. 20% of the remaining amount will go to his wife and whatever is left will go to his charitable foundation. What percent of the millionaire's wealth will go to his foundation? 4% 5% 10% 16% none
16% 80% goes to his son so 20% remains. 20% of 20% = 4% goes to his wife This means 100% - 80% - 4% = 16% will be left for his foundation.
Cindy, a coin collector, had x coins. His friend James gave 80% of his coins to Cindy as a gift so that Cindy would have a total of 1000 coins. If the previously Cindy and James had 1200 coins in total, how many coins are left with James after giving most of it to Cindy? 1000 800 600 400 200
200 Originally, Cindy had x coins and James had y. Such that, x + y = 1200. James gave 80% of his coins to Cindy, which is 0.8y. Cindy now has 0.8y + x = 1000 coins. We then have two equations: x + y = 1200 x + 0.8y = 1000 Solving for y by subtracting one equation from the other gives us 0.2y = 200 Luckily, this is the number of coins left with James since he already gave 0.8y of his coins.
The area of the space bounded by the x and y axes and between the lines described by the equations y = -2x + 10 and y = -2x + 2, in square units, is 1 4 10 24 25
24 These two equations are in slope-intercept form; note that the lines have the same slope (they are parallel) with different intercepts. Make a quick sketch for yourself on your scratch paper -- place the two y-intercept points (0,2) and (0,10) and the two x-intercept points (5,0) and (1,0) found by making y=0. Connect the points; you should have two parallel lines that slope from the upper left down to the lower right. Each line forms a triangle with the x and y axes. The area between the lines is the area of the larger triangle minus the area of the smaller triangle. The formula for the area of a triangle is ½(bh). For the larger triangle, this is ½(5)(10) = 25; for the smaller, ½ (1)(2) = 1. Subtracting gives an area of 24.
A group of six students run in an election for the two positions of class president and vice president. How many different possible president-vice president combinations can be elected? *** 12 15 30 45 720
30 Six students have a chance at the president slot, and then the vice president will be chosen from the remaining five. So there are (6)(5)=30 possibilities. You do not have to divide by 2, because the president-vice president combination is ordered; President Smith and Vice President Jones is not the same combination as President Jones and Vice President Smith.
(0.00020/0.00001)·(0.030/0.002)
300 Clear the decimals by multiplying the numerators and enominators by the same powers of 10 and things become more clear: For the term in the first parentheses, moving the decimal to the right five times results in 20/1 = 20, while for the second parentheses moving the decimal to the right 3 times results in 30/2 = 15. The final answer is 20(15) = 300.
For what value of x is it true that 0.2x = 3.3 - 0.5y and 0.6y = 0.2x + 2.2? 0.4 0.5 4 5 20
4 These are two equations in two unknowns, conveniently with the same x-term in each. Solve the second equation for the x-term, set it equal to the first equation. This is easier on the eyes if you multiply everything through by 10 first: 33-5y = 6y-22, so 55 = 11y and 5 = y. Don't pick answer choice D; the question asks about x. Plug the y back in to either equation to find x = 4.
Two trains leave from the same station, travelling in opposite directions. One train travels at a speed 15 km/hr slower than the other. After 16 hours they are 1680 km apart. Find the average speed of the slower train. 15km/hr 30km/hr 45km/hr 60km/hr The answer cannot be determined from the information given
45 km/hr An old algebra class standby, the distance-rate-time problem. Let f = rate of the faster train; then the rate of the slower train will be f-15. The faster train travels a distance of dfast = f (16), while the slower train travels dslow = (f-15)(16) = 16f -240. Together these must combine for the 1680 km, so 16f + 16f - 240 = 1680, so 32f = 1920. Divisibility rules come in handy to simplify this calculation. Solving, f = 60 km/hr; the speed of the slower train = f-15 = 45 km/hr.
The length of a side of a certain square T is twice the length of a side of another square S. If the sum of their perimeters is 444 units, what is the length of a side of square T? 37 55.5 74 148 296
74 Let s = the length of a side of the smaller square, and 4s its perimeter. Let 2s = the larger square side, and so its perimeter is 8s. Adding the perimeters, 4s + 8s = 444, so s = 37. DON'T choose answer choice A, because it doesn't answer the question. You want the length of a side of square T, or 2s = 37*2 = 74.
A physicist drives from his house to his office every 9:00 am. His office is 30 miles away. If his average speed is 60 miles per hour, what time will he arrive at his office? 9:00 9:30 10:30 11:00 11:30
9:30 Since his speed is 60 miles per hour, the physicist could travel on the average 60 miles in an hour or 30 miles in a half hour. Thus, he will arrive at his office 30 minutes later, which is at 9:30 am.
If y is a positive integer, and x = y(5)(6)(7)(8), then x is evenly divisible by*** I 12 II 25 III 15 I only II only II and III only I and III only I, II, and III
I and III only The variable x must be divisible not only by the numbers listed as its factors along with y, but also by the factors of those numbers if they are not prime. So, x must be divisible by 2, 3, 4, 5, 6, 7, and 8, and by products of two or more distinct numbers from the list - but NOT by the product of any of the numbers with itself. So x is divisible by 3 * 4 = 12, and by 3 * 5 = 15, but not by 5 * 5 = 25.
The sum of the digits of a certain two-digit number is 7. Reversing its digits creates a second number which is larger than the first by 9. The first number is I. 43 II. 25 III. 34 I only I and II only III only II and III only I and III only
III only If you are handy with algebra you can let a = the first digit and b = the second; the value of the first number will then be 10a+b. We know that a+b = b+a = 7. The value of the second number can be written 10b+a, which will equal 10a+b + 9. That's two equations in two unknowns. The second reduces to b-a=1; add this to b+a=7. The a's cancel and you find that b=4, and then a=3, so the first number is 43 and the second is 34. The way to fake it is to first realize that only one number can be the answer, so eliminate choices B, C, and E. Check A and D to see if the digits sum to 7; they do. Next, reverse the digits and see if the second number is 9 more than the first. This is only true of 34. Be careful not to pick choice E; 34 and 43 have an absolute difference of 9, but only 43 (the reverse of 34) is larger by 9. This second method of approaching the problem may be a bit faster than the first.
At a school event four kinds of ice cream were served: vanilla, chocolate, swirl, and strawberry. 26% of the students requested vanilla, and 30% requested chocolate. 500 servings of ice cream were given out. How many students requested strawberry? 47 110 220 280 The answer cannot be determined from the information given
The answer cannot be determined from the information given It is possible to find the number of servings of vanilla and chocolate: 0.26(500) = 130, and 0.30(500) = 150, respectively. However, there is not enough information given in the problem to determine how the swirl and strawberry divided up the other 220 servings.
In 2008, Company A produced 20,000 mobile phones a day, while Company B produces 18,000. In 2009, Company A increased their average daily production by 20% while Company B increased by 30%. Column A: The average daily production of Company A in 2009 Column B: The average daily production of Company B in 2009
The quantity in Column A is greater. Column A is = 20,000 * 120% = 24,000 Column B is = 18,000 * 130% = 23,400
y2 - z2 = 21 y = 5 ** Column A: z2 Column B:2
The quantity in Column A is greater. Since y = 5; y2 = 25 then, the first equation becomes 25 - z2 = 21; or z2 = 4 Which is greater than the quantity in Column B, 2.
The circle is inscribed inside an equilateral triangle. The angle bisectors pass through the center of the circle. Column A: The measure of angle ADO Column B: The measure of angle AOC divided by two
The quantity in Column A is greater. ADO forms a right triangle making angle ADO a right triangle. The half of angle AOC is part equal to angle AOD. From the figure, AOC obviously forms an acute angle.
Column A: (7!)(3!) / (5!)(2!) Column B: 5!
The quantity in Column A is greater. Do not calculate 7!, then multiply it by 3!, etc. Instead, write out the factors and cancel factors found in both the numerator and denominator. For Column A: (7 x 6 x 5 x 4 x 3 x 2)(3 x 2)/(5 x 4 x 3 x 2) (2) =(7 x 6 x 3) = 126. For Column B: (5 x 4 x 3 x 2) = 120.
Peter left home and drove toward the family vacation house at an average speed of 40 km/hour. Sophia left home some time later and, driving at 48 km/hr for five hours, caught up with Peter. At that time Peter had been driving for h hours. Column A 336 km Column B The distance Sophia could drive in h hours at her current speed
The quantity in Column A is greater. One way to handle distance-rate-time problems is to ask yourself "What equals what?" In this case, it is the distance Peter has driven and the distance Sophia has driven at the time they meet up. The expression for Peter's distance = (40)(h), and for Sophia it is (48)(5). Solve for h = (48)(5)/(40); save yourself calculation time by pulling the factor 8 out from the numerator and the denominator, leaving (6)(5)/(5); now the 5's cancel, leaving the 6 (ain't factoring wonderful?) Now, the distance Sophia could drive in 6 hours at 48 km/hr = (6)(48) = 288 km.
The number n is an integer, 7^n < 3000, and 5^n > 600. Column A: 5 Column B: n
The quantity in Column A is greater. The best approach for this problem is to "plug and chug". Remember that the number must work for both equations. Since it is easier to do powers of 5 than of 7, start there and see if we can narrow things down a bit. You know from the second equation that n will be positive. Let's start with n=2. You know that 5 x 5 = 25, and 25 x 5 = 125; 125 x 5 is the same as half of 125 x 10, so 1250/2 = 625. So, n has to be at least 4 in order for 5n to be greater than 600. Now let's look at the powers of 7, but save yourself scratch paper time by estimating. Since we know that n has to be at least 4, let's find 74 and go from there: 7 x 7 = 49; 49 x 7 is almost like 50 x 7, which is 350; 350 x 7 will be between 300 x 7 = 2100 and 400 x 7 = 2800. Another power of 7 would exceed the 3000 limit. So n = 4 is the only number that works for both equations.
The square is inscribed inside the circle. Column A: The diameter of the circle Column B: The side length of the square
The quantity in Column A is greater. The diameter is a diagonal of the square. Even by just looking, the diagonal is longer than the side for a square.
Parts at a factory are packed into shipping containers. The weights of the boxes are approximately normally distributed with an average of 50 pounds and a standard deviation of 7 pounds.*** Column A: The approximate percentage of boxes that weigh less than 43 pounds Column B: The approximate percentage of boxes that weigh more than 64 pounds
The quantity in Column A is greater. The values of 43 and 64 fall at one and two standard deviations, respectively, away from the mean. In a normal distribution, approximately 34% of the data falls within one standard deviation above, and another 34% below, the mean. Approximately 14% of the data falls between the first and the second standard deviation on each side of the mean, and approximately 2% of the data is found between the second and third standard deviation marks on each side of the mean. Altogether, across six standard deviations, 2% + 14% + 34% + 34% + 14%+ 2% = 100% of the data. The percentages below 43 (one standard deviation below the mean of 50) are 14% and 2%, or 16%. The percentage above 64 (two standard deviations above the mean) is only 2%. The theoretical normal distribution extends to infinity in both directions, but we must be talking about a reasonably finite number of packages.
5x - y = 3, and -x - y = -39 Column A: x2 Column B: y + 16
The quantity in Column A is greater. These are two simultaneous equations. Use standard algebraic procedures to solve for x and y. For example, you could solve the second equation for y and plug it into the first equation to get x. Or, you could subtract the second equation from the first to cancel the y's and solve for x. Either way, once you have x = 7, plug it into the second equation to get y = 32. Now you can calculate Column A: (x)(x) = 49 and compare it to Column B: y + 16 = 48.
Column A: √(42 + 72) Column B: √(52 + 62)
The quantity in Column A is greater. This question tests whether you know the rules for adding exponents. Even if you don't, though, you can just calculate your way through. Although 4 + 7 = 5 + 6, (4)(4) + (7)(7) ≠ (5)(5) + (6)(6). Column A can be simplified to √(16 + 49) = √65, while Column B becomes √(25 + 36) =√61
m and n are prime numbers less than 100. m divided by n gives a remainder of 2. n divided by m gives a remainder of 3 *** Column A: m Column B: n
The quantity in Column A is greater. m = 5, n = 3 The trick here is to note that dividing a prime number with a smaller prime number will always give you a remainder which is actually their difference. Dividing a smaller prime number with a bigger prime number will give you back the the smaller prime number. So, our first option is 2 or 3. However, we cant find a prime number divided by 2 and will give 2 as a remainder. Thus, we are left with 3. 5 is the only prime number when divided by 3 gives a remainder of four
Column A: Liters of 10% acid to be added to 16 liters of 25% acid to achieve a 20% solution Column B: Liters of 20% acid to be added to 10 liters of 30% acid to obtain a 25% solution ***
The quantity in Column B is greater A mixture problem. Let A = the liters to be added in the first problem. The equation for Column A is: A(0.10) + 16(0.25) = (A+16)(0.20). Multiply through by 100 to simplify the decimal, and solve, for A = 8. For the second problem, let B be the number of liters to be added. B(0.20) + 10(0.30) = (B+10)(0.25). Solving, B = 10.
Column A: 5^2 / (5 + 0.5)^2 Column B: (5 + 0.5)^2 / 5^2
The quantity in Column B is greater Both differ slightly; their numerator and denominator are switched. The one with a greater numerator therefore has the greater value.
A six-sided die is rolled several times and the results recorded.*** Column A: The probability of rolling a prime number on each of two rolls Column B: The probability of rolling an even number or a 1 on each of three rolls
The quantity in Column B is greater Each roll is independent of the others. The answer hinges on knowing that 1 is NOT a prime number. The three possible prime numbers, then, are 2,3, and 5, or ½ of the six total possibilities for one roll. For Column A: (1/2)(1/2) = ¼ = 25%. For Column B: The even numbers are 2, 4, and 6, and with 1 represent 4/6, or 2/3, of the six total possibilities. (2/3)(2/3)(2/3) = 8/27. Estimate this to be close to 30%. Or, you could compare ¼ to 8/27 by looking at what their numerators would be if they were to have the same denominators. To do this, multiply each fraction's numerator by the other's denominator. Since (1)(27) < (8)(4), the 8/27 "wins".
Column A: 315 + 313 +314 Column B: 316
The quantity in Column B is greater Here's the trick: Look at Column B first. Recognize that 3^16 = 3 x 3^15 = 3^15 + 3^15 + 3^15. Subtract 3^15 from both Column A and Column B. Now Column A is: 3^13 + 3^14 and Column B is: 3^15 + 3^15. The two terms in Column B are both bigger than either term in Column A, so whatever those gnarly numbers are, the ones in Column B add up to more.
a = -2.942 and [a] is the greatest integer that is less than a Column A: [a] Column B: -2
The quantity in Column B is greater The greatest integer that is less then -2.942 is -3 not -2. Thus, the quantity in Column B, -2 is greater than the quantity in Column A, -3.
p is a multiple of 6 Column A: The remainder when p is divided by 3 Column B: 1
The quantity in Column B is greater. Since 6 is a multiple of 3, any multiple of 6 is also a multiple of 3. Any multiple of 3 when divided by 3 has zero remainder
Column A: The price of m grams of mineral which costs q dollars per gram Column B: The price of m/5 grams of jewelry which costs 10q dollars per gram
The quantity in Column B is greater. Column A is = qm dollars. Column B is = m/5 * 10q = 2qm
Column A: The area of a circle with diameter 26 Column B: The area between two concentric circles with radii 15 and 6
The quantity in Column B is greater. First, note that the radius of the circle in A is 13, not 26. Its area will be 132π = 169π . Do not calculate further; ignore the π , it will be a common factor for all the circles' areas. The area between the concentric circles (it may be helpful to draw a diagram) is π152 - π62 = 225π - 36π = 189π.
Jessie looks in her closet to pick one of her 30 shirts to wear. She has 6 green shirts, 4 blue shirts, 2 red shirts, 10 yellow shirts, 1 white shirt, and the rest are other colors. 16 of the shirts are woven; the others are knit. ** Column A minimum number of woven shirts that are not green or yellow Column B number of knit shirts that are a color other than red, yellow, or white
The quantity in Column B is greater. Of the 16 woven shirts, all could be green or yellow, so Column A = 0. For Column B, of the 30-16 = 14 knit shirts, a maximum of 13 of them can be red, yellow, or white (2 + 10 + 1); the last shirt must be something else. The range of possibilities is 1 to 13, all of which are greater than Column A.
Column A: Number of square tiles needed to cover a space on a wall of 72" x 108", using tiles that each have a perimeter of 36 inches. Column B: Number of square tiles needed to make a rectangular border one tile wide around the outside of the same 72" x 108" space, using tiles that each have an area of 9 square inches.
The quantity in Column B is greater. Read carefully - did you notice that the tiles specified in Column A had a perimeter of 36 inches, while the tiles in Column B were specified using their area? This is especially easy to miss since 36 is a perfect square. For Column A, the tiles have side length of 36/4 = 9". You will need 72/9 = 8 tiles along the width, and 108/9 = 12 tiles along the length; (12)(8) = 96 tiles. Now, since the square border tiles have area of 9 square inches, they must have side length of 3". The width can be bordered by 72/3 = 24 tiles, while the length can be bordered by 108/3 = 36 tiles. Don't forget there are two lengths and two widths, for a total of 24 + 24 + 36 + 36 = 120 tiles. (You might also want to add another 4 tiles, for the corners, but it doesn't make a difference for this problem.) So more 3" small tiles are needed for the border (120) than large 9" tiles for the area (96). If you had used 6" tiles for the area as the problem attempted to mislead you, you would have needed 216 of them and mistakenly chosen A.
Colored marbles are drawn at random from a bag without replacing any. In the bag are four clear marbles and seven black marbles. Column A: The probability of drawing two clear marbles Column B: The probability of drawing three black marbles
The quantity in Column B is greater. The total number of marbles is 4 + 7 = 11. In order to draw 2 clear marbles, you would have to get one the first try (4/11 chance) and one of the remaining three clear marbles out of the ten marbles left in the bag on the second try (3/10). Multiply these together - simplifying the 4 in the numerator and 10 in the denominator to 2/5 makes it easier - and get 6/55. Estimate that this is about a 10% chance (you can figure it out exactly later if you need to.) For the black marbles, on the first draw you have (7/11) chance, (6/10) the next, and (5/9) on the third, as you are successful in drawing black marbles and the number of marbles left in the bag decreases. Multiplying these three fractions together - simplifying the 6 and 5 in the numerator against the 9 and 10, respectively, in the denominator to speed things along - to get 7/33. Estimate that this is close to a 20% chance, enough bigger than the 10% chance found earlier so that you don't need to actually calculate them out.
Candy pieces of the same size but differing colors are mixed to make 5-pound bags for bulk sale. The colors red, yellow, blue, and green are included in the ratio 4 : 5 : 3 : 3. Column A: The approximate number of pounds of red and blue candy in three bags Column B: The approximate number of pounds of yellow candy in five bags
The quantity in Column B is greater. The total number of parts is 4 + 5 + 3 + 3 = 15. Red and blue account for 4 + 3 = 7 parts, and yellow accounts for 5. In three 5-pound bags, 7/15 of the total pounds are red and blue: (3 bags) (5 pounds per bag)(7/15) = 7 pounds. In five 5-pound bags, 5/15 of the total pounds are yellow: (5 bags)(5 pounds per bag)(5/15) = 8.3 pounds. Note that for comparison purposes, you can streamline the calculation by ignoring the weight of a bag (5 pounds) since it is a common factor.
m©n = n2/(n-2m)*** Column A: 3©-2 Column B: -2©3
The quantity in Column B is greater. This is an "unknown operator" problem, and it is just another way of saying "given variables m and n, arrange them in this way and evaluate". So, for Column A, m©n = (-2)2/[(-2)-2(3)] = 4/-8 = -1/2. For Column B, m©n = (3)2/[3-2(-2)] = 9/7.
x2 + y2 < 4 *** Column A: x2 - y2 Column B: 4
The quantity in Column B is greater. x2 and y2 are always positive. Since their sum is greater than 4, their difference should be less than 4. Think of x2 and y2 as ordinary positive numbers. [NOTE: In GRE General Test, all numbers are assumed real. This will not be true when the numbers are complex.]
?PQR is a scalene triangle and the measure of angle PQR = 80° Column A: The sum of the measure of any two angles of the triangle Column B: 100°
The relationship cannot be determined from the information given. Remember that there are three angles. We only have information for one angle, i.e. 80° but we don't know the other two angles. All we know is that the other two angles has a total measure of 100° but the sum of ANY two angles cannot be precisely known. It can be 100°, or greater than 80° but less than 100°, or greater than 100°. Thus, the information given is insufficient.
Column A: The area of a rectangle with a perimeter of 10 Column B :The area of a triangle with a perimeter of 10
The relationship cannot be determined from the information given. The rectangle can be very long but very thin creating a small area while the triangle is a regular one creating a bigger area. The rectangle can be a regular, a square creating a bigger area while the triangle can be very thin creating a smaller area. Thus, we cannot tell which has a greater area.
x2 - x - 20 = 0 Column A: x Column B: 5
The relationship cannot be determined from the information given. This is a problem where blind plugging in of the answer choices could lead you astray. While it is true that the equation works for x = 5, there is a second root to the equation x = -4, which you could have found by factoring: (x - 5)(x + 4) = 0 or Viete's Formulas x1x2 = -20 and x1 + x2 = 1. Since x = 5 or -4, you do not have enough information to say whether x is less than or equal to 5.
(-1)-x = -1 and x is positive integer ≠ 0 Column A: 3 Column B: x
The relationship cannot be determined from the information given. You may be tempted to pick answer choice C, since when x = 3 the equation holds true. But it would hold true for any odd number.
|-1| ≥ q Column A: q Column B: -1
The relationship cannot be determined from the information given. |-1| ≥ q is actually 1 ≥ q. Thus, q can be greater than -1, less than -1 or equal to -1.
The two circles intersect each other at exactly one point. Column A: The shortest distance between the centers of the two circles Column B: The sum of the circumference of both circles divided by 2π
The two quantities are equal The center to center distance is the sum of radi of both circles, (r + R). The sum of the circumference of both circles is 2π(r + R). Dividing by 2π will give us (r + R). Thus, both are equal.
Column A: (√7 + 13) (√7 - 13) Column B: - (13 + √7) (13- √7)
The two quantities are equal. Evaluate each expression using FOIL. Note that the central terms drop out in each case. Don't neglect the external negative in Column B. Alternatively, multiply the second term in Column B by the leading -1, and you can rearrange to match Column A. No FOIL required!
Elephants in a zoo require approximately 150 pounds of food per day. Elephants in the wild require approximately 315 pounds of food per day. Column A: The amount of food required by three zoo elephants for one week Column B: The amount of food required by five wild elephants for two days
The two quantities are equal. For the zoo elephants, the total amount of food is (3 elephants)(150 pounds per day per elephant)(7 days) = 3150 pounds. For the wild elephants, the total amount of food is (5 elephants)(315 pounds per day per elephant)(2 days) = 3150 pounds.
A right triangle ABC has hypotenuse 20 and base 16. Column A: The area of triangle ABC Column B: Two times the perimeter of triangle ABC
The two quantities are equal. Fun fact: the areas and perimeters of triangles with legs 12 and 5 or 6 and 8 (talking whole numbers here) are equal. This triangle has a related relationship; its legs are twice as long as 6 and 8, and its area is equal to twice its perimeter. But you don't need to know all that for this problem. Just use the Pythagorean Theorem to find the height of the triangle: 162 + h2 = 202, so h2 = 144, so h = 12. (Note that knowing higher squares off the top of your head is useful.) So the area of the triangle = ½ bh = ½ (16)(12) = 96, and twice the perimeter is 2(20 + 16 + 12) = 96.
x2 + 4x = 32*** Column A: 4 Column B: 8 + sum of all possible values of x
The two quantities are equal. Just looking at it, it looks like whatever B is, it's going to be larger than A. However, this is a quadratic equation equivalent to x^2 + 4x - 32 = 0. Factor into (x + 8)(x - 4) = 0. In order for this equation to equal 0, either x+ 8 = 0 or x - 4 = 0. So, x must equal to either -8 or 4, the sum of which is -4. -4 added to 8 gives 4.
Column A: The greatest common factor of 36, 54, and 90 Column B: least common multiple of 18, 3, and 6
The two quantities are equal. The factors of 36 are 3, 4, 6, 9, 12, 18, and 36; for 54 (up to 36): 2, 3, 6, 9, 18, 27 ...; for 90 (up to 36): 1, 2, 3, 5, 6, 9, 10, 15, 18, 30 ... The largest number that appears on both lists is 18. You can also arrive at the GCF by prime factorization. Since 36 = 2232, 54 = 2×33, and 90 = 2×325, the largest distinct powers that appear in all three factorizations are 2 and 32, so 2×32 = 18. For the multiples of the numbers in Column B, first note that 3 and 6 have 18 as a multiple, so 18 is the least common multiple. Note that Column B was a lot easier to figure out than Column A. If you had glanced over both and done B first, its result of 18 would have helped you decide about A. It's easy to see that 18 is a factor of 36, 54, and 90, so you wouldn't have had to bother with considering smaller ones; and that 36 isn't a factor of the other two. So the answer for A is 18 also.
Column A: (√[64y3(xy)6])1/3 Column B: √[16y2(xy)4]
The two quantities are equal. These look gnarly at first, but a closer look shows that the quantity under the radicand in Column A is a perfect cube, while the quantity under the radicand in Column B is a perfect square. For Column A, taking it piecewise might clarify things: : 3√[64y3(xy)6] = (3√64)(3√y3)( 3√(xy)6) = 4y(xy)2. For Column B: (√16)(√y2)(√(xy)4) = 4(xy)2
Andy and Brandon are outside on a sunny late afternoon. Andy, who is 1.6 m tall, casts a 400 cm shadow, while Brandon's shadow is 300 cm.** Column A twice Brandon's height Column B The absolute value of the difference between the length of Andy's shadow and Andy's height
The two quantities are equal. This is a ratio. Andy's height is to Brandon's height as the length of Andy's shadow is to the length of Brandon's shadow: 160/B = 400/300; so B = 120 (don't forget to change Andy's height into centimeters like the other measures.) Now, Column A = 2(120)=240, and Column B = 400-160 = 240.
x2 - 2x + 1 = 1 Column A: 2x Column B: x^2
The two quantities are equal. Transposing all terms except x2 to the right gives x2 = 1 - 1 + 2x x2 = 2x
Column A: 200 times larger than 0.063 Column B: 30 times larger than 0.42
The two quantities are equal. You shouldn't need your scratch paper for calculations like these; improve your mental math (and you can ALWAYS improve your mental math!) so you can save the time for other work. For Column A, first multiply by the 100, by moving the decimal point over twice: 6.3. Now multiply that by 2, to get 12.6. For Column B, multiply by the 10 by moving the decimal point over once: 4.2. Then multiply by the 3 to get 12.6.
S = {y-z, x}. The mean of set S is 3y. What is the unknown member of the set, x, in terms of y and z? 2y z-5y z+5y z-3 4y+3
z+5y The mean is the average of the set, so the given mean 3y equals the sum of the given first term and an unknown second term x divided by two: 3y = [(y-z)+x]/2. Solving, x = 6y - y + z = 5y + z = z + 5y.