GRE Math Practice Problems

♠n denotes the number obtained when n is rounded to the nearest tenth. For example ♠4.31 = 4.3 ♠0.089 - ♠1.135 = A. 1.05 B. 1.04 C. -1.05 D. -1.0 E. -0.1

0.089 to the nearest tenth = 0.1 1.135 to the nearest tenth = 1.1 0.1 - 1.1 = -1

For how many integer values of n will the value of the expression 4n + 7 be an integer greater than 1 and less than 200? A. 48 B. 49 C. 50 D. 51 E. 52

1 < 4n + 7 < 200 n can be 0, or -1 n cannot be -2 or any other negative integer or the expression 4n + 7 will be less than1. The largest value for n will be an integer < (200 - 7) /4 193/4 = 48.25, hence 48 The number of integers between -1 and 48 inclusive is 50 *I didn't include -1 and 0

Helpers are needed to prepare for the fete. Each helper can make either 2 large cakes or 35 small cakes per hour. The kitchen is available for 3 hours and 20 large cakes and 700 small cakes are needed. How many helpers are required? A. 10 B. 15 C. 20 D. 25 E. 30

20 large cakes will require the equivalent of 10 helpers working for one hour. 700 small cakes will require the equivalent of 20 helpers working for one hour. This means if only one hour were available we would need 30 helpers. But since three hours are available we can use 10 helpers.

A straight line cuts across a circle with radius 3 such that the area of the circle to one side of the line is 2Pi. What is the area of the circle on the other side of the line? 9Pi 7Pi 4Pi 2Pi There is not enough information to solve the problem.

A circle with radius 3 has area 9Pi. So if to one side of the line the area is 2Pi, then it follows that there is 7Pi on the other side.

(6^5 - 6^4)/5 A. 1/5 B. 6/5 C. 6³ D. 64 / 5 E. 64

6^5 = 6^4x 6 (6^4 x 6) - 6^4 = 6^4(6 - 1) = 6^4 x 5 Now, dividing by 5 will give us 6^4

230 + 230 + 230 + 230 = A. 8120 B. 830 C. 232 D. 230 E. 226

All four terms are identical therefore we have 4 (230). But 4 = 22, and so we can write 22. 230 Which is equivalent to 232

A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle? A. 2.5π B. 3π C. 5π D. 4π E. 10π

Draw the diagram. The diagonal of the rectangle is the diameter of the circle. The diagonal is the hypotenuse of a 3,4,5 triangle, and is therefore, 5. Circumference = π.diameter = 5π

( √2 - √3 )² = A. 5 - 2√6 B. 5 - √6 C. 1 - 2√6 D. 1 - √2 E. 1

Expand as for (a + b)2. (√2 - √3)(√2 - √3) = 2 - 2(√2 + √3) + 3 = 5 - 2 √6

5A _BC_ D43 In the following correctly worked addition sum, A,B,C and D represent different digits, and all the digits in the sum are different. What is the sum of A,B,C and D? A. 23 B. 22 C. 18 D. 16 E. 14

First you must realize that the sum of two 2-digit numbers cannot be more that 198 (99 + 99). Therefore in the given problem D must be 1. Now use trial and error to satisfy the sum 5A + BC = 143 A + C must give 3 in the units place, but neither can be 1 since all the digits have to be different. Therefore A + C = 13. With one to carry over into the tens column, 1 + 5 + B = 14, and B = 8. A + C + B + D = 13 + 8 + 1 = 22

Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps is 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to British stamps? A. 5 : 1 B. 10 : 5 C. 15 : 2 D. 20 : 2 E. 25 : 2

Indian stamps are common to both ratios. Multiply both ratios by factors such that the Indian stamps are represented by the same number. US : Indian = 5 : 2, and Indian : British = 5 : 1. Multiply the first by 5, and the second by 2. Now US : Indian = 25 : 10, and Indian : British = 10 : 2 Hence the two ratios can be combined and US : British = 25 : 2

For each of x people, Bunto bought a gyro and a beer at a pub. For each of x people, Kristi bought 4 gyros and a beer at the same pub. If Bunto spent a total of $6.20 and Kristi spent a total of $17.60, how much did Kristi spend just for gyros? (Assume that all gyros cost the same and all beers cost the same). A $15.20 B $11.40 C $9.20 D $6.80 E $3.80

For Bunto, we can make an equation so that g+b = $6.20 (where g = price of one gyro for each member of the group and b = price of a beer for each member of the group) and an equation 4g+b=$17.60 for Kristi. Now, we must subtract the first equation from the second, to get 3g=$11.40. => 4g+b=$17.60 - (g+b = $6.20) 3g = 11.40 Dividing both sides of this new equation by 3, we get: => g=11.40/3 = $3.80. To get the total that Kristi spent on her 4 gyros, we simply multiply 4(3.80)=$15.20. This is our answer. As a check, we can find the price of a beer by taking $6.20-$3.80=$2.40 and then taking $15.20+$2.40 to make sure we get $17.60 for Kristie's total spending.

After being dropped a certain ball always bounces back to 2/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (in inches) will it reach after its fourth bounce? A. 20 B. 15 C. 8 D. 5 E. 3.2

If after each bounce it reaches 2/5 of the previous height, then after the second bounce it will reach 2/5 x 125. After the third it will reach 2/5 x 2/5 x 125. After the fourth it will reach 2/5 x 2/5 x 2/5 x 125. This cancels down to 2 x 2 x 2 = 8

If an object travels at five feet per second, how many feet does it travel in one hour? A. 30 B. 300 C. 720 D. 1800 E. 18000

If an object travels at 5 feet per second it covers 5x60 feet in one minute, and 5x60x60 feet in one hour. Answer = 18000 (E)

A circular logo is enlarged to fit the lid of a jar. The new diameter is 50 per cent larger than the original. By what percentage has the area of the logo increased? A. 50 B. 80 C. 100 D. 125 E. 250

If the diameter of the old lid is 100, the diameter of the new lid will be 150. This gives a ratio of 2 : 3 The areas of the lids will be in the ratio (2)2 : (3)2 . or 4 : 9 This represents an increase in area of 5units for every 4 units 5/4 is equivalent to 125 percent.

Two sets of 4 consecutive positive integers have exactly one integer in common. The sum of the integers in the set with greater numbers is how much greater than the sum of the integers in the other set? A. 4 B. 7 C. 8 D. 12 E. it cannot be determined from the information given.

If two sets of four consecutive integers have one integer in common, the total in the combined set is 7., and we can write the sets as n + (n + 1) + (n + 2) + (n + 3 ) and (n + 3) + (n + 4) + (n + 5) + (n + 6) Note that each term in the second set is 3 more than the equivalent term in the first set. Since there are four terms the total of the differences will be 4 x 3 = 12

If a + b = c, then a^2 + 2ab + b^2 = ... 2c c(a - b) (a - b)(a + b) c^2 a + b

If we factor a^2 + 2ab + b^2 we get (a + b)(a +b) or (a + b)^2. This should be an equation that you can recognize easily on the GRE, as it will appear often. Another one to remember is the difference of squares, or a^2 - b^2 which is equal to (a + b)(a - b), though you don't need to know this for this question. If we use the first equation to substitute for a + b we get: (a + b)^2 = c^2

A cubical block of metal weighs 6 pounds. How much will another cube of the same metal weigh if its sides are twice as long?

If you double the sides of a cube, the ratio of the surface areas of the old and new cubes will be 1: 4. The ratio of the volumes of the old and new cubes will be 1: 8. Weight is proportional to volume. So, If the first weighs 6 pounds, the second weighs 6x8 pounds =48. [need to remember to double depth too!]

If 3d - 2r = 5r - 4d, what is d in terms of r? r/5 r/3 r 2r 4r

In order to solve for d in terms of r, we must first get the r's and the d's on the same side of the equation, respectively. 3d - 2r = 5r - 4d -->> 3d + 4d = 5r + 2r -->> 7d = 7r -->> so d = r.

Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ? A. 11/P + 6 B. P/11 +6 C. 17 - P/6 D. 17/P E. 11.5P

Let Ben's age now be B Anita's age now is A. (A - 6) = P(B - 6) But A is 17 and therefore 11 = P(B - 6) 11/P = B-6 (11/P) + 6 = B

On a two-dimensional coordinate plane, line A is represented by the equation y = 3x + 1 and line B is represented by the equation y = 2x + 2. At which of the following coordinate pairs do lines A and B intersect? (1, 2) (1, 4) (2, 5) (3, 7) (3, 10)

Lines A and B intersect at a single point (we know this because any two straight lines on a coordinate plane that are not parallel must intersect at one, and only one, point). Since lines A and B have different slopes (line A has a slope of 3 and line B has a slope of 2) we know that they are not parallel, and that they must intersect at some point (x, y). This point (x, y) lies on both lines; this means that the values of x and y must satisfy both line equations. Since we must find a single value for x and a single value for y that will work with both equations, we can use both line equations to set one pair of variables equal to each other, and then solve for the other variable. In this case, it's easiest to set the equations themselves as equal (thereby saying that y = y): q=q --> 3x+1 = 2x+2 And solving for x gives us: 3x+1=2x+2 -->x=1 Now that we know x, we can substitute this value into either of our equations to get y: 3x + 1 = (3)(1) + 1 = 4. The two lines intersect at (1, 4).

Of the following, which is greater than ½ ? Indicate ALL such fractions. A. 2/5 B. 4/7 C. 4/9 D. 5/11 E. 6/13 F. 8/15 G. 9/17

One way to deal with fractions is to convert them all to decimals. In this case all you would need to do is to see which is greater than 0.5. Otherwise to see which is greater than ½, double the numerator and see if the result is greater than the denominator. In B, one of the correct answers, doubling the numerator gives us 8, which is bigger than 7. In F, doubling 8 gives 16, which is greater than 15. In G doubling 9 gives 18, which is greater than 17.

If n ≠ 0, which of the following expressions could have a value less than n? Indicate ALL such expressions. A. 2n B. n² C. 2 - n

Remember that n could be positive negative or a fraction. Try out a few cases: In case A, if n is -1, then 2n is less than n. In case B, if n is a fraction such as ½ then n2 will be less than n. In case C, if n is 2, then 2-n = 0, which is less than n. Therefore, all of the choices could have a value less than n

The average (arithmetic mean) of four numbers is 30. After one of the numbers is removed, the average of the remaining three numbers is 10. What number was removed? 10 30 40 90 120

Since Average = (sum of terms)/(number of terms), set up the average equation as follows: Average * number of terms = sum of terms Since the original average of 4 terms was 30, we know that the sum of the 4 original terms is: 30 * 4 = 120 ...and since the final average of 3 terms is 10, we know that the final sum is: 10 * 3 = 30. The difference between the original sum and the final sum will be the amount representing the term that was removed, so 120 - 30 = 90. The correct answer is D.

A certain travel agency organizes sightseeing tours. Each client may select the cities she wishes to visit and the order in which she visits them. If each tour must include exactly 4 cities, and if there are exactly 10 cities available, how many different tours are possible? 24 210 420 2,400 5,040

The correct answer is 5,040. We can use the Permutation formula, [nPk = n!/(n-k)!], here to find an ordered group of 4 out of a group of 10: 10P4 = 10!/(10-4)! = 10!/6! = 10 x 9 x 8 x 7 = There are 5,040 different ways to choose and sequence 4 different cities out of a group of 10.

A straight fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet long. If a certain fence begins and ends with a post, which of the following could be the length of the fence in feet? (12 inches = 1 foot). Indicate ALL such answers. A. 17 B. 28 C. 35 D. 39 E. 50

The fence will consist of one more post than there are chains. (e.g. P-c-P-c-P). Therefore, a total length has to be a multiple of the length of the chain plus one post (5.5) plus one post extra.We have length = (5.5n + 0.5), where n can be any positive whole number. If n= 3, length =17; if n= 5, length = 28, etc.but there is no whole number that can give 35. Hence all the answers except C are correct.

n and p are integers greater than 1 5n is the square of a number 75np is the cube of a number. The smallest value for n + p is A. 14 B. 18 C. 20 D. 30 E. 50

The smallest value for n such that 5n is a square is 5. 75np can now be written as 75 x 5 x p. This gives prime factors.... 3 x 5 x 5 x 5 x p To make the expression a perfect cube, p will have to have factors 3 x 3 , and hence p =9 n + p = 5 + 9 = 14

In one country's currency, there are 40 dribbles in a gibble, and 4 gibbles in a zabble. How many dribbles are there in half a zabble?

There are 2 gibbles in half a zabble. This means that there are 2*40 = 80 dribbles.

What is the perimeter, in feet, of a rectangular swimming pool 16 feet wide, that has the same area as a rectangular pool 32 feet long and 10 feet wide? 20 72 84 160 320

To solve this we should set up an equation and solve for the missing variable x. This equation depends on knowing the equation for the area of a rectangle l*w. So x*16 = 32*10 x*16 = 320 x = 320/16 x = 20 Now that we know that x = 20, we know the perimeter of the pool is 2l + 2w or 2*16 + 2*20 32 + 40 = 72. The correct answer is B.

If d + d = c + c + c + c and d + c = 6, then d = 0 2 4 6 12

We can solve this system of equations by substitution. To do this, we can first get d in terms of c and solve for c. We are given d + c = 6 => d = 6 - c. Next, we plug this into the other equation to yield (6 - c) + (6 - c) = c + c + c + c => 12 - 2c = 4c => 12 = 6c => c = 2. Knowing that c = 2, we can substitute that value for c into d + c = 6: => d + 2 = 6 => d = 4.

√5 percent of 5√5 = A. 0.05 B. 0.25 C. 0.5 D. 2.5 E. 25

We can write the statement mathematically, using x to mean �of� and /100 for �per cent�. So ( √5/100) x 5√5 = 5 x 5 /100 = 0.25

If the degree measures of the angles of a triangle are in the ratio 6:8:10, what is the degree measure of the smallest angle? 15 degrees 45 degrees 60 degrees 75 degrees 90 degrees

We know that 6:8:10 is a ratio that can be simplified to 3:4:5, which is a classic triangle ratio. We also know that the three angles of a triangle sum to 180. So 3x + 4x + 5x = 180, where x is a multiplier that we can use to find the actual angles in the proportion 3:4:5. 3x + 4x + 5x = 180 => 12x = 180 => x = 15. If x = 15 then the angles associated with each ratio are: 3: 3x => 3 * 15 = 45 4: 4x => 4 * 15 = 60 5: 5x => 5 * 15 = 75 The correct answer is B.

If 30% of a certain number is 900, what is 10% of that same number? 27 30 270 300 3,000

We need to determine the certain number, lets call it 'x' and to do this we can set up the following equation with the information given. x*.30 = 900 => divide both sides by .30 (it might be easier if you don't like decimals to multiply both sides by 10 and divide by 3) x = 3,000 Now that we have x, we need to take 10% of it x*.10 => 3,000*.10 = 300

A piece of licorice of length L inches is cut into two pieces, resulting in one of the two pieces having a length of two inches more than three times the length of the other piece. Which of the following is the length, in inches, of the longer piece? 3L + 2)⁄4 (3L - 2)⁄4 (4L - 2)⁄3 (L - 2)⁄2 (L + 2)⁄2

We need to set up the equations in this question. Let's assign X and Y to the the two cut pieces, where X > Y. The full length of the licorice, L, is the total of the two cut pieces, so: L = X + Y Next, we must translate the fact that one piece is two inches longer than three times the length of the other piece. The first of these is clearly the longer piece, so X is two inches longer than three times Y: X = 3Y + 2 => X - 2 = 3Y => (X - 2)/3 = Y Since the question asks us to provide X, let's replace Y in the first equation and solve for X in terms of L. L = X + (X - 2)/3 => 3L = 3X + X - 2 => 3L + 2 = 4X => (3L + 2)/4 = X The correct answer is A.

What is the average (arithmetic mean) of all the multiples of ten from 10 to 190 inclusive?

You could add up all the multiples of 10 (10 + 20 + 30 ....+190), and divide by the number of terms (19). Or you could realize that the average of an evenly spaced series of numbers is equal to the value of the middle term (or the average of the two middle terms if there are an even number of terms). The middle term out of 19 is the tenth term in the series = 100. [I just took the mean of 10 and 190]

In a class of 78 students 41 are taking French, 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either course? A. 6 B. 15 C. 24 D. 33 E. 54

You could solve this by drawing a Venn diagram. A simpler way is to realize that you can subtract the number of students taking both languages from the numbers taking French to find the number taking only French. Likewise find those taking only German. Then we have:Total = only French + only German + both + neither 78 = (41-9) + (22-9) + 9 + neither. Not enrolled students = 24