Quantitative Practice Problems

If ((0.0024∗10^j))/((0.08∗10^r))=3∗10^6, then j−r equals which of the following? A. 9 B. 8 C. 7 D. 6 E. 5

0.0024=24∗10^−4,and 0.08=8∗10^−2. Rewrite the left side of the given equation as ((24∗10^−4)∗10^j)((8∗10^−2)∗10^r). Dividing and applying the rule that (x^a)(x^b)=x^(a−b), the equation simplifies to 3∗10^−2∗10^(j−r)=3∗10^6. Collecting the 10s on the left and dividing by 3 gives 10^(j−r−2)=10^6, so j−r−2=6 and j-r=8. This is B. CAT Rating 50

(√11+√11+√11)²= A. 363 B. 121 C. 99 D. 66 E. 33

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or y =√(11)'s, or whatever). How many √(11)'s are there here? Three. This expression can be rewritten as (3∗√(11))². Now the exponent rules will apply; (3∗(√(11))²=3²∗(√(11))²=9∗11=99. The answer is C. CAT Rating 46

If m is an integer such that (−2)^2m=2^(9−m), then m= A. 1 B. 2 C. 3 D. 4 E. 6

Anything raised to an even power results in a positive number, so (−2)^(2m)=2^(2m). At this point, the bases on each side of the equation are the same, so their exponents must be equal: 2m = 9-m. 3m = 9, and m = 3. The answer is C. CAT Rating 9

Is x>0? 1. x^2=9x 2. x^2=81

Be careful not to divide both sides of statement (1) by x without considering the x=0 possibility. Statement (1) can be rewritten as x²−9x=0 and factored as x(x-9)=0. This gives x=0 or x=9. Since we cannot determine whether x>0, statement (1) is insufficient. Statement (2) gives x=+/-9, which is also insufficient. Putting both statements together, x must be 9. This is sufficient, so the answer is C. CAT Rating 62

If m and n are integers and 36/3^4=1/3^m+1/3^n, what is the value of m+n? A. -2 B. 0 C. 2 D. 3 E. 5

Begin by reducing the left side of the equation. 36/3^4=4/3^2=4/9. On the right side of the equation, 1/3^m+1/3^n=3^n/3^(m+n)+3^m/3^(m+n)=(3^n+3^m)/3^(m+n). Returning to the left side, the numerator, 4, must be the sum of two powers of 3, and conveniently 4=3+1, so the left side can be rewritten as 4/9=3/9+1/9=1/3+1/9=1/3^1+1/3^2. In turn, m+n=1+2=3. This is D. CAT 67

The sum of the heights of two high-rises is x feet. If the first high-rise is 37 feet taller than the second, how tall will the second high-rise be after they add an antenna with a height of z feet to the top? A. (x+z)/(2+37) B. 2x−(37+z) C. (x−37)/2+z D. x/2−37+z E. (2x−37)/z

Call the high-rises f and s. Then, f+s=x and f=s+37. Because the second equation is pre-solved for f, substitute it into the first equation to obtain (s+37)+s=x. Solving for s gives s=(x-37)/2. Finally, the question calls for s+z, which is ((x-37)/2)+z. Answer C is correct. CAT Rating 42

If x is a positive integer, what is: (2ⁿ/2⁻ⁿ)ⁿ A. 1 B. 2^(2n²) C. 2^(2n)² D. 4²ⁿ E. 4^(2n)²

Dividing like bases means subtracting their exponents, so this expression reduces to (2²ⁿ)ⁿ. Nested powers multiply, so we have 2^(2n∗n)=2^(2n²). This is B. CAT Rating 52

2(x−2)/5+(24−12x)/6+2x=4x/5+3. Solve for x. A. 0 B. .5 C. 1 D. 1.5 E. 2

Do the same to both sides. Multiplying both sides by 5 * 6=30 will eliminate all denominators, but make sure to multiply that 30 onto every term on each side of the equation. The result is 6 * 2 * (x-2) + 5 * (24-12x) + 30 * 2x = 6 * 4x + 30 * 3. Simplifying, 12x-24+120-60x+60x=24x+90. Collecting terms, 12x+96=24x+90, so 12x=6, and x=1/2. This is B. CAT Rating 47

If k≠0 and z²/k+4z+3=z/k, then k= A. (z²−3)/4 B. (−z²+4z)/3 C. −3z(z+4) D. (z−z2)/(4z+3) E. z²+4z−3

Eliminate the denominators by doing the same to both sides - multiply everything by k. Then z²+4zk+3k=z, and 4zk+3k=z−z². Factoring the left side, (4z+3)k=z−z². k=(z−z²)/(4z+3). This is answer D. CAT Rating 48

15/(2(⁻⁵)+2(⁻⁶)+2(⁻⁷)+4(⁻⁴)) = A. 2⁷ B. 2⁸ C. 2⁹ D. 15(2⁷) E. 15(2⁸)

Factoring the denominator is the way to go here, and it's best to factor out the smallest term. That could be misleading given the negative powers and the 4⁻⁴ here, though. First prime factor 4⁻⁴ to get (2²)⁻⁴, or 2⁻⁸. The smallest term in the denominator is this one, 2⁻⁸. Factoring out 2⁻⁸ from each term in the denominator restates the expression as 15/(2⁻⁸∗(2³+2²+2¹+1)= 15/(2⁻⁸∗(8+4+2+1))= 15/(2⁻⁸∗15)=1/2⁻⁸=2⁸. CAT Rating 55

If a and b are integers and ab−a is odd, which of the following must be odd? A. b² B. b C. a²+b D. ab E. ab+b

If the test authors give an algebraic expression in one of its two forms, it's a pretty good bet that you need to use that expression in the other form. In this case, we're given ab-a=odd; it's a good bet that we'll get more value out of factoring this statement into a(b-1)=odd. Two integers multiply out to be odd if and only if both of those integers are themselves odd, so we can determine that a is odd and b-1 is odd. In turn, because b-1 is odd, b is even. So a is odd and b is even. Turning to the answer choices: Answer A is b^2, which is even. Answer B is b, which is also even. Answer C is a^2 + b. a^2 is odd, because a is odd and odd * odd=odd. And odd + even=odd, so a^2 + b is odd and C is correct. Just to check, we can also rule out D and E. D is odd * even=even. E can be rewritten as b(a+1)=even*even=even. The only answer that works is C. CAT Rating 33

x^−5*(x^4)^2*(x^3)/x*(x^−7)*(x^−5)^4 The above equation may be rewritten as which of the following? A. 1/x^25 B. 1/x^8 C. x^6 D. x^20 E. x^32

Nested powers multiply: (x^a)^b=x^(a∗b). Products of the same base to different powers add exponents: x^a∗x^b=x^(a+b). Also recall that x=x^1. Applying those rules, this expression reduces to (x^−5∗x^8∗x^3)/(x^1∗x^−7∗x^−20) and then to x^6/x^−26. Finally, dividing like bases means subtracting their powers, so we end up with x^32. That's answer E. CAT Rating 44

If 4x²+9y²=100 and (2x+3y)²=150, then what is the value of 6xy? A. 5(2+√6) B.10√6 C. 25 D. 50 E. 100

Once again a question provides its truth in the least-convenient form. Whether or not the solution is immediately apparent, it can't hurt (and will probably help) to rewrite the given common algebraic expression (2x+3y)²=150 in its other form: 4x²+12xy+9y²=150. It is also given that 4x²+9y²=100. Subtracting these two equations will conveniently eliminate both the x² and the y² terms. The result is 12xy=50. Since the question asks about 6xy, divide both sides by 2 to find that 6xy=25. This is answer C. When a GMAT question asks for an expression (like 6xy) instead of a single variable or quantity (like x), remember that you may be able to solve directly for that expression even if it's impossible to determine the values of the individual variables the expression involves. In this question, it is not possible to determine values for either x or y, but the expression 6xy is obtainable directly. In fact, whenever a question asks the value of an expression, students should generally assume that they are meant to solve for that expression directly rather than from its component parts. CAT Rating 47

If 5x=4y, which of the following is NOT true? A. (x+y)/y=9/5 B. (y)/y−x=5 C. (x−y)/y=1/5 D. (4x)/5y=16/25 E. (x+3y)/x=19/4

One approach to this question is to solve 5x=4y for x and for y. This gives x=(4/5)y and y=(5/4)x. For each answer choice, substitute for the variable that occurs fewer times. Answer A becomes ((4/5)y+y)/y=((9/5)y)/y=9/5 (true). Answer B becomes y/(y-(4/5)y)=y/((1/5)y=5 (true). Answer C becomes ((4/5)y-y)/y=(-1/5)y/y=-1/5, which is not equal to 1/5, so C is correct. D and E, if checked, also produce true statements. Another approach is to pick numbers. Choosing x=4 and y=5, the answers come out to 9/5, 5, -1/5, 16/25, and 19/4, respectively. Either way, the answer is C. CAT Rating 33

If (2^x)(2^y)=8 and (9^x)(3^y)=81, then (x,y) equal which of the following? A. (1,2) B. (2,1) C. (1,1) D. (2,2) E. (1,3)

Prime factor, prime factor, prime factor. To solve an exponent question of this type, there are really four steps: (1) Prime factor the bases. (2) Use the exponent rules to collect like bases. (3) Set powers of like bases equal. (4) Solve the resulting equation(s). Following that formula in this case, prime factoring gives 2^x∗2^y=2^3 and 3^2x∗3^y=3^4. Collecting bases gives 2^(x+y)=2^3 and 3^(2x+y)=3^4. Setting powers of like bases equal gives x + y = 3 and 2x + y = 4. Finally, this system can be solved by elimination. Subtracting gives x=1, and plugging that value into either equation gives y=2. The answer is (1,2), which is A. CAT Rating 33

0<x<y, and x and y are consecutive integers. If the difference between x² and y² is 12,201, then what is the value of x? A. 6,100 B. 6,101 C. 12,200 D. 12,201 E. 24,402

Since x and y are consecutive integers and y>x, we can write y = x+1.Then the question asks the value of x for which (x+1)²−x²=12,201, and we can factor this difference of squares as (x+1+x)(x+1−x)=12,201. This simplifies to (2x+1)(1)=12,201, or simply 2x+1=12,201. Thus, 2x=12,200 and x=6,100. Answer A is correct. CAT Rating 52

Data Sufficiency What is the value of x-y? 1. x-y=y-x 2. x-y=x²-y²

Solution: A This questions asks for a specific value for "x - y" Statement 1) "x - y = y - x." This statement can be simplified to "2x = 2y" and finally to "x = y." You can then finish the Algebraic Manipulation by subtracting y from both sides and you get "x - y = 0" This clearly gives you a specific value for "x - y." This statement is sufficient. The answer is either A or D. Statement 2) "x - y = x2-y2" The left side of the equation is the difference of squares so we can rewrite this as "x - y = (x + y) (x - y)." This statement leaves you with two possibilities. In order for this to be true, either "x - y = 0" so that both sides of the equation = zero. Or x + y = 1 so that both sides of the equation = x - y. These are two different possibilities allow infinite values for "x - y" since if "x + y = 1" "x - y" could equal anything. This statement is not sufficient. The correct answer is A. CAT 55

If 5∗x^√125 = 1/5^(1/x), then x = A. −4 B. −√(1/2) C. 0 D. √(1/2) E. 1

Start by prime factoring the bases, and rewrite the root as a power: 5∗(5^3)^(1/x)=1/5^(1/x). On the left side, combine like bases by multiplying the nested powers, recognizing that 5=5^1, and adding the powers of like bases multiplied together: 5^(1+(3/x)). As for the right side, a power in the denominator is equivalent to a negative power in the numerator: 5^(−1/x). With the bases prime factored and collected, we can set the powers equal: 1+(3/x)=−1/x. To simplify, do the same to both sides and eliminate the denominators by multiplying through by x to obtain x+3 = -1, or x =-4, which is answer A. CAT Rating 55

If x≠0 and x=√(4xy−4y²), then in terms of y, x= A. 2y B. y C. y² D. (−4y²)(1−4y) E.−2y

Start by squaring both sides to give x²=4xy−4y². Attempting to collect the x terms won't get us anywhere because of the x2 term. The equation x²−4xy=−4y² cannot be solved for x directly. But if we view y as a constant, then we have a quadratic equation in terms of x, so let's collect all of the terms on one side. The resulting equation is x²−4xy+4y²=0. Now recognize that we have a perfect square equation of the form (a−b)2=a²−2ab+b². Let a=x and b=2y, and our equation becomes 0=x²−4xy+4y²=x²−2∗x∗(2y)+(2y)²=(x−2y)². Since (x−2y)2=0,x−2y=0 and x=2y. Answer A is correct. CAT Rating 55

If x does not equal 3 and (x²−9)/2y=(x−3)/4, then in terms of y, x= A. (y−6)/2 B. (y−3)/2 C. y−3 D. y−6 E. y+6/2

Students should immediately spot the difference of squares on the left side of this equation. Factoring produces (x+3)(x-3)/2y=(x-3)/4. Divide both sides by (x-3) to eliminate that term: (x+3)/2y=1/4. Cross-multiply to get 2y=4(x+3)=4x+12. Then 4x=2y-12 and x=(2y-12)/4=(y-6)/2. This is A. CAT Rating 49

What is the value of (³√(√(³√512)))² A. 1/16 B. 1/8 C. 1/2 D. 2 E. 8

The first step to solving most exponent questions is to prime factor your bases. The base here is 512, and prime factoring reveals that 512=29. Express the roots as exponents using the rule a-root(x) = x(1/a). The expression becomes ((((29)(1/3))(1/2))(1/3))2. Nested powers multiply, so this simplifies to 2(9∗(1/3)∗(1/2)∗(1/3)∗2)=21=2. CAT Rating 48

(.08)^(−4)/(.04)^(−3) = A. −16/25 B. −16/25 C. 25/16 D. 16 E. 25

The given expression is ((8∗10^−2)^−4)/((4∗10^−2)^−3). There are multiple options to simplify from here, but one good method is to factor the 8 and the 4 and then distribute the nested exponents. That is, ((8∗10^−2)^−4)/((4∗10^−2)^−3) = ((2^3∗10^−2)^−4)/(2^2∗10^−2)^−3)= (2^−12∗10^8)/(2^−6∗10^6). (Be sure to distribute the negatives carefully!) Dividing means subtracting the exponents, which gives 2^−6∗10^2. This, in turn, is 100/(2^6)=(25)/2^4=25/16. Answer C is correct. CAT Rating 55

A number x is multiplied by 3, and this product is then divided by 5. If the positive square root of the result of these two operations equals x, what is the value of x if does not equal 0? A. 25/9 B. 9/5 C. 5/3 D. 3/5 E. 9/25

The question gives us √(3x5)=x. Square both sides to obtain 3x/5=x². This is a quadratic, so put it in the standard ax²+bx+c form and then factor. x²−(3/5)x=0, x(x−(3/5))=0, and thus x=0 or x−(3/5)=0. The question states that x≠0, so x=3/5. This is D. CAT Rating 49

The size of diamonds is measured in carats. If the price of a diamond doubles for every 0.5 carats, which of the following is worth the most? A. One 4-carat diamond B. Fourteen 1-carat diamonds C. Four 2-carat diamonds and eight 1-carat diamonds D. One 1-carat diamond, two 2-carat diamonds, and three 3-carat diamonds E. Three 1.5-carat diamonds and three 2.5-carat diamonds

There's not much to do here besides working out and comparing each answer choice. Estimating can help to make this process reasonably painless. Let the price of a one-carat diamond be p. Then a 1.5-carat diamond costs 2p, a 2-carat diamond costs 4p, a 2.5-carat diamond costs 8p, a 3-carat diamond costs 16p, a 3.5-carat diamond costs 32p, and a 4-carat diamond costs 64p. The exponential growth rate means that higher-carat diamonds dwarf smaller diamonds in value. Answer A, the 4-carat diamond, is worth 64p. Answer B, a pile of 1-carat diamonds, doesn't even come close at 14p. Similarly, answer C is a few diamonds worth 4p each and some little ones - nowhere near 64p. Answer D looks closer, so let's calculate: p+2 * 4p+3 * 16p=57p. It comes up just a little bit short. Finally, answer E is way too small. The three 2.5-carat diamonds alone come out to 24p, not even halfway to 64p, and the three 1.5-carat diamonds can't possibly add enough. Give us the BIG diamond. That's A. CAT Rating 50

Which of the following equations is NOT equivalent to 25x²=y²−4? A. 25x²+4=y² B. 75x²=3y²−12 C. 25x²=(y+2)(y−2) D. 5x=y−2 E. x²=(y²−4)25

There's pretty much nothing to do here except to check each answer choice and try to use algebra to convert it into the equation 25x²=y²−4. Answer A works; simply subtract 4 from both sides. Answer B works too; divide both sides by 3. Answer C has simply factored the difference of squares; it's also equivalent. Answer D, however, is not equivalent. You can't just square root everything and expect equality - both because the square roots require plus-or-minus and more importantly because y²−4 is not the same as (y−2)². Because the question asks for the equation that is NOT equivalent to the original, answer D is correct. For completeness, note that answer E is simply the original equation divided by 25 on both sides. Only D is not equivalent to 25x²=y²−4. CAT Rating 39

The expression ((√8+63)+(√8−63))² = A. 20 B. 19 C. 18 D. 17 E. 16

This expression is a classic inconvenient truth; it represents a common algebraic expression in one form when we instead need to use the other. Using our knowledge of common algebraic expressions, then, we can rewrite the given expression as (√(8+(√63)))²+2(√(8+(√63))) (√(8(√63)))+(√(8−(√63)))². It looks awful, but it simplifies quickly: 8+√(63)+8−(√63)+2(√(8+(√63)))(√(8−(√63)))= 16+2(√(8+(√63)))(√(8−(√63))= 16+2∗√((8+(√63)(8−(√63))). From here, recognize the factored form of the difference of squares. Switching to the other form gives 16+2∗(8²−(√(63))²)= 16+2∗√(64−63)= 16+2∗√(1)= 16+2=18 . And so, at last, the answer is C. CAT Rating 61

Data Sufficiency Is x^2>x^3? 1.x^3 > 0 2. x≠1

This is a yes/ no question "is x^2>x^3?" Statement 1) x^3 is positive. Given that x^3 is positive x must be positive as well. However, x could still be a non-integer between 0 and 1. For example, if x = ½ then x^2 = ¼ and x^3 = 1/8. In this case x^2 >x^3 so that would be a "yes." If x is a positive integer like 2 then x^3> x^2, which means is a "no." Since this statement allows for both a "yes" and a "no" it is not sufficient. Eliminate choices A and D. Statement 2) "x does not equal 1." Given the possibility of fractions between 1 and 0, this statement is clearly not sufficient on its own. Each of the values that were used in the analysis of statement 1 works with this statement as well. This is also not sufficient. Eliminate choice B. Together) The same values work for both statements giving you a "yes" and a "no" even with both statements together. If x is between 0 and 1 the answer is yes. If x is greater than 1 the answer is no. The correct answer choice is E. CAT 53

Data Sufficiency Is n an integer less than 5? 1. 5n is a positive integer. 2. n5 is a positive integer.

This is a yes/no question "Is n and integer less than 5?" Notice that the statement does not tell that n is an integer, but asks you if it is. Read the question stem carefully each time. Statement 1) "5n is a positive integer." This statement is not sufficient. If you play "Devil's advocate" you can get both integers and non-integers. For example, n could equal 1, which is an integer less than 5. Or n could = 1/5 and 5n = 1. This would give you a "no" since 1/5 is not an integer. This statement is not consistent so it is not sufficient. Eliminate choices A and D. Statement 2) "n/5 is a positive integer." In order for n/5 to be a positive integer n must be at least equal to 5 and must be a multiple of 5. For example, 5/5 = 1 and 10/5 = 2 etc. So this statement is sufficient because the answer is "no." "N" is an integer but it not less than 5. It must be at least 5. Because you can answer definitively "no" to this question then this statement is sufficient and the answer is B. CAT 51

Data SUff If x+y=6, then what does x-y equal? x² - y²=12 2y+x=8

This question asks for a specific answer for "x - y?" Facts) x + y = 6. Statement 1) "x²-y²=12." You should recognize this as the difference of squares. Factoring this will make it fit the question and the information that you already have. You can rewrite this to say "(x + y) (x - y) = 12. Given that in the fact you know that "x +y = 6" then it must be the case that "x - y = 2." This statement is sufficient. The answer is either A or D. Statement 2) "2y + x = 8." From this statement and from the equation in the fact, you can determine that y = 2. If you plug this number back into the equation in the fact, then x + 2 = 6. Therefore x = 4 and "x - y = 2." This statement agrees with statement 1 and is also sufficient. The correct answer is D. CAT 49

Data Sufficiency (46) Is a^2>3a−b^4? 1. 3a−b^4=−5 2. a>5 and b>0

This question asks y/n is "a^2>3a−b^4 ?" Statement 1) "3a - b^4 = -5" This statement may not appear sufficient at first. You cannot manipulate it algebraically to mirror the question stem as might be the first impulse. However, the important thing is that "3a−b^4 equals a negative number." Remember that a^2 cannot be less than zero. A squared number cannot be negative. The answer is "yes, a^2>3a−b^4." This statement is consistent so it is sufficient. The correct answer is either A or D. Statement 2) "a > 5 and b > 0." Again, this statement may not appear to be sufficient. It does not give specific values for a or b. However, if you "Just Do It" and plug in the numbers then you will see that it is sufficient. A good strategy for these greater than statements is to use the actual numbers given. Although the statement says a > 5 and b > 0 you can just use 5 and 0 in the inequality. The inequality becomes "25 > 15 - 0?" The answer is clearly yes 25 > 15. And as you increase both "a" and "b" the result becomes a stronger "yes." For example, if "a = 6 and b = 2" the inequality is "36 > 18 - 16?" Conceptually it looks like this, so long as a is greater than 3 then a^2 will be greater than 3a. Whatever number "b" is can only take away from the 3a it cannot add to it. In fact, statement 2 gives more information than strictly necessary; "a > 3" would be sufficient. The correct answer is D. CAT 63

If x is positive, then 1/(√(x+1)+√x) = A. 1 B. x C. 1/x D. √(x+1)−√x E. √(x+1)+√x

This question is really tricky if you haven't seen one like it before. However, a little bit of ingenuity and a knowledge of common algebraic equations can lead to the solution. Look at the answer choices; there isn't a denominator in the bunch. But how can we eliminate the denominator in this case? Well, it would be great if we could eliminate the roots, to begin with - perhaps by squaring them somehow. We use the technique of multiplying by 1, pretty much the only tool available in the case of an expression. Many students will try (√(x+1)+√(x))/(√(x+1)+√(x)), squaring the denominator. Unfortunately, the resulting denominator is x+1+2∗√(x+1)∗√(x)+x - the middle term makes things even more complicated. If only we knew a common algebraic expression to square each term without introducing a middle term... The difference of squares, perhaps? Recall from You Oughtta Know that the difference of squares is The Great Transformer, and try multiplying top and bottom by (√(x+1)−√(x))/(√(x+1)−√(x)). The numerator becomes (√(x+1)−√(x)). The denominator, on the other hand, is (√(x+1))²−(√(x))²=x+1−x=1. As a result, the entire expression simplifies to (√x+1)−√(x), which is answer D. CAT Rating 52

The variable x is inversely proportional to the square of the variable y. If y is divided by 3a, then x is multiplied by which of the following? A. 1/9a B. 1/9a² C. 1/3a D. 9a E. 9a²

Two quantities are proportional if there is a constant factor relating them. That is, "p and q are proportional" means that p=k * q for some constant k. Inverse proportionality means proportionality to the reciprocal; p and q are inversely proportional if p=k/q for some constant k. Here, x is inversely proportional to y^2. This means that x=k/y^2 for some constant k. Replacing y with y/(3a) gives x = k/(y/3a)^2 = k/(y^2/(9a^2)). To simplify this complex fraction, recall the technique of multiplying by 1. Multiply by (9a^2)/(9a^2) to obtain (9a^2) * k/y^2=(9a^2) * x. x has been multiplied by 9a^2, so the answer is E. CAT Rating 57

Is x²−y² a positive number? 1. x−y is a positive number. 2. x+y is a positive number.

When the test authors give a common algebraic expression in one of its two forms, we almost certainly will need to use the other form instead. Students should be extremely comfortable spotting the frequently-tested "difference of squares" formula. Here, we're asked about x²−y², but this can be written more conveniently as (x+y)(x-y). This will be positive whenever (x+y) and (x-y) are either both positive or both negative. Neither statement alone provides all of the needed information, but together the premises are sufficient. The answer is C. CAT Rating 51