#15 Quantitative Comparison - Algebra and Applied Mathematics
a) sqrt 7*8 b) 2 * sqrt 14
both are equal We can simplify Quantity A because a factor of one of the radical's terms is a perfect square: sqrt 7*8 = sqrt 7 * 2 * 4 = sqrt 14 * sqrt 4 =2 * sqrt 14
for x > 0 a) (7sqrtx^2)^3 b) (sqrt(x^3/7)^4
both are equal We can start by simplifying each exponent into fraction form. We will start with Quantity A: (7sqrtx^2)^3 = (x^2/7)^3 = x^6/7 now simplify B sqrt(x^3/7)^4 = (x^3/7)^4*1/2=(x^3/7)^2= x^6/7 Although the two quantities appear different at first glance, they simplify into the same exponent.
x is greater than or equal to 0 a) sqrt x^2 b) (sqrtx)^2
equal Quantities A and B are similar expressions but differ in the order of root and square. Since the square and square root functions are inverse functions, they should return x when done in either order. Quantity A simplifies to x. Since we know x ≥ 0, the function in Quantity B is always defined and also simplifies to x.
c > 0, d>0 a) c+d b) |c+d|
equal both Since both c and d are both positive numbers, c + d must be a positive number. The absolute value of a positive number is the same as the original number, so answer choice [C] is correct.
a) 0.015 * 1.872 b) 0.586 * 0.7281
B is greater
-1 < x < 0 0 < y < 1 z > 1 a) xz+y b) xy+z
b is greater
a) 1/6 of 700 b) 95
A is greater Quantity A is equal to 700/6, which is greater than 600/6 = 100. Since Quantity A is larger than 100, it must be larger than 95. Therefore, Quantity A is larger than Quantity B. Answer choice [A] is correct. [DAT Tip!] Go straight to the calculator to double check your answer, or mark the question to come back and check with the calculator later if you have time.
f(x) = x + 7 g(x) = (x+7)^2 x+6 > 0 a) f(g(x)) b) g(f(x))
B is greater Our initial constraint tells us that x + 6 > 0, or x > -6. This more than satisfies the condition above, so Quantity B must be greater Quantity B is larger than Quantity A for all x where x + 10 > 0, or x > -10.
a) five tens b) six nines
B is greater Quantity A equals 5 * 10 = 50. Quantity B equals 6 * 9 = 54.
x > 7 a) 7/x b) x/7
B is greater The question stem tells us that x > 7. This means that in Quantity A, the numerator will be less than the denominator. Thus, Quantity A is less than one. In Quantity B, the numerator will be greater than the denominator. Thus, Quantity B is greater than one.
x>1 a) 2^2x+2 b) (8^x)^2
B is greater We can rewrite Quantity B so that we have a base of 2 raised to some power to compare it to Quantity A. We can use the power law: (a^m)^n = a^mn: ((2^3)^x)^2=2^6x We solve for x such that Quantity A and B will be equal. 2x+2=6x 2=4x 1/2=x A and B are equal only when x = 1/2. Since x > 1, Quantity B will be greater and answer choice [B] is correct. DAT Pro-tip: We can plug in x = 2 and calculate the value of each quantity. If x = 2, Quantity A = 2^6 and Quantity B = 8^4, which is greater.
a) 82*37 b) 71*48
B is greater Note that both numbers in Quantity A and Quantity B have the same sum: 119. Similar to comparing the areas of two rectangles with the same perimeters - if two pairs of numbers have the same sum, the pair that is closer together will have the greater product. Alternatively, this is a problem where it could be faster just to use the calculator!
A = {3, 4, 10, 10, 27, 31, 47} B = {1, 1, 3, 6, 11, 12, 16, 29, 32} a) median of A b) median of B
B is greater Recall that the median value of a set of numbers is the middle value when all values are ordered from lowest to highest. Both A and B are already ordered, so we can look at the middle numbers. For set A, 10 is the median. For set B, 11 is the median. Q: What if there are an even number of elements in a set? A: If there are an even number of elements in a set, the median will be the average of the two central elements. In a set of 8 elements, the median would be the average of the fourth and fifth numbers.
A) -|-|-|-72||| B) - |72|
Both are equal Absolute value signs have no effect on the magnitude of a number; it only effects the sign.
A) 1/40,000 B) .02/800
Both are equal Rewrite .02/800 * 100/100 = 2/80,000 = 1/ 40,000
a) 3^7 b) 7^3
a is greater
a) 48^5/6^4 divided by 16^6/36^2 b) (3/2)^5
a is greater
x > a) (x+3)(x-1)/(x-1)(x+2) b) (x+3)(x-1)/(x+2)(x+3)
a is greater For Quantity A, we cancel out (x - 1) from both the numerator and denominator: (x+3)/(x+2) For Quantity B, we cancel out (x + 3) from both the numerator and denominator: (x-1)/(x+2) Now that Quantity A and B have the same denominator, we can compare their numerators. x + 3 is greater than x - 1 for all values of x, so Quantity A is greater than Quantity B.
x, y, z >0 x-y > 0 x-z < 0 a) x/y b) y/z
a is greater From the first statement, we know x, y, and z, are all positive numbers. From the next two statements, we know how y and z relate to x: x > y x < z If x is greater than y and z is greater than x, than z must be greater than y as well: y < z Since x > y and both are positive numbers, Quantity A is greater than one. Since y < z and both are positive, Quantity B is less than one.
0 < x < y < z a) y/x+z b) x/z+y
a is greater From the initial problem statement, we know the numerator in Quantity A is greater than the numerator in Quantity B, since y > x. Similarly, we know that the denominator of Quantity A is smaller than the denominator of Quantity B: x < y x + z < y + z Therefore, since Quantity A has both a larger numerator and a smaller denominator, Quantity A must be the larger fraction
A: 2y+7x=8 B: 5x=3y+17 a) y intercept of line a b) x intercept of line b
a is greater Let's first rewrite our Equation A in slope-intercept form: 2y+7x=8 2y=-7x+8 y=-7/2x+4 Therefore the y-intercept of Equation A is 4. To find the x-intercept of Equation B, we can substitute 0 in the equation for y: 5x=3y+17 5x=0+17 x=17/5 4 is greater than 17/5, so Quantity A is greater
a) half of 3 dozen b) 15
a is greater One dozen is equal to 12, so 3 dozen equals 36. Quantity A is equal to half of 36, or 18. Since 18 is larger than 15, Quantity A is greater than Quantity B. Answer choice [A] is correct.
a) 18 divided (7-50 b) 18 divided 8-18 divided 5
a is greater Quantity A equals 9 and Quantity B equals a negative number
a) principal square root of 73 b) 8
a is greater Recall that the principal square root is the unique positive (non-negative) square root of a non-negative real number. For example, the principal square root of 4 is 2, although both -2 and 2 are square roots of 4. The value of √73 must be between 8 and 9, since 73 is between 8^2 = 64 and 9^2 = 81.
m + 9 > p-8 a) m+10 b) p-7
a is greater Simplify the inequality by adding 1 to both sides: m+9+1> p-8+1 m+10>p-7
a) 3^4 b) 4^3
a is greater Since the bases and exponents are small whole numbers, we can calculate the exact values of Quantity A and B. Quantity A = 3^4 = 81 Quantity B = 4^3 = 64
j/k > k/j > 0 a) j^2 b) k^2
a is greater The given inequality tells us that j and k have the same sign because their quotients are positive. We can manipulate this inequality by cross-multiplying: j/k>k/j j^2> k^2 Note that when multiplying a negative number across an inequality, we would need to flip the sign of the inequality. However, since we know that the signs of j and k are the same, we are either multiplying by positive numbers, or twice by negative numbers. In either case, we leave our inequality sign unchanged (flipping it twice for negative numbers results in the original sign).
a) 10/13 b) 70%
a is greater We can best directly compare these two quantities by expressing both as fractions: 10/13 & 7/10 Note the difference between the numerator and denominator is the same in each quantity. When the difference is the same, and the value is less than one, the fraction with the larger numerator and denominator will be larger. For instance, 99/100 is larger than 1/2. Since the numerator and denominator of Quantity A are larger, Quantity A is larger than Quantity B.
p * q = 15 a) p/q b) 0
a is greater We don't know anything about the exact values of p and q or their quotient. However, since their product is positive, we know p and q must have the same sign (they are either both positive or both negative). The quotient of two numbers that have the same sign is positive, so Quantity A must be positive and therefore larger than 0.
x>0 a) x^2 + 2x+ 1 b) x(x+1)+1
a is greater The question tells us that x > 0, or that x is positive. Now, simplify the expression given in Quantity B: x(x+1)+1 x^2+x+1 If we compare Quantity A (x^2 + 2x + 1) to Quantity B (x^2 + x + 1), we can see that Quantity A is always x greater than Quantity B.
a) 5 / 4/ 3/ 2/ 1 b) 1/ 2 / 3/ 4/ 5
a is greater a) 5/24 b) 1/120 DAT Pro-tip: We can also intuitively realize that Quantity A is larger, because we start with a larger number and divide by smaller and smaller numbers, versus dividing small numbers by increasingly larger numbers!
x < 3 a) 2x+13 b) 5x-1
a is greater Both quantities can be represented as straight lines. If two lines are not parallel, then they will cross at exactly one point. We solve for their intersection by setting the two expressions equal to each other: 2x + 13 = 5x - 1 13 = 3x -1 3x = 14 x = 14/3 When x = 14/3, Quantity A and B are equal. When x < 3 as given in the question, Quantity A is greater than Quantity B.
35% of 900 285
a is greater 35% of 900 is greater than 33% (or 1/3) of 900, which is equal to 300. Since 300 is greater than Quantity B, and Quantity is greater than 300, answer choice [A] is correct.
a) x<(13*3)-30 b) 9
b is greater Quantity A simplifies to x < 9. Since Quantity B is equal to 9, it must be greater than Quantity A.
x>0 a) 4(x+8)/16 b) x/4+4
b is greater Simplify Quantity A by eliminating by a factor of 4 from the numerator and denominator: 4(x+8)/16 = (x+8)/4 = x/4 + 2 Now, we see Quantity B is greater
a + 5 = b a) a b) b
b is greater The given inequality means that b is 5 larger than a, so Quantity B is larger. Answer choice [B] is correct. a = 1 & b=6 a=0 & b=5 a=-1 & b=4
a) one third of five dozen b) 24
b is greater We know that 1 dozen equals 12, so Quantity A simplifies to: 1/3 × 5 × 12 = 20. Quantity B is 24, so it is greater than Quantity A
7x+43y=15 43x+7y=15 A) x b) y
both are equal We could determine exact values for x and y using substitution or elimination, but we don't actually need to solve the system of equations. Notice the symmetry of the two equations, and how x and y are merely swapped between the two equations. This means that the system of equations is satisfied when x = y.
a) one hundred hundreds b) one thousand tens
both equal Quantity A is equal to 100 × 100 = 10,000. Quantity B is equal to 1,000 × 10 = 10,000. Quantity A and Quantity B are equal. Answer choice [C] is correct.
a) .72 * 10^3 b) 72,000*10^-2
both equal We can evaluate scientific notation by shifting our decimal points. For Quantity A, we move our decimal place 3 places to the right, which equals 720. For Quantity B, we move our decimal place 2 places to the left, which equals 720.
a) log(x^3)+log(x/2)-log(8) b) 4log(x/2)
both equal We can simplify Quantity A to make it easier to compare to Quantity B by using several properties of logarithms. One way to simplify Quantity A is to first combine the first and third terms using the quotient rule: log(x^3)-log8=log(x^3/8) log(x^3/8) = log(x^3/2^3)= log(x/2)^3 We can then rewrite the simplified term using the power rule: log(x/2)^3=3log(x/2) Finally, we can add the result to our remaining second term: 3log(x/2)+log(x/2)=4log(x/2) Note that this is only one way to approach simplifying Quantity A. There are several valid approaches, but all lead to the same result.
a) area of a square with perimeter = 32 b) 64
both equal In Quantity A, since we know the perimeter of the square is 32, the length of one side must be 32/4 = 8. Its area is 8 × 8 = 64.
f ≠ 9 a) 3f+1 b) 27 (f is the function symbol)
cannot be determined The question tells us that f is not equal to 9, so it may be less than or greater than 9. We can look at two example cases. If f = 0, then Quantity A = 1, which is less than Quantity B. If f = 10, then Quantity A = 31, which is greater than Quantity B. Quantity A could be either less than or greater than Quantity B depending on the value of f, so answer choice [D] is correct.
for non-zero value of x & y: xy=z a) xz/y b) yz/x
relationship cannot be determined a) xz/y = x(xy)/y=x^2 b) yz/x = y(xy)/x = y^2 We are comparing x2 and y2. We only know x and y are inversely related to z, but we don't know if the absolute value of x is larger or smaller than y. Therefore, we cannot determine how these quantities compare.