GRE QUANT PRACTICE
2 − [1 − (1 − [2 − 3] − 2) + 3] =
-2 Start on the inside, and move outward: 2 - [1 − (1 − [2 − 3] − 2) + 3] = 2 − [1 − (1 − [−1] − 2) + 3] = 2 − [1 − (1 + 1 − 2) + 3] = 2 − [1 − (0) + 3] = 2 − [4] = −2
Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?
176 If the ratio of Yankees fans to Dodgers fans is 5 to 3, then we can say 5x = the original number of Yankees fans and 3x = the original number of Dodger fans. Then 22 Yankees fans change teams: (5x - 22)/(3x + 22) = 1/1 Let's simplify this: 5x - 22 = 3x + 22 2x = 44 x = 22 Now we plug in x to find the original number of fans: Original number of Yankees fans = 5x = 5*22 = 110 Original number of Dodger fans = 3x = 3*22 = 66 Total: 110 + 66 = 176
For all numbers x and y, the operation ϕ is defined by x ϕ y = (x + y)(x - y) + (y - x)(y + x) + xy. What is the value of √12 ϕ √3?
6 First of all, some algebraic simplification. Notice that (y - x) = -(x - y). Thus (x + y)(x - y) + (x + y)(y - x) + xy = (x + y)(x - y) - (x + y)(x - y) + xy = xy That's a BIG simplification. Now, plug in the numerical values: xy = (√12)(√3) = √36 = 6 Answer = 6
Compounding Interest Formula
A=P(1+r/n)^nt A=final amount P=initial principal balance R=interest rate N=number of times interest applied per time period t=number of time periods elapsed
(1*1/6)(1*11/21)
1*7/9 For multiplication, division, and raising to an exponent, mixed numerals are worse than useless. We have to change to improper fractions, do the calculation, and then change back to mix numerals to identify the answer.
The college that is drawing the most investment income in 2008 takes in approximately how much in mean total income per student in 2008? (Total income = tuition + investments)
The college that is drawing the most investment income in 2008 is the college all the way at the left side of the graph, with a very high "dot" and a very low "X." This sounds like some prestigious small school, perhaps an Ivy League school, with a low enrollment and mountains of money. It's dot, the investment income, is about $650 million, and the X, the tuition income, is about $160 million. The total income would be $650 million + $160 million = $810 million We want the income per student at this place. There are about 3500 students at this place. So: 810,000,000/3500 = 8,100,000/35 = 231428
If x is the greatest common divisor of 90 and 18, and y is the least common multiple of 51 and 34, then x + y =
This is like two little problems inside a big one. First of all, what is the Greatest Common Divisor or Greatest Common Factor of 18 and 90? Well, the largest factor of 18 is of course 18 itself. Notice that 18 is also a factor of 90, because 18 \times× 5 = 90 Since that's a common factor, and 18 doesn't have any bigger factors than itself, this must be the GCF, so x = 18. Now, what is the LCM of 34 and 51? Well, let's find the prime factorizations of both. Each one is simply a product of two prime numbers. 34 = 2 \times× 17 51 = 3 \times× 17 Any common multiple of 34 and 51 has to be divisible by these three prime numbers {2, 3, 17}. The LCM would simply be the product of those three prime numbers. LCM = 2 * 3 * 17 = 2 * 51 = 102 = y x + y = 18 + 102 = 120
The functions f(x) and g(x) are defined by f(x) = x2 - 1 and g(x) = 1 - 2x. Given that f(g(k)) = 3, which of the following could be the value of k?
k = 3/2 Let's start from what we are given f(g(k)) = 3 and substitute g(k) to try to create an equation and solve for k. It is g(x) = 1 - 2x so, g(k) = 1 - 2k. Substituting above yields f(g(k)) = 3 f(1 - 2k) = 3 (1 - 2k)^2 - 1 = 3 [ since f(x) = x^2 - 1 ] (1 - 2k)^2 - 4 = 0 (1 - 2k)^2 - 2^2 = 0 ((1 - 2k) - 2)((1 - 2k) + 2) = 0 [ difference of squares ] (- 2k - 1)(3 - 2k) = 0 From this we get - 2k - 1 = 0 or 3 - 2k = 0 Which gives k = -1/2 (not included in the options) or k = 3/2
For integers p and q, (2^p)(5^q) = 40000. Which of the following is the value of p + q?
10 40000 = (4)(10000) 40000 = (4)(10^4) 40000 = (2^2)[(2×5)^4] Remember that we can distribute exponents over multiplication. 40000 = (2^2)(2^4)(5^4) 40000 = (2^6)(5^4) Thus, p = 6, q = 4, and p + q = 10 Answer = (B)
Today, Bill is thirteen times as old as Pete. In nine years, Bill will be four times as old as Pete. How old will Pete be 2 years from today?
Let's let Pete's present age be x. In 9 years, Pete's age will be x+9. Bill's present age is 13 times Pete's present age, or 13x. If we add nine years on to that age, Bill's age in 9 years will be 13x + 9. So now we have Pete's age in 9 years and Bill's age in 9 years: P = x +9 B = 13x + 9 We are told that in 9 years, Bill's age will be 4 times Pete's age. So, 4P = B. If we plug in our ages above, we get: 4(x+9) = 13x+9 4x+36 = 13x+9 36 = 9x+9 27=9x x = 3 So, Pete's present age is 3. Remember that the question asks how old he will be 2 years from today, so our answer is 5.
Jack has 5 cats and 1 dog. If the dog's weight is 3 times the average (arithmetic mean) weight of the cats, then the dog's weight is what fraction of the total weight of all 6 animals?
3/8 The problem tells us that there are five cats. We don't know what they weigh, so let's just say that all weigh x pounds where x equals the average weight. (As you can see in the FAQ below, we don't know that the cats weigh the same, but it doesn't actually matter for this problem.) If the average weight of a cat is x pounds, then the weight of the dog is 3x: 5x + 3x = 8x Add them up and the total weight is 8x pounds. Now we just have to find the dog's weight as a fraction of the total weight: 3x/8x = 3/8
If x is a positive integer and x+2 is divisible by 10, what is the remainder when x2+4x+9 is divided by 10?
5 A quick way to solve this problem is to split the 9 into 4 and 5. That gives us x2 + 4x + 4 + 5. The reason to do this is that we can factor x2 + 4x + 4 into (x+2)(x+2): If (x+2) is divisible by 10, then (x+2)(x+2) must also be divisible by 10. If this doesn't feel intuitive, try some test numbers. If 8 is divisible by 4, then 64 is also divisible by 4. 6 is divisible by 3, and so is 36, etc. (x+2)(x+2) is divisible by 10, so (x+2)(x+2) + 5 must have a remainder of 5 when divided by 10. The remainder is right there in the arithmetic! Again, this is easier to see with an example. Pretend that x=8. (8+2)(8+2) = (10)(10) = 100. Then we have 100+5. 105 /10 = 10 remainder 5. The answer is 5
If the mean of list A is 6.8 and the standard deviation is 3.6, then how many elements of list A are within 1 unit of standard deviation of the mean? A = {2, 9, 2, 6, 9, 10, 7, 4, 5, 14}
7 We are given a set of numbers and the mean and standard deviation of that set: A = {2,9,2,6,9,10,7,4,5,14} mean=6.8 SD=3.6 When dealing with a set or list of numbers and properties such as mean and standard deviation, it is often useful to rewrite the numbers in order from least to greatest: A = {2,2,4,5,6,7,9,9,10,14} This will be especially helpful here because we are looking at how many elements of A are within one unit of standard deviation from the mean. We can translate this into a range with a maximum and minimum. Then, any element of A within this range will qualify!The maximum will be one unit of standard deviation above the mean: max = mean + units * SD max = 6.8 + 1 * 3.6 = 10.4 The minimum will be one unit of standard deviation below the mean: min = mean - units * SD min = 6.8-1 * 3.6 = 3.2 So the elements of A that are within one SD of the mean are in the range (3.2,10.4) (that is, greater than or equal to 3.2 and less than or equal to 10.4).When we apply this to our ordered version of A, we see that the 7 elements from 4 to 10 are in range.
x is between Note: Figure not drawn to scale
7 and 8 Since we know this is a right triangle, we can use the Pythagorean theorem: a^2 + b^2 = c^2 (sqrt(10))^2 + x^2 = (2*sqrt(15))^2 10 + x^2 = 2^2 * (sqrt(15))^2 10 + x^2 = 4 * 15 10 + x^2 = 60 x^2 = 60 - 10 x^2 = 50 x = sqrt(50) We know that 7 × 7 = 49, so the answer must be greater than 7, but just slightly.
Bills monthly cable bill is 2/7 of his monthly rent. If Bill pays $300 in cable each month, how much is his monthly rent?
Let c be Bills cable bill, Let x be Bills monthly rent. A. c = x(2/7) B. 300 = x(2/7) 300 = 2x/7 2100 = 2x 1050 = x
Kathy's salary is 3/7 Nora's salary, and is 5/4 Teresa's salary. Nora's salary is what fraction of Teresa's salary?
Let k be kathy's salary, let n be nora's salary, let t be teresa's salary. A. k = n3/7 B. k = t5/4 C. n3/7 = t5/4 12n = 35t n = 35/12
Seven years ago Bob was k times as old as Ann. If Ann is now 11 years old, what is Bob's present age in terms of k?
Let's let B = Bob's present age. We know that 7 years ago, Bob's age was k times Ann's age, and Ann is currently 11. So: B - 7 = k(11 - 7) B - 7 = 4k B = 7+4k
If $5,000,000 is the initial amount placed in an account that collects 7% annual interest, which of the following compounding rates would produce the largest total amount after two years?
The smaller the compounding period is, the greater the number of times the interest will be compounded. Of course, if we compound monthly instead of quarterly, then we are compounding by 1/12 of the annual rate each time, instead of 1/4. The number of times we compound goes up, but the percentage by which we compound each time goes down. Naively, you may think that those two would cancel out, but they don't. As discussed in this blog, as the compounding period gets smaller, the total amount of interest earned goes up. Therefore, we will get the most with the smallest compound period, daily. Answer = (D)
Increasing the original price of a certain item by 25 percent and then increasing the new price by 25 percent is equivalent to increasing the original price by what percent?
When dealing with percents when we're not given specific numbers, 100 is a good number to use. If we set the original price at 100, increasing it by 25% makes it 125. Increasing 125 by 25% adds 31.25, making the final number 156.25. That's an increase of 56.25% from the original. The correct answer is (E)
It takes 1 pound of flour to make y cakes. The price of flour is w dollars for x pounds. In terms of w, x and y, what is the dollar cost of the flour required to make 1 cake?
w/xy dollars We need the price of flour to make 1 cake. To find the price, we must first find out how much flour is needed to make 1 cake. Given: It takes 1 pound of flour to make y cakes. Let p pounds of flour be needed to make 1 cake. Then we have, 1 pound/y cakes = p pounds/1 cake We need p and therefore we should multiply both sides with 1 cake (and cross out common units from numerator and denominator just as we cross out numbers). Therefore, p = 1/y meaning 1/y pounds of flour is needed to make 1 cake. Now we need to find the price of 1/y pounds of flour, given x pounds costs w dollars. Let the price of 1/y pounds be d dollars. Then we have, w dollars/x pounds = d dollars/(1/y) pounds Multiplying both sides with 1/y pounds and simplifying we have d dollars = (w dollars)(1/y pounds)/x pounds = (w/y)/x dollars = w/xy dollars
| x-1 | = 4
x = 5, -3 or (1+4), (1-3)
The hypotenuse of a right triangle is 20 inches long, and one of the legs of the triangle is 10√3 inches long. If the measure of the smallest angle in the triangle is xº, which of the following expressions correctly represents the value of x?
x = 30 While this question may be tempting us into calculating, few if any calculations should be needed to solve here. This core concept that this question is testing us on is related to one of our special right triangles, the 30-60-90 triangle. Let's review the rule for the ratio of sides of a 30-60-90 triangle. Our ratio is as follows: x:\text{x}\sqrt{3}x√3:2x x represents the length of the shortest leg, opposite the 30º angle \text{x}\sqrt{3}x√3 represents the longer leg, opposite the 60º angle 2x represents the hypotenuse As we read this question, we should notice that we have one leg and a hypotenuse that meet the conditions of this ratio. Since this is a right triangle, we can calculate the value of the third leg using pythagorean theorem, but recognizing that we have two pieces that satisfy the 30-60-90 ratio should indicate that the other leg must fit the ratio as well. Having now determine that this is in fact a 30-60-90 triangle, we know that x, which represents the smallest angle of the triangle must equal 30, leading us to answer choice B. Takeaway: It is extremely important to feel comfortable recognizing and applying the 30-60-90 triangle ratio. In this problem, while we can calculate the third side using pythagorean theorem, this still will not help us in calculating angle measurements. In fact, without knowing this ratio, there is no simple way to obtain the angle measurements in this triangle.
If (3^y)^2 = (1/9)*3^(2x) where x and y are integers, what is the value of y in terms of x?
x-1 To get y in terms of x, we need to compare the exponents on both sides of the equation. In order to compare the exponents on both sides of an exponential equation, we must have them on the same base. For example, 2^x = 3^y gives us no information about the relationship between x and y. However, 2^x = 2^y tells us x = y. The left-hand side of the equation has a base of 3. To make the right-hand side have a base of 3 as well, we'll simplify that expression by making use of the quotient rule: (1/9)*3^(2x) = (1/3^2)*3^(2x) = (3^(2x))/(3^2)) = 3^(2x-2)(as per the quotient rule (x^a)/(x^b) = x^(a-b) The left-hand side can be simplified as follows: (3^y)^2 = 3 ^(2y) [as per the product rule (x^a)^b = x ^ab Therefore we have 3^(2y) = 3^(2x-2) Now that both sides have the same base, we can set the exponents equal to each other. Therefore we have 2y = 2x − 2 Dividing both sides by 2 we get y = x - 1
Nina has exactly enough money to purchase 6 widgets. If the cost of each widget were reduced by $1.25, then Nina would have exactly enough money to purchase 8 widgets. How much money does Nina have?
$30 Set T = Nina's money, and x = current price of 1 widget. We can set up: 6x = T 8(x - 1.25) = T Substitute in for T: 6x = 8(x - 1.25) 6x = 8x - 10 -2x = -10 x = 5 Plug back in: 6(5) = T T = 30
If the retail price of a shirt is R dollars, and the price including sales tax is T dollars then the sales tax, as a percent, is
(100(T-R))/R Let's walk through this scene with some actual numbers first. If the price of a shirt at a store is $10, then without tax, you would just pay $10 for that shirt. However, when the tax gets included, the final amount you have to pay increases. So, if the tax happened to be 20%, the final amount you would have to pay is 10 + 0.20 × 10 = $12. This means that the dollar amount of tax can be calculated by subtracting the original retail price from the final amount, and we'll use this later on to determine the tax as a percent: dollar amount of tax = final amount − original retail price Since the problem tells us that the "original retail price" is R and since the "final amount" is the same thing as "the price including sales tax" (which the problem represents with the variable T), we would rewrite the above equation as: dollar amount of tax = T−R Now, even though this problem is talking about "tax," it's really just another type of "percent increase" problem, and we can make use of the following formula to calculate the tax as a percent change: percent change = amount of change/starting amount*100 Or, in the context of this problem involving tax: tax as a percent = dollar amount of tax/original retail price*100 Replacing those words with our variables gives us: ((T - R)/R) *100 or (100(T-R))/R
At Joe's candy store, the total cost of 1 gumball and 1 lollipop is $0.74. The total cost of 1 chocolate bar and 1 lollipop is $0.92. The total cost of 1 gumball and 1 chocolate bar is $1.24. What is the cost in dollars of 1 chocolate bar?
.71 Start by naming variables that refer to the three items: Now translate each phrase from the prompt into an algebraic expression. The total cost of 1 gumball and 1 lollipop is $0.74: (We're converting dollars to cents so we can work with whole numbers to make the arithmetic easier. We can convert back to dollars at the end. This is optional, so just do whatever is comfortable for you!) The total cost of 1 chocolate bar and 1 lollipop is $0.92: C + L = 92 The total cost of 1 gumball and 1 chocolate bar is $1.24 G + C = 124 We have three variables and three equations, so we just need to solve for C to find the cost of 1 chocolate bar. Adding and subtracting two equations at a time is probably the fastest method of solving this system of equations, though you could also isolate a variable and plug it into another equation. We chose the adding and subtracting method. First subtract the G+C=124 and G+L=74 to get C-L=50. Then add this to C+L=92 to cancel out the L's and get the value for C: Remember that we converted dollars to cents, and 71 cents is $0.71. The answer is (E).
Sam was successful on 80% of his first 50 free throw attempts to start the basketball season. He was then successful on m of his next t free throw attempts. Which of the following represents Sam's overall proportion of successful free throws to free throw attempts?
(40+m)/(50+t) This question is asking us to create a proportion from the total number of free throws Sam made and the total number of free throws Sam attempted. Sam's made and attempted free throws are broken into two groups, so we will need to combine these groups to find Sam's overall totals. Let's start by looking at Sam's first 50 free throw attempts. We are told that he was successful on 80% of these first 50 attempts. This translates to 40 (80% of 50 = 40) successful free throws in 50 attempts. We are then told Sam was successful on m of his next t free throws. To find Sam's overall total of successful free throws, we must add 40 and m together, giving us 40+m. This represents all of Sam's successful free throws. To find Sam's overall total free throw attempts, we must add 50 and t, giving us 50+t. This represents all of the free throws Sam attempted. Our final step here is to use these two expressions to create a proportion representing "Sam's overall proportion of successful free throws to free throw attempts." To do this, we will put his successful free throw attempts in the numerator and his total free throw attempts in the denominator, giving us: (40+m)/(50+t)
If ak - b = c - dk, then k =
(b+c)/(a+d) To start, we are provided with the expression: ak - b = c - dk We need to solve for k. So, let's start by isolating terms with k in them on one side of the equation and terms without k in them on the other side of the equation. We can add "dk" to both sides, and add "b" to both sides. We obtain: ak + dk = c + b Now, on the left side of the equation we can do some manipulation to further isolate k. We see that "ak" and "dk" have a common factor of k, so we can pull that common factor out, like so: k(a + d) = c + b We know that k(a + d) = ak + dk because when we distribute the k throughout the parentheses on the left side of the equation, we obtain the original expression ak + dk. Now, we can divide both sides of the equation by (a + d), completely isolating k: k = (c + b) / (a + d) And, we have found the value of k!
If the average (arithmetic mean) of a and b is j, and the average of c, d, and e is k, what is the average of a, b, c, d, e and j ?
(k+j)/2 The mean = (sum of values)/(number of values). Remember that it's often much more useful to rewrite that equation in the form (sum of values) = (mean)(number of values). We can't add or subtract means, but we can add and subtract sums. For the first set, (a + b)/2 = j so, a + b = 2j For the second set, (c + d + e)/3 = k so, c + d + e = 3k Now, we want to take an average of the final six numbers average = (a + b + c + d + e + j)/6 average = ((a + b) + (c + d + e) + j)/6 average = (2j +3k +j)/6 average = (3j +3k)/6 average = (j + k)/6
If the line passes through the origin, what is the value of k?
-24.5 To find the value of k, we can solve for the equation of the line in y = mx + b form. To do so, we need to find the slope, m. The formula to find the slope is (y2 - y1)/(x2 - x1). So, let's plug in the two points whose values we know for certain: (-4, 7), which is given in the problem, and (0, 0), specified in the problem as "the origin." From there, we calculate (7 - 0)/(-4 - 0), which gives us -7/4 as the slope of the line, which we plug in for m in the formula y = mx + b. In this formula, b is the value of y when x = 0. Any line that passes through the origin will have a b value of 0, since the value of y is 0 when x is 0. So our equation is: y = -7/4x + 0 then plug in x to find y: y = -7/4(14) + 0 y = -24.5
Line k is in the rectangular coordinate system. If line k is defined by the equation 3y = 2x + 6, and line k intersects the x-axis at point (a,b), then what is the value of a?
-3 Any point that exists on the x-axis has a y-coordinate of zero. Since the question tells us that line k intersects the x-axis at the point (a, b), this means that b must equal zero. In other words, line k passes through the point (a, 0). We can now use this point (a, 0) as the (x, y) values in the equation of the line and solve for a: 3y = 2x + 63(0) = 2(a) + 60 = 2a + 6−6 = 2a−3 = a Therefore, (A) −3 is the correct answer.
5a + 4b = 8 4a + 5b = 10 What is the value of a² - b²?
-4 The expression a² - b² can be factored as (a + b)(a - b). So if we know, or can find, the values of (a + b) and (a - b), we can get the value of a² - b². The given equations are: 5a + 4b = 8 4a + 5b = 10 If you have done much work with linear equations outside of exam prep, you may expect that we will solve these by substitution to obtain the individual values of a and b. However, this problem—like many GRE problems—has an important feature that allows us to avoid this lengthy process. We do not need to solve for a and b directly. Notice that the coefficients of a and b are swapped between the two equations. When this happens, we can easily find the value of any multiple of a + b and any multiple of a - b by simply adding and subtracting the two equations. Here's how: Adding the two equations, we get 9a + 9b = 18 and dividing all terms by 9 gives a + b = 2 If we subtract the two original equations instead, we obtain a - b = -2 Putting these two expressions together, we can solve: a² - b² = (a + b)(a - b) = 2(-2) = -4
s1, s2, s3, s4, s5, .... In the sequence above, each term after the first term is equal to the preceding term divided by a positive number p, such that p > 1. If s3 = 24 and s5 = 6, which of the following is the value of s8?
.75 In sequence problems, we often have to work term-by-term: we need often the value of one term to get the value of the next. In this sequence, each term (after the first term) equals the preceding term divided by positive number p. We know that s3 = 24, so we can find an expression for the next term. s_4=24/p From that, we can find an expression for the next term: s_5= (24/p)/p = 24/p^2 = 6 Multiply both sides of that last equation by p^2. 24 = 6p^2 4 = p^2 2 = p That's the value of p! Each term is the previous term divided by two! a3 = 24 a4 = 12 a5 = 6 a6 = 3 a7 = 1.5 a8 = 0.75 Answer = .75
Machine A can make 350 widgets in 1 hour, and machine B can make 250 widgets in 1 hour. If both machines work together, how much time will it take them to make a total of 1000 widgets?
1 hour and 40 mins The first step here is to recognize that if machine A can make 350 widgets per hour, and machine B can make 250 widgets in 1 hour, then combined they can make 350+250=600 widgets in 1 hour. So working together, they can make 600 widgets per hour. Now, to solve this question we can use equivalent ratios. We can compare the number of widgets made to the number of minutes required to make those widgets. We already know that the two machines working together can make 600 widgets in 1 hour (or 600 widgets in 60 minutes). So: number of widgets made/minutes elapsed = 600/60 SO: 600/60 = 1000/t 600t = 60,000 t = 100 mins = 1 hour, 40 mins
If four numbers are randomly selected without replacement from set {1, 2, 3, 4}, what is the probability that the four numbers are selected in ascending order?
1/24 If we pick all four numbers in ascending order, that means that we select number 1 first and then number 2 second and then number 3 third and then number 4 last. So we need to calculate the individual probabilities of these events (taking into account the fact that there is NO replacement between selections), and then multiply them all together. For example, for the probability of selecting the 1 first, there's a 1 in 4 chance of selecting a 1 on the first draw. That 1 is then removed from the pool of numbers, leaving {2, 3, 4}. There is now a 1 in 3 chance of selecting the 2 from those three numbers. And so on. Here is the math: P(all four numbers in ascending order) = P(1first AND 2second AND 3third AND 4fourth) = P(1first) * P(2second) * P(3third) * P(4fourth) = 1/4 * 1/3 * 1/2 * 1 = 1/24
Car X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y, but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm, Car X reaches B, and at that moment, Car Y is still 35 miles away from B. What is the speed of Car X?
100mph At 1:30 pm, the two cars are in the same place: Car X is passing Car Y at that moment. One hour and 45 minutes later, at 3:15 pm, Car X is 35 miles ahead of Car Y. We can use a D = RT equation for the gap between the cars. We have a gap of D = 35 miles in a time of 1 hr 45 min, that is, T = 7/4 hours. R = D/T = 35m/(7/4)h = 35/1*7/4 = 20 mph That's the speed at which the gap is increasing, once Car X is in front of Car Y. In other words, Car X is going 20 mph faster than Car Y. RX = RY + 20 (Equation #1) Now, we have to use that curious fact: "Car X's speed is 1.25 times Car Y's speed." In other words, RX = 1.25(RY) (Equation #2) Substitute equation #2 into equation #1. 1.25(RY) = RY + 20 1.25(RY) - RY = 20 0.25(RY) = 20 1/4(RY) = 20 RY = 80 Car Y is going 80 mph, and Car X is going 20 mph faster than Car Y, so Car X is going 100 mph!
What is the sum of all integers from 45 to 155 inclusive?
11,100 This is ultimately the exact same process and thinking as above; however, it's all packaged in a quick and easy formula! An arithmetic series is a series where there is a constant difference between terms. For example, 45, 46, 47, ..., 153, 154, 155 is an arithmetic series where the constant difference is 1. The formula for the sum of an arithmetic series is: sum = n * (A1 + An) / 2, where n is the number of terms in the seriesA1 is the first term in the series (45)An is the last term in the series (155) The value for n in this problem is determined by using inclusive counting: "The number of integers from x to y inclusive is equal to y - x + 1". n = 155 - 45 + 1 = 111 Plugging in all the values, we get: sum = 111 * (45 + 155) / 2sum = 111 * 200 / 2sum = 111 * 100sum = 11,100
If 10^a x 10^b x 10^c = 1,000,000, and a, b, and c are different positive integers, then 10^a + 10^b + 10^c =
1110 When numbers with exponents are multiplied and they have the same base, add the exponents. So, 10^a×10^b×10^c = 10^a+b+c. Also, we know that 1,000,000 = 10^6, so 10^a+b+c=10^6. The same base 10 is raised to a power on both sides of the equation, so those powers are equal. That means that a+b+c=6. If a, b, and c are all different positive integers, then a=1, b=2, c=3. (It could have been a different combination, but the point is that 1, 2, and 3 are the numbers, whichever variables we assign them to.) Next we have to evaluate 10^a+10^b+10^c, so plug in the values we got for a, b, and c. Just like before, order doesn't matter when we add numbers, so it doesn't matter which variable gets which value, as long as we're adding 10^1, 10^2, and 10^3:
Which of the following is a root of the equation 2x2 - 20x = 48?
12 The equation 2x^2 - 20x = 48 is a quadratic, meaning we can set it equal to zero: 2x^2 - 20x - 48 = 0 We can factor out a 2, which leaves us with 2(x^2 - 10x - 24) = 0. Now we need to factor the part in parentheses the way we would any quadratic equation, giving us: 2(x - 12)(x + 2) = 0 Now we can conclude that either x - 12 = 0, or x + 2 = 0. This leaves us with the following when we solve: x = 12 x = -2 12 is one of our answer choices, so we choose (E).
Appleton's population is 400 greater than Berryville's population. If Berryville's population were reduced by 900 people, then Appleton's population would be 3 times as large as Berryville's population. What is Berryville's current population?
1550 Set A = Appleton's current population, and B = Berryville's current population. We know: A = B + 400 A = 3*(B - 900) Substitute in for A: B + 400 = 3*(B − 900) B + 400 = 3B − 2700 400 = 2B − 2700 3100 = 2B 1550 = B
If the hypotenuse of an isosceles right triangle has length of 8, then the area of the triangle is
16 This triangle is an isosceles right triangle, a.k.a the 45°-45°-90° triangle. It can be helpful to remember that the ratio of the sides are 1-1-√2. Even if you don't remember that, let AC = BC = x. Then, using the Pythagorean Theorem: x^2 + x^2 = 82 2x^2 = 64 x^2 = 32 There's no reason to find the square root, because we are looking for the area. Area = (0.5)bh = (0.5)(BC)(AC) = (0.5)(x)(x) = (0.5)x^2 = (0.5)(32) = 16
A is the center of the circle, and the length of AB is 4√2. The blue shaded region is a square. What is the area of the shaded region?
16(4−π) Start by drawing segments from point A that are perpendicular to the bottom and to the right side: Notice that these points, with A & B, form a square that is a quarter of the big square, and AB = 4√2 is the diagonal of this square. The ratio of the diagonal of a square to the side of a square is √2: 1 This is the same as the ratio of the hypotenuse to a leg in a 45-45-90 triangle. If the diagonal is AB = 4√2, then the side of that little square is 4. That one number unlocks this problem. First of all, the side of that little square is half the side of the big square, so the big square has a side of s = 8 and an area of s^2 = 64. The side of that little square is also the radius of the circle, so r = 4 and the area of the circle is A = πr2 = 16π blue shaded region = (square) - (circle) blue shaded region = 64 - 16π = 16(4 - π)
If the sum of two numbers is 6, and the sum of their reciprocals is 15/8, what is the product of the two numbers?
16/5 In this question we are given information about two numbers. So let's let x = one of the numbers, and y = the other number. Now, first we are told that the sum of the two numbers is equal to 6. So we can write: x + y = 6 We are also told that the sum of the numbers' reciprocals is equal to 15/8. So we can write this equation: (1/x) + (1/y) = 15/8 We now have two equations and two unknowns, and our goal is to find the value of xy. There are several ways to accomplish this. Let's take the second equation, and combine the two fractions on the left hand side. To do this, we'll need a common denominator. So we can take the first fraction and multiply the top and bottom by y, and take the second fraction and multiply the top and bottom by x: (y/xy) + (x/xy) = 15/8 Now that both of the fractions on the left hand side have the same denominator, we can combine them to get the following: (y +x)/xy = 15/8 At this point, we should recognize that we have the sum of x and y in the numerator, which we know is equal to 6 (we know this because of the first equation: x + y = 6). So we can replace x + y with 6: 6/xy = 15/8 Now, all we need to do is solve for xy. To do this, we can cross multiply: 6 × 8 = xy × 15 48 = 15 xy 48/15 = xy This can be simplified to: 16/5 = xy So the answer is C. 16/5
In the xy-coordinate system, a circle with radius √30 and center (2, 1) intersects the x-axis at (k, 0). One possible value of k is
2 + √29 Notice that we have given letter names to the points: the center of the circle (2, 1) is A; the point on the x-axis directly beneath the center, (2, 0), is C; and the unknown x-intercept (k, 0) is B. Notice that ABC is a right triangle, and notice that we know two sides: we know hypotenuse AB = √30 and we know that shorter leg AC = 1. Thus, we can find the length of the third side with the Pythagorean Theorem. Remember that whenever you need to find distances in the x-y plane, the Pythagorean Theorem is the most direct way to solve: it's far more direct than applying the Distance Formula. AC^2 + BC^2 = AB^2 1^2 + BC^2 = (√30)^2 1 + BC^2 = 30 BC^2 = 30 - 1 = 29 BC = √29 That's the distance from (2, 0) to (k, 0). Remember that if we have two horizontal points, we can find the distance simply by subtracting. We can set this subtraction equal to the distance we just found. k - 2 = √29 k = 2 + √29
If 5x - 3y = 7 and 2y - 4x = 3, then 2x - 2y =
20 We can solve this system of equations by using either the substitution method or elimination method. However, something interesting to note is that the problem asks us to solve for a very specific expression (2x - 2y) instead of something more commonplace like just the variable x or the variable y. This is most often a hint that we can directly solve for that specific expression or something close to it. Using the elimination method, let's first organize the terms in the two equations to make them easier to add together: a. 5x - 3y = 7 b. -4x + 2y = 3 Why are we going to add the two equations together? Well, by doing so, we will end up with something close to the very specific expression that the problem wants us to solve for: 5x - 3y = 7 + -4x + 2y = 3 --------------- x - y = 10 Then: 2x-2y = ? 2(x-y) = 2(10) 2x-2y = 20
What is the average (arithmetic mean) of all multiples of 10 from 10 to 400 inclusive?
205 We're asked to find the average value of the multiples of 10 from 10 to 400. We should recognize right away that this is a list of evenly spaced numbers.We can apply an extremely useful property of lists of evenly spaced numbers: Any two elements that are equally far from the median (middle) of the list will have a mean value equal to the mean of the entire list! Using sequence notation, the simplest example is taking the first and last elements of a list with (a1+an)/2 = (a1+a2+.....an)/N Let's apply this property with our list of multiples of 10. The outermost pair of numbers consists of the first and last elements, 10 and 400. Their average is: M=(10+400)/2 = 205 This is, in fact, our answer! The average of the entire list is 205. We can confirm it again by testing the next pair of elements, 20 and 390. Their average is: M=(20+390)/2 = 205
If sqrt{2x^2+2xy+13y^2}=x+3y, then x =
2y First, we can square both sides. This will remove the square root on the left side. We can then FOIL the right hand side of the equation. This leaves us with a quadratic on each side of the equal sign. Next, we can subtract the terms on the right hand side from the terms on the left (combine like terms). This leaves us with zero on the right hand side of the equation. This is an important strategy that will make it easier for us to solve the quadratic equation on the left hand side. Next we factor the quadratic expression on the left hand side, resulting in (x-2y)(x-2y) = 0. We know that in order for the left hand side to be equal to zero, it must be the case that x-2y = 0. We can add 2y to each side, to get x = 2y.
What is the value of x?
3 So, from our diagram, we can gather that our triangle is a right triangle. That means we can relate the three sides together with the Pythagorean theorem and use it to solve for the value of x. :) The Pythagorean theorem equals: a2 + b2 = c2 Now, let's assign the sides: Side "c" should be the hypotenuse of the right triangle. That means c = 2x + 7. "a" can be x + 2 and "b" can be 3x + 3. Now, let's plug them into the Pythagorean theorem. (x+2)2 + (3x+3)2 = (2x+7)2 We can use the FOIL method to expand our squares. If you aren't comfortable with "FOILing", I highly recommend watching the lesson video (you can find it below). x2 + 4x + 4 + 9x2 + 18x + 9 = 4x2 + 28x + 49 Now, combine like terms: 10x2 + 22x + 13 = 4x2 + 28x + 49 Move all terms to one side: 6x2 - 6x - 36 = 0 We can factor out a 6 here: x2 - x - 6 = 0 Factor the quadratic: (x - 3)(x + 2) = 0 From there, we'll get that x = 3, x = -2. 3 must be our answer because we can't have 0 be the length of a side of the triangle (one side of the triangle is x + 2, if x = -2, then -2 + 2 = 0).
When positive integer k is divided by 5, the remainder is 2. When k is divided by 6, the remainder is 5. If k is less than 40, what is the remainder when k is divided by 7?
3 The first thing we should understand is that K has three requirements: - When positive integer k is divided by 5, the remainder is 2. - When k is divided by 6, the remainder is 5. - k is less than 40. The possible values of k according to that first rule are all multiples of 5, plus 2. 5*0 + 2 = 2 5*1 + 2 = 7 5*2 + 2 = 12 5*3 + 2 = 17 ... We don't need to do much math, really; we just need to see a pattern: we're looking at every integer which ends in 2 or 7. In a perfect world, this might only take a few seconds to see. In reality, that exploration phase might take closer to a minute. Once we know that, we simply need to follow through on the second requirement. In order to do that, we should subtract the remainder of 5 from each number, then check if it's divisible by 6. Is 5 less than 2 divisible by 6? That's 2 - 5 = -3, so no. Is 5 less than 7 divisible by 6? That's 7 - 5 = 2, so no. Is 5 less than 12 divisible by 6? That's 12 - 5 = 7, so no. Is 5 less than 17 divisible by 6? That's 17 - 5 = 12, so yes! We can check the rest of the values up to k = 40 to find that 17 is the only possible value. Alternatively, a pattern might emerge to make this even quicker--each number in the list is 5 greater than the previous number, so we just need to look one left of the number and see if that's divisible by 6. In the answer video, you can see that 17 is next to 12, and 12 is divisible by 6. Therefore, k is 17.
If a triangle in the xy-coordinate system has vertices at(-2 , -3), (4, -3) and (28, 7), what is the area of the triangle?
30 To find the area of the triangle is, the formula is: Area of triangle = ½ (base) x (height) In order to find the base, we need to find the distance between (-2, -3) and (4, -3). Luckily, both points lie on the same y-coordinate. The segment between them is therefore a horizontal line. That length can be the base. The distance between (-2, -3) and (4, -3) is a horizontal distance of 6. We can ignore the y-coordinates since they are the same. 4 - (-2) = 6 So the base is 6. Secondly, in order to find the height we have to determine the distance from the base to the third vertex (28, 7). Well, the base is on the horizontal line y = -3. The point (28, 7) is shifted way over to the right, but that doesn't matter: a point with y = 7 is 10 above the line y = -3, because 7 - (-3) = 10 Therefore, the height is 10. So we have now calculated a base of 6 and height of 10. If we plug that into our formula, we get: Area of triangle = ½ (6) x (10) = 30
At the moment there are 54,210 tagged birds in a certain wildlife refuge. If exactly 20 percent of all birds in the refuge are tagged, what percent of the untagged birds must be tagged so that half of all birds in the refuge are tagged?
37.5% The fastest way to solve this problem doesn't involve any arithmetic. We can do this short cut because the problem asks us for a percentage. It doesn't actually matter how many birds there really are. Pretend that there were only 10 birds. If 20% are tagged, then two are tagged. If we want half of the birds to be tagged, that would mean we would need to tag three more to get a total of five tagged birds. Here's the tricky part. The problem asks us what percentage needs to be tagged of the remaining untagged birds. That means that we need to tag three of the remaining eight (because two were already tagged, leaving eight untagged). That's three over eight, or: 3/8 = .375 or 37.5%
k is a positive number. If k is twice its reciprocal, and j is twice k, then jk =
4 Let's write some equations based on the information we've been given. We know that k is twice its reciprocal. We can write this as: k = 2 × 1/k We can simplify the right hand side, and then multiply both sides of the equation by k: k = 2/k k^2 = 2 This means that k is equal to either √2 or −√2. However, the first sentence in the question tells us that k is a positive number. That means that k cannot be equal to −√2, so it must be the case that k = √2 The question also tells us that j is twice k. So: j = 2k We know that k = √2, so we can write: j = 2 * √2 We now have values for j and k, so we can find the value of jk: j × k = (2 * √2) *√2 j × k = 4 So jk = 4, and the answer is B.
In order to qualify for the year-end tennis tournament, Sam must win at least 60 percent of his matches this year. Presently Sam has won 14 of his 18 matches. Of Sam's 13 matches remaining in the year, what is the least number that he must win in order to qualify for the year-end tournament?
5 We are told that Sam has played 18 matches, and he has 13 matches remaining. So we know that he has 31 matches in total. In order to qualify for the tournament, Sam needs to win 60 percent of those 31 matches. 60% of 31 = 0.6 \times× 31 = 18.6 So we know that he needs at least 18.6 wins in order to qualify for the tournament. So far, he already has 14 wins. So he needs an additional 4.6 wins to qualify (18.6 - 14 = 4.6). Since the number of wins must be an integer, we know that Sam needs at least 5 more wins to qualify.
A necklace is made up of three different colored beads: red, blue, and green. If the ratio of blue to red beads is 1:3 and red to green beads is 2:3, what is the lowest number of beads that could be on the necklace if the total number of beads on the necklace is greater than forty?
51 First of all, notice that even if you don't have a clue how to do this, this problem is ripe for guessing. We want a number that is greater than 40, so even if we can't do anything else, we can eliminate (A) and (B), and then guess from the three remaining answers. We need to put all the information together into one combined ratio, so we can figure out something about the total number of bead. Blue to red B : R = 1 : 3 Red to green R : G = 2 : 3 The problem here is that the common element, red, is represented by two different numbers in the two ratios: it's currently represented by 3 in the first ratio and by 2 in the second. To combine the two ratios, we would need these numbers to be the same. Remember that a ratio is essentially a fraction, and we always can multiply the numerator and denominator of a fraction by the same number and the value of the fraction stays the same. We can use that trick here: if we multiply both the values in a ratio by the same number, the value of the ratio stays the same. We want to be strategic, to get the two "red" numbers to the same value. The LCM of 2 and 3 is 6, so that will be a target value. The "red" value in the first ratio is 3, so we will multiply both values in that ratio by 2. The "red" value in the second ratio is 2, so we will multiply both values in that ratio by 3. Here are the results of those multiplications: Blue to red (multiply each by 2) B : R = 2 : 6 Red to green (multiply each by 3) R : G = 6 : 9 Notice that the values of the individual ratios didn't change, because 1:3 and 2:6 are the same ratio, and 2:3 and 6:9 are also the same ratio. In this form, though, both "red" values are the same, 6. This allows us to combine the two separate ratios into one big ratio. Blue to red to green B : R : G = 2 : 6 : 9 OK, now we have a single combined ratio. One of the things this means is that, for some positive integer n, we could say: the number of blue beads = 2n the number of red beads = 6n the number of green beads = 9n Those are the only three colors on the necklace, so we can add those three to get the total number of bead. total number of beads = 2n + 6n + 9n = 17n Very interesting: we know n is a positive integer, so the total number of beads must be a multiple of 17. What are the multiples of 17? Well, 17 × 1 = 17, but that's too low. Then, 17 × 2 = 34, which is an answer choice, but it's not greater than 40. The next is 17 × 3 = 51: this is an answer choice, and this is the lowest multiple of 17 that is greater than 40. Of the answer choices listed, this is the only multiple of 17 greater than 40. Thus, the total number of beads = 51
Joan has 100 candies to distribute among 10 children. If each child receives at least 1 candy and no two children receive the same number of candies, what is the maximum number of candies that a child can receive?
55 We want to know the maximum number of candies that one child could receive. This means we essentially want one child to get as many candies as is possible given the limitations above. So, we start with 100 candies. Let's start by satisfying our second condition, which is that all children must have at least 1 candy. Once we give all 10 children 1 candy, we have 90 remaining, because we just distributed 10. But we can't give the remaining 90 to the one child to maximize his share. We haven't yet met our first condition, which is that each child receive a different number of candies. Right now, they all have 1 candy. We can fix this by giving each child a different number of candies, but we can still maximize the number given to one student by using the smallest numbers possible, starting from 1 and moving upwards. Like so: Child 1: 1 candy Child 2: 2 candies Child 3: 3 candies Child 4: 4 candies Child 5: 5 candies And so on, until we find that child 9 has been given 9 candies. We now want to subtract all of the candies assigned to the first 9 children from our 100 candies, and give the rest to child 10. Like so: 100 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 = 55 candies Therefore, the maximum number of candies that child 10 can receive is 55.
Three friends are buying a gift for a friend. Declan contributes 4 dollars more than 1/4the cost of the gift, Ed contributes 1 dollar less than 1/3 the cost of the gift, and Frank contributes the remaining 22 dollars. What is the cost of the gift?
60 Let's begin by defining 'x' as the total cost of the gift. If Declan "contributes 4 dollars more than 1/4 the cost of the gift," then: Declan's contribution = (x/4) + 4 If Ed "contributes 1 dollar less than 1/3 the cost of the gift," then: Ed's contribution = (x/3) - 1 If Frank "contributes the remaining 22 dollars," then: Frank's contribution = 22 These three values must sum to the total cost of the gift, so we can write: ((x/4) + 4) + ((x/3) - 1) + 22 = x Combining like terms on the left side yields: (x/4) + (x/3) +25 = x From here, we can clear the denominators if we multiply both sides of the equation by the LCM of 4&3 -- 12. 12((x/4) + (x/3) +25) = 12x 3x + 4x + 300 = 12x 7x + 300 = 12x 300 = 12x - 7x 300 = 5x 60 = x
A certain school district has specified 5 different novels and 4 different non-fiction books from which teachers can choose to assemble a summer reading list. Each summer reading list must have exactly 3 novels and 2 non-fiction books. How many different summer reading lists are possible?
60 This is a question in which we have to count all possible ways to complete a process in phases. This means that we have to count the number of ways that each phase is done with and then, according to the Fundamental Principle of Counting, multiply them to find the total number of ways. In here, the process is assembling the reading list and this process is split into two phases: Selecting the 3 novels and selecting the 2 non-fiction books. So, we need to count all the ways that we have to select 3 novels out of 5 and to select 2 non-fiction books out of 4. As there are two formulas that we use in counting problems, the formula for permutations when we are interested about the order of the selected objects and the formula for combinations when we do not care about the order, we first need to determine if the order matters here. Let's think of the nature of the problem for a minute: We want to choose 3 novels out of 5 available and put them in the reading list. Do we care about their order that will be selected? Is there any special reason for one of them to be first, another one second and the last one third. No, we do not care about that. We just want the 3 novels to be in the reading list that we make, we do not care what is first, second, third, nor do we make any suggestions about the order that these should be read by the students. So, the order does not matter and thus we will use the combination formula. The same for the non-fiction books, we do not care about their order too. So, for novels we use the formula for combinations: Therefore, the total number of ways to complete the reading list: 6 ⨉ 10 = 60
Sue planted 4 times as many apple seeds as she planted orange seeds. 15 percent of the apple seeds grew into trees, and 10 percent of the orange seeds grew into trees. If a total of 420 apple trees and orange trees grew from the seeds, how many orange seeds did Sue plant?
600 Set A = number of apple seeds, and O = number of orange seeds. We know: A = 4*O number of apple trees = 0.15*A number of orange trees = 0.1*O 0.15*A + 0.1*O = 420 --> 15*A + 10*O = 42000 Substitute A = 4*O: 15*(4*O) + 10*O = 42000 60*O + 10*O = 42000 70*O = 42000 O = 600
How many three-digit numbers are there such that all three digits are different and the first digit is not zero?
648 First digit: 9 options (1, 2, 3, 4, 5, 6, 7, 8, 9). The first digit cannot be zero. Second digit: 9 options (0-9, minus the number that we chose for the first digit). Third digit: 8 options (0-9, minus the numbers that we chose for the first and second digits). 9 x 9 x 8 = 648
Point A in the xy-coordinate system is shown below. Given two other points B (4a, b) and C (2a, 5b), what is the area of triangle ABC in terms of a and b?
6ab Think about what we know. We know that both a and b are positive, because point A is in the first quadrant. Point B is at (4a, b). It has the same y-coordinate as point A, so the line segment AB must be horizontal. It is is a horizontal line from x = a to x = 4a. To find distance along a horizontal or vertical line in the x-y plane, we simply subtract. Length of AB = 4a - a = 3a That's the length of the base of the triangle. Now consider point C. Point C = (2a, 5b) has an x-value that is between A & B, and it's above them. A & B are at a height of y = b, and C is at a height of y = 5b. To find the distance, again, we simply subtract: height = 5b - b = 4b That's the height of the triangle. We have base and height, so we can find the area. Area = 1/2B*H Area = 1/2(3a)(4b) Area = 1/2(12ab) Area = 6ab
What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15?
6x^8y^15 First of all, the GCF of 18 and 24 is 6, so that has to be the numerical coefficient. One common mistake folks make: when trying to figure out, say, the GCF of x^8 and x^12, people mistakenly think this is the same as finding the GCF of 8 and 12. That's totally incorrect. The term x^8 has eight factors of x, and the term x^12 has twelve factors of x, so the most they have in common is eight factors of x --- in other words, x^8 is the common factor. As a general rule, if p is any variable, and if b > a, then the GCF of p^a and p^b has to be the lower power, p^a. Thus, the GCF of y^20 and y^15 is y^15. Putting all this together, the GCF of the two given expressions is 6x^8y^15. Answer = D
(2√3+√5)(2√3−√5)=
7 Here's the expression we are asked to evaluate: (2√3 + √5) (2√3 - √5) Notice that the two binomials we want to multiply together are almost identical. The only difference between them is that the first has a plus sign, while the second has a minus sign. This means that we can use the following rule: (a + b) (a - b) = a^2 - b^2 In the expression we are asked to evaluate, a = 2√3, and b = √5. So we have to square (2√3), and square √5, and subtract the second from the first: (2√3 + √5) (2√3 - √5) = (2√3)^2 - (√5)^2 = 12 - 5 = 7 So our answer is B.
A box contains 4 red chips and 2 blue chips. If two chips are selected at random without replacement, what is the probability that the chips are different colors?
To get the probability that the two chips are different colors, we want to consider the probability of getting first a red chip then a blue chip, or first a blue chip then a red chip. The chips are not replaced, so the second chip we pick will be out of 5 total chips, not 6. Probability of getting red first then blue = (4/6)(2/5) Probability of getting blue first then red = (2/6)(4/5) Now we add them together, since either scenario would give us two chips that are different colors: (4/6)(2/5) + (2/6)(4/5) = 8/30 + 8/30 = 16/30 = 8/15 Answer: (B)
If 3x < 2y < 0, which of the following must be the greatest?
We know that both 3x and 2y are negative numbers and that 3x is "more negative"—i.e., it has a greater absolute value. Let's say 2y = -4 and 3x = -9. These satisfy the original inequality. (A) 2y - 3x = (-4) - (-9) = -4 + 9 = +5 (B) 3x - 2y = (-9) - (-4) = -9 + 4 = -5 (C) -(3x - 2y) = -[(-9) - (-4)] = -(-9 + 4) = -(-5) = +5 (D) -(3x + 2y) = -[(-9) + (-4)] = -(-13) = +13 (E) 0 If one choice must be the greatest, then it will be the one that is the greatest for any numerical choice. This single numerical choice is enough to demonstrate that choice (D) is the greatest. Answer = (D)
