GRE MATH
Solve for w: 22w = 8^w ‐ 5
We must first obtain the same base on both sides. Convert the 8 into a power of 2: 22^w = (23)^w - 5 22^w = 23^w - 15 Now that the bases are equal, we can set the exponents equal to each other: 2^w = 3^w - 15 → w = 15. ManhattanGMAT Equations, Inequalitiesv
If 2 is one solution to the equation x2 - 9x + c = 0, where c is a constant, what is the other solution?
Work backwards - even though we do not know the value of c, since 2 is one solution, we know the factored form of the quadratic is (x - 2)(x − ?). We also know that the two numbers in parentheses must add to −9. Therefore the factored form is (x - 2)(x - 7) and the other solution is x = 7. This problem can also be solved by plugging x = 2 into the original equation and solving
If an integer that is divisible by 6 is squared, then which (nonzero) onedigit integers is this squared result definitely divisible by?
Answer: 1, 2, 3, 4, 6, and 9 Call the original integer n. Since n is divisible by 6, we can say n = 6m, where m is any integer. Squaring n yields n2 = (6m)2 = 36m2. Since 36 is divisible by 1, 2, 3, 4, 6, and 9, they are all factors of n2 as well. Any combination of 5, 7, and/or 8 may also divide n2, but we can't say for sure whether they do without knowing what m is.
Simplify 5 85 .
Answer: 17 5 When a square root lurks in the denominator, we can rationalize the denominator by multiplying by the appropriate form of 1 - in this case, , and 85 divided by 5 is 17, so the simplest form is 17 sqrt 5 .
What is the greatest number of primes that could be included in a set composed of four consecutive integers? Name the elements of the set.
Answer: 3, in the set {2, 3, 4, 5} Any set composed of four consecutive integers will contain two even and two odd integers. Since 2 is the only even integer that is prime, no such sets can have four primes, and sets that do not contain 2 can have, at most, two primes. The only set with three primes is {2, 3, 4, 5}. Why isn't {1, 2, 3, 4} acceptable as another solution to this question?
Is the statement sufficient? A group of rabbits multiplies at a constant rate. By what factor does its population increase every day? (1) The population grows from 200 to 5,000 in one week.
Answer: Sufficient Remember, we just need to know that we can calculate the rate of growth. They've given us the initial and final numbers of rabbits, as well as the time span. That is enough to calculate the rate of growth. For example, in 7 days, the population increases by a factor of 5,000/200 = 25. In one day it increases by a factor of 7 25 . (We do not, however, need to actually do this calculation on a Data Sufficiency question!)
What is 35% of 120?
Although 35% of a number is not easy to find without some calculation, 10% and 5% are usually easier. 35% = 3 × 10% + 5% 10% of 120 is 12 and 5% is half of 10%, so 5% of 120 is 6. 3 × (12) + 6 = 36 + 6 = 42
What error has been made? x2 36 x2 36 x 6
Answer: Remember, x 2 x . So after we take the square root of both sides, we have x 6 . This gives two possibilities: x = 6 or x = −6. Alternatively, simply recall that there are always two possible solutions in exponential equations with an even exponent. Thus when x2 = 36, x = 6 or -6. ManhattanGMAT Equations, Inequalities
Dinner cost $230 including a 15% tip. How much was dinner without the tip?
Answer: $200 If $230 includes the cost of the dinner plus an additional 15%, then it is 115% of the cost of the dinner, so 230 = (115/100)x. 100 230 115 100 115 100 115 x 200 = x
Calculate 5 16 4 .
Answer: 32 Using the rules of exponents, 5 1 1 5 16 4 165 4 16 4 . Since it is easier to calculate 1 16 4 than it is to calculate 165, the latter representation will be easier to simplify. 1 16 4 = 2, and 25 = 32.
Factor: x2 - 11x + 30 = 0
Answer: (x - 5)(x - 6) = 0 Since the last sign is positive, set up 2 parentheses with the sign of the middle term. (x - )(x - ) Find two numbers that multiply to 30 and add to 11 and place them in the parentheses. (x - 5)(x - 6) What values for x solve the equation?
What is the minimum value of f(x) = -5 + (x + 7)^2, and at what value of x does it occur?
Answer: minimum value = 7, x = 5 The squared expression will always be non-negative, so to make f(x) as small as possible, make the squared expression as small as possible - set it equal to zero. If x + 7 = 0, x = -7. Once you have the x value, plug it back into the original equation to solve for the minimum value. f(x) = -5 + (0)2. Therefore, the minimum value is -5. Remember, f(x) and y are synonymous. ManhattanGMAT Equations, Inequalities, & VICs
What are the roots of x3 - x = 0?
Answer: x = 0, −1, or 1 Factor the equation, since we already have 0 on one side: x ( x2 - 1 ) = 0 x ( x + 1) (x - 1) = 0 x = 0, −1, or 1. The temptation is to move x to the other side and divide both sides by x, leaving us with x2 = 1. Avoid dividing away a variable unless you know it does not equal 0.
Simplify: ( 2 + 3)( 2 − 3)(2 − 3 )(2 + 3 )
Answer: −7 Remember, (a + b)(a - b) = a2 - b2. Therefore, our expression is equal to: (2 - 9) × (4 - 3) = (−7)(1) = −7
Is the statement sufficient? Is x/y even? 1) x and y are both even.
Even numbers can be represented as 2m or 2n, where m and n are integers. (Think about why this is.) Dividing would give (2n)/(2m), or just n/m. Not only can it not be determined whether this result is even (e.g., x = 40 and y = 4) or odd (e.g., x = 44 and y = 4), we cannot even determine that it will be an integer! (e.g, x = 42 and y = 4.) The statement is insufficient.
Which fraction is greater in each pair? 5 8 or 6 10 ? 132 300 or 89 170 ?
For the first set of fractions, we can cross multiply and compare the numerators. 8 5 10 6 50 is greater than 48, so 8 5 is greater. For the second set of fractions, estimate. 132 300 is less than half, whereas 89 170 is more than half. 89 170 is thus larger.
Is the statement sufficient? Is xy < 25? (1) x and y are both less than 5.
We cannot simply multiply x < 5 and y < 5 to get xy < 25. If x and y are both negative, xy could be greater than 25. Example: (−10)(−4) = 40. Could we multiply x > 5 and y > 5 to get xy > 25?
Bottle 1, with capacity x, is half full of water. Bottle 2, with capacity y, is one sixth full. If Bottle 2 is three times the size of Bottle 1 and the contents of Bottle 1 are emptied into Bottle 2, how many liters of water, in terms of y, are in Bottle 2? a) 1 2 y b) 1 6 y c) 2 3 y d) 1 3 y e) 5 6 y
When problems involve many fractions and no specific quantities, it is best to pick numbers that are multiples of all the denominators in the problem. The least common multiple of 6 and 2 is 6. Thus, let the capacity of Bottle 1 = 6 and the capacity of Bottle 2 = 18. Bottle 1 holds 3 liters and bottle 2 holds 3 liters. Bottle 1 is dumped into Bottle 2, which then contains 6 liters. Test each answer choice with y = 18 and notice that d) is the solution, since 1(18) 6.
How would you factor each of the following expressions? a) x5 - x3 b) 48 + 49 + 410 c) mn−2 - 3mn + 4mn+1
a) The GCF is x3, the smaller power. x3(x2 - 1) = x3(x + 1)(x - 1). b) The GCF is 48. 48(1 + 41 + 42) = 48(21). c) The smallest power of m is the GCF. Here it is mn−2: mn−2(1 - 3m2 + 4m3).
Set up an appropriate equation to describe the given scenario: The elasticity (e) of a material is directly proportional to the square of its density (d) and inversely proportional to the cube of its mass (m).
Answer: 2 3 kd e m A constant k is used in expressions of direct or inverse proportionality. e is directly proportional to d2 , which means e = kd2. e is also inversely proportional to m3, so e = k/m3. Putting these two equations together, we get 2 3 e kd . m Note that k in the final equation must be the product of the k constants in the first two equations, but since k could be any value, we can repeat the use of k for simplicity.
Calculate (-1)789.
Answer: -1 Since (-1) (-1) = 1, -1 raised to any even power is 1. If you multiply by -1 one more time, you end up with -1, so -1 raised to any odd power will equal -1. 789 is an odd number, so (-1)789 = -1.
What is the 25th term of this sequence? Sn = Sn-1 - 10 and S3 = 0.
Answer: -220 First, we need to convert the recursive sequence definition provided into a direct sequence formula. Each term is 10 less than the previous one. Therefore Sn = -10n + k, where k is some constant that we must determine. Use S3 to find a value for k: 0 = -10(3) + k. Thus, k = 30, so Sn = -10n + 30. Now we plug in 25 for n: S25 = -10(25) + 30 = -220. Alternatively, we could plug in 0 for S3 and find that S4 = -10, S5 = -20, S6 = -30, etc. Thus, S25 = -220.
What is the units digit of (27)(74)(56)?
Answer: 0 Although you could multiply everything out, that is too timeconsuming. Notice that 2 × 5 = 10. That means the units digit is 0. Anything multiplied by 0 is 0, so we know that the units digit of the final product will be 0
(104)(0.000001) =
Answer: 0.01 or 102 An easy shortcut when dealing with powers of 10 is to simply move the decimal over the same number of units as the exponent. In this case, the exponent is 4, so we move the decimal to the right 4 places. Alternatively, .000001 can be rewritten as 106, and (104 )(106) = 102 .
(57)(10 )(0.001) (10 )(10 )
Answer: 0.57 First, change 0.001 to 10 3 . Now, combine the terms on the top and the bottom. (103)(10-3) = 100 = 1 (104)(10-2) = 102 We are left with 2 (57)(1) (10 ) . To divide by 102, just move the decimal to the left 2 places. 57 becomes 0.57.
What is the value of x? 5^3x =5^7x‐4
Answer: 1 Since the bases are equal, we can simply set the exponents equal to each other. 3x = 7x - 4 4 = 4x 1 = x
Simplify the following expression: (4(6(8(90))1)‐1)2
Answer: 1/144 PEMDAS dictates the order of operations to perform. We must always calculate the innermost parentheses first, then work our way outwards. Calculate 90 = 1 first; then 81 = 8. Next we have (6(8))-1 = 1/48.; then (4/48)2 = 1/144. It's easy to remember PEMDAS with this saying: Please Excuse My Dear
What is x + y + z? x + y = 8 x + z = 11 y + z = 7
Answer: 13 There is often a faster method than solving for the value of each variable. In this case, we can simply add all the equations together! x + y = 8 x + z = 11 y + z = 7 2x + 2y + 2z = 26 x + y + z = 13 Remember, x + y + z is a "combo." In this type of problem there is a good chance you will not need to determine the individual values of the variables
Consider the formula 2 3 . a H b If a is doubled and b is increased by a factor of 4, by what factor is H increased?
Answer: 2 The exponent of 3 on a means when we double a, the whole formula gets multiplied by 23, or 8. b has no exponent, but it is in the denominator, so quadrupling it is the equivalent of multiplying the formula by 1/4. Thus, H gets multiplied by 8 × 1/4 = 2
Identify the error: 8!2 x 8!10 implies that 2 x 10.
Answer: 2 x 10 is incorrect In a compound inequality, you must perform the same operation to all 3 expressions, not just the outside expressions. If you subtract 8! from all 3 expressions, you get 2 x 8! 10 .
If the units digit of an integer is 7, then which one‐digit integers is it definitely NOT divisible by?
Answer: 2, 4, 5, 6, and 8 Integers that are divisible by 2, 4, 6, or 8 end in 2, 4, 6, 8, or 0; those divisible by 5 end in 5 or 0. As an exercise, try to provide examples of integers with a ones digit of 7 that are divisible by 1, 3, 7, and 9.
The price of a television increased from $180 to $216. What is the percent increase in the price?
Answer: 20% Percent change is equal to change divided by original value. The change is 216 - 180 = $36. The original price is $180. 36/180 reduces to 1/5, which is the same as 20%.
The original price of an iPhone® was increased by 25%. A sale brought the price of the iPhone® back down to its original price. The sale reduced the new price of the iPhone® by what percent?
Answer: 20% Start with a smart number. Assume the price of the iPhone® is $100. 25% of 100 is 25, so the increased price was $125. We know the sale then reduced the price of the phone to its original price, $100, so the sale reduced the price by $25, because 125 - 100 = 25. The percent decrease is the difference in prices divided by the original price. 25/125 reduces to 1/5, which is 20%.
A bag of jellybeans contains 4 flavors: watermelon, cherry, orange and pear. 1/4 of the jellybeans are watermelon, 1/3 are cherry, 1/6 are orange, and the rest are pear. What percent of the jellybeans are pear?
Answer: 25% First we need to find out what fraction of the jellybeans are not pear flavored. We have to add the fractional amounts of the other flavors. The common denominator is 12, so 3 4 2 9 3 12 12 12 12 4 . Thus,1 - 3/4 = 1/4 of the jellybeans must be pear. 1/4 expressed as a percent is 25%
Simplify 6,300.
Answer: 30 7 Whenever simplifying an expression under the square root sign, factor the expression. In this case, 6,300 = 22 32 52 7. For every pair under the square root sign, move one outside the radical, and throw the other away: 22 32 52 7 becomes (2)(3)(5) 7 , or simply 30 7.
What percent of 1.5 × 107 is 4,500,000?
Answer: 30% 1.5 × 107=15,000,000. We can use benchmark values to estimate. 10% of 15,000,000 is 1,500,000. This is too small. But notice that 1,500,000 is 1/3 of 4,500,000, so if we triple 10% of 15,000,000, we'll have our answer. Therefore, 4,500,000 is 30% of 1.5 × 107.
What is the units digit of (5³)(7²)(3²)?
Answer: 5 When solving for the units digit of a number, you can ignore all the other digits. 5³ = 125. Drop the other digits and keep the 5. 7² = 49. Drop the other digits and keep the 9. 3² = 9. Keep the 9. 5 × 9 = 45. Keep the 5. 5 × 9 = 45. The units digit is 5.
The price of a refrigerator is increased by 50%. It then goes on sale, with the new sale price equaling 75% of the original price. The sale price is what percent of the increased price?
Answer: 50% When solving word problems involving percents, it's usually helpful to pick 100 as your starting value. If the price is increased by 50%, the new price is $150. The sale reduces the price to 75% of the original price. $100 is the original price, so the sale reduces the price to $75. The question asks what percent the sale price is of the increased price. 75/150 = 1/2 = 50%.
4 3 4 4 3
Answer: 53 Instead of multiplying out everything, look for ways to reduce. On the top of the fraction, 64 can be separated into (24)(34). This can be cancelled with the 24 and 34 on the bottom of the fraction, so we are left with 3 3 50 10 , which can be reduced to 53. (Note that 53 = 125.)
Simplify: 2 + 3 3
Answer: 6 3 3 To remove a square root from a denominator of the form a + b , multiply the fraction by a b a b . The form is the same whether you are dealing with numbers, variables, or a combination of the two. 3 2 3 3 2 3 6 3 3 2 3 2 3 2 3 2 3 4 2 3 2 3 3 6 3 3 6 3 3. 1
What is the only two‐digit number that is both a perfect square and a perfect cube?
Answer: 64 We need a 2-digit integer that is both a perfect square and a perfect cube. This set includes all integers of the form m3 = n2, where both m and n are integers. Manipulating the equation tells us that n = m3/2. Thus we can only choose integers for m that will make n an integer—so m must be a perfect square. The only perfect square that works is 4: 43 = 64, a 2-digit integer. 9 doesn't work, because 93 = 729, a 3-digit integer. 1 doesn't work either, because 13 = 1, a 1-digit integer
What is the greatest common factor of 990 and 924?
Answer: 66 To find the Greatest Common Factor of 2 or more numbers, you need to figure out all the factors they share in common. In this case, 990 and 924 each have one 2, one 3 and one 11. That means that the GCF will be 2 × 3 × 11, or 66.
What is x? (Hint: Try a method other than substitution) x + y = 10 3x 5y = 6
Answer: 7 One way to solve for a variable when you have two equations is to combine the equations in a way that eliminates one variable. In this case, we can multiply the first equation by 5, and then add it to the second equation, giving us: 5x + 5y = 50 3x - 5y = 6 8x + 0y = 56 → x = 7 On the GMAT, combination is often faster than substitution.
If both x and y are odd, is x2 + y odd?
Answer: No, x2 + y is even Odd numbers can be represented as 2m + 1 or 2n + 1, where m and n are integers. (Think about why this is.) (2m + 1)2 = 4m2 + 4m + 1, and adding 2n + 1 would yield 4m2 + 4m + 2n + 2. This is always even, since a 2 can be factored from all four terms. More simply, we could just recall: an odd number times an odd number is always odd, and an odd plus an odd is always even. When in doubt, try it out! Pick numbers to test properties.
If the ones digit of an integer is 0, then which (nonzero) one‐digit integers is the integer definitely NOT divisible by?
Answer: None It could be divisible by any of the one-digit integers! (Except for 0; dividing by 0 is always off limits.) To verify, take any nonzero one-digit integer, multiply it by ten, and the product will end in zero and be divisible by the original one-digit integer.
Is the statement sufficient? Is x < 0? 1) xy2 < 0
Answer: Sufficient Any number, except for 0, raised to an even power will be positive. If y were 0, the inequality would not be true, so we know that y2, regardless of the sign of y, will be positive. For xy2 to be less than zero, that means that x must be negative. The statement is sufficient.
Are the two statements sufficient when combined? What is x? (1) 3 3 5 x y z = 8 (2) 6y + 10z = 18
Answer: Sufficient Divide the equation in (2) by 2 and get 3y + 5z = 9. Substitute 9 for the denominator of the fraction in (1). This leaves an equation with one variable, x. Remember, when you see 3 variables and only 2 equations, you should not automatically assume that you cannot solve for a particular value.
Is the statement sufficient? Is x < 0? 1) 13 x
Answer: Sufficient Don't let the 13 confuse you; the only thing that matters is that 13 is an odd number. Odd roots, as well as odd exponents, preserve the sign of the number inside. If 13 x 0 , then x is also less than 0. The statement is sufficient.
Is the statement sufficient? Is x < 0? 1) 13 x 0
Answer: Sufficient Don't let the 13 confuse you; the only thing that matters is that 13 is an odd number. Odd roots, as well as odd exponents, preserve the sign of the number inside. If 13 x 0 , then x is also less than 0. The statement is sufficient.
Is the statement sufficient? Given that x2 - y2 = 20, what is y? (1) x + y = 5
Answer: Sufficient Factor the special product. We know that (x + y)(x - y) = 20. Since (x + y) = 5, (x - y) = 4. We have two linear equations, so we know we can solve for x and y individually. The equations are linear because there are no squared terms, no xy terms, and no x/y terms. The solutions are x = 4.5, y = 0.5.
Is the statement sufficient? Last year, John earned a combined $150,000 from his salary and bonus. This year, the amount he earned from salary was the same percentage of his total earnings as it was last year. How much was John's salary this year? 1) Last year, John earned twice as much from his salary as he did from his bonus
Answer: Insufficient We do have enough information to determine the amount John earned from salary and from bonus last year. ($100,000 comes from salary and $50,000 from bonus.) But we are only told that the same percentage of his total earnings this year came from salary. Lacking the actual amount of salary, bonus, or total earnings, we do not have enough information to answer the question. The statement is insufficient
Is the statement sufficient? Is x > y? (1) ax < ay
Answer: Insufficient We do not know the sign of a, so we cannot simply divide by a on both sides. We must consider two possible scenarios when rephrasing statement (1). If a > 0, then we can divide by a on both sides and x < y. However, if a < 0, after dividing we flip the inequality sign and get x > y. The statement is insufficient.
Is the statement sufficient? What is the units digit of 9x? 1) x is a prime number
Answer: Insufficient When trying to find the units digit of a number, ignore all the other digits in a number. 91 = 9, 9² = 81, 9³ = 729, 94 = 6,561. The units digit of 9 raised to the first four powers is 9, 1, 9, 1, etc. We see that the pattern repeats: odd exponents yield a units digits of 9, while even exponents yield a units digit of 1. We know that x is prime. Although all other primes are odd, 2 is even. Thus we cannot determine the units digit, and the statement is insufficient
Is the statement sufficient? Is x y ? 1) x - y > 0
Answer: Insufficient When variables are inside absolute values, a big unknown is whether the variables are positive or negative. If x and y are both positive, then the answer to the question will be yes. But now suppose that x is 3 and y is -7. 3 - (-7) = 10. In this case, the answer to the question is no. We have a yes case and a no case. The statement is insufficient.
Is the statement sufficient? What are the solutions to the equation x2 + kx - 10 = 0, where k is a constant? (1) One of the solutions is -5.
Answer: Sufficient If one solution is -5, we know one of the factors of the quadratic expression is (x + 5). We now know the other factor is (x - 2) because the two numbers in parentheses must multiply to -10. Therefore the other solution is x = 2. The statement is sufficient.
Is the statement sufficient? xy < 0. Is y < 0? 1) y2 x > 0
Answer: Sufficient If we know that xy < 0, then we know that x and y have different signs - one must be positive and the other negative. From the statement, we know that x must be positive, because we are not allowed to take an even root of a negative number. If x is positive, then y must be negative. The answer to the question is yes, and the statement is sufficient
Is the statement sufficient? The positive integer x is a prime number. What is x? 1) x + 11 is a prime number.
Answer: Sufficient If you tested numbers to answer this question, you probably figured out pretty quickly that 2 is a possible value of x. If you continue to test numbers to make sure there are no other possible values for x, you may notice a pattern emerging. 11 + 3 = 14, 11 + 5 = 16, 11 + 7 = 18, etc. 11 plus any prime besides 2 will yield an even number. 2 is the only even prime, because every other even number has 2 as a factor. Therefore, x must equal 2. The statement is sufficient
Is the statement sufficient? Is a < 0? 1) ab < 0
Answer: Sufficient In order for ab < 0, a must be negative. (This is equivalent to saying that a < 0.) If a were nonnegative, then the minimum value ab could take would be 0, regardless of the value of b.
Is the statement sufficient? The combined revenue for a company for 2006 and 2007 was $700,000. What percent of the combined revenue was earned in 2006? 1) Revenue dropped 25% from 2006 to 2007.
Answer: Sufficient Let's label the revenue for 2006 as x and the revenue for 2007 as y. From the question, we know that x + y = 700,000. From the statement, we know that revenue dropped 25% from 2006 to 2007, which means the revenue from 2007 is only 75% of the revenue for 2006. Thus 0.75x = y. We can substitute this into the original equation to find x + (0.75x) = 700,000 and solve for x.
Is the statement sufficient? If x is divisible by y, is x/y odd? 1) x and y are both odd.
Answer: Sufficient This question is tricky, because an odd divided by an odd can yield an odd integer or a non-integer. However, the question stem states that x is divisible by y. Therefore, x/y is an integer, and the result must be odd.
Is the statement sufficient? Carla earns a base salary of $30,000 plus 10% commission on her total sales revenue exceeding $50,000. How much did she make on commission this year? 1) If her total sales revenue had been 25% higher, her commission would have been 20% higher
Answer: Sufficient. First, label total commission c and total sales revenue r. The key is to realize that we have 2 different ways to express the relationship between our two variables. From the question, we know that c = 0.1(r - 50,000). From the statement, we know that 1.2c = 0.1(1.25r - 50,000). We know that we have 2 linear equations relating our 2 variables, so we will get one unique solution. For extra practice, what is the value of c and r?
If both x and y are even, is x - y even?
Answer: Yes, x - y is even Even numbers can be represented as 2m or 2n, where m and n are integers. (Think about why this is.) Subtracting two numbers of this form would yield 2n - 2m, or 2(n - m), which has 2 as a factor, so it is even. More simply, we could just recall: an even number minus an even number is always even. When in doubt, try it out! Pick numbers to test properties.
If both x and y are odd, is xy odd?
Answer: Yes, xy is odd Odd numbers can be represented as 2m + 1 or 2n + 1, where m and n are integers. (Think about why this is.) Multiplying two numbers of this form together would yield 4nm + 2m + 2n + 1, which is always odd; the 1st, 2nd, and 3rd terms are multiplied by 2 (or 4), so they are even, as is their sum. An even number plus one is odd. Thus xy is odd. More simply, we could just recall: an odd number times an odd number is always odd. When in doubt, try it out! Pick numbers to test properties.
1 863 471 , , , a) 3 b) 4 c) 5 d) 30 e) 35
Answer: a) 3 We are only asked for an approximate answer, so use the heavy division shortcut. 1,863,471 18 3 626,502 6
x is divisible by 144. If 3 x is an integer, then which of the following is 3 x definitely divisible by? (Choose all that apply) a) 4 b) 8 c) 9 d) 12
Answer: a) 4 and d) 12 Remember that when we complete a prime box for a variable, that variable could still have additional factors. For the cube root of a number to be an integer, the original number must have 3 of each prime factor, or some multiple of 3 (3, 6, 9, etc.). In this case, that means the factors of x that we can't see must include at least two additional 2s and one additional 3. From this information, we can definitively conclude that 3 x must have two 2s and one 3 as factors. 4 and 12 are the only numbers in the list we can guarantee are factors of 3 x .
Simplify: a) 4^5 + 4^5 + 4^5 + 4^5 b) xw + yw + zx + zy
Answer: a) 46; b) (w + z)(x + y) a) The greatest common factor is 45. 45(1 + 1 + 1 + 1) = 45(4) = 46. Make sure to look for common terms that can be factored out of an expression. Factoring is often a crucial step toward solving an equation. b) Factor by grouping: (xw + yw) + (zx + zy) = w(x + y) + z(x + y) = (w + z)(x + y). If you have 4 expressions and 4 variables, look to factor by grouping
x is divisible by 42. Which of the following numbers is definitely a factor of x2? (Choose all that apply.) 2 21 3 7 9 10 63 7 9 3
Answer: a) 63 and c) 36 If x definitely has 2, 3 and 7 as factors, then when we square x, we know that x2 will have two 2s, two 3s and two 7s as factors. 63 is 7 × 3 × 3, and 36 is 2 × 2 × 3 × 3. Using the factor foundation rule, we can guarantee that all numbers that solely use those factors are factors of x2. Both 63 and 36 use only prime factors found in x2.
Solve for each of the following: a) If 2 7 y x , What is 2x + y? b) If 2t r 5, What is 3r + 6t?
Answer: a) 7; b) 75 a) Multiply both sides by 2 and add y to each side. 2x + y = 7 b) Square both sides and multiply by 3. 6t + 3r = 75 ManhattanGMAT Equations, In
For each of the following, could the answer be an integer if x is an integer greater than 1? a) x10 + x-10 = b) x1/6 + x1/2 =
Answer: a) No; b) Yes a) No. x-10 = 1/x10. For any x > 1, this won't be an integer. b) Yes. This is equivalent to 6 x x , so if x has an integer sixth root this will be an integer. For example, if x equals 64, the sixth root of x is 2, and the square root is 8. Any number with an integer sixth root will have an integer square root. Why?
Is it possible to solve for a single value of x in each of the following systems of equations? a) 2x + 3y = 8 b) x2 + y - 17 = 0 2x - y = 0 y = 2x c) 2x - 4y = 13 -6x + 12y = -39
Answer: a) Yes; b) No; c) No a) Yes. We are given 2 linear equations. There are no xy terms or x/y terms. b) No. There is an x2 term. Even if 2x is substituted into the first equation for y, 17 isn't a perfect square, so we should expect the quadratic to have 2 distinct solutions. c) No. The two equations are equivalent. The second equation is just the first equation multiplied by -3.
5 5 n is always equal to which of the following? a) n b) n25 c) n1/5 d) 1
Answer: a) n Try it: 5 5 n is the same as 1 n5 5 , which is equal to n, since (na)b = nab, and 5 times 1/5 equals 1. Alternatively, 5 5n 5 n 5 n 5 n 5 n 5 n n. You can try this out if you need convincing. Pick a few numbers and see what happens!
7 - ≥ 4 b
Answer: b ≤ −28 To isolate b, multiply both sides by −7 and flip the direction of the inequality sign. When multiplying or dividing an inequality by a negative number, remember to switch the direction of the inequality sign.
21,267 is approximately what percent of 106? a) 0.2% b) 2% c) 20%
Answer: b) 2% Use benchmark values to estimate. 106 = 1,000,000. Finding 1% is the same as dividing by 100, so 1% of 106 is 104 or 10,000. Since 21,267 is a little more than twice 10,000, so 21,267 is approximately 2% of 106. You could also use heavy division to estimate your answer: 6 21,267 21,267 2% 10 1,000,000
Which of the following is closest to 23% of 41/60 of 240 rounded to the nearest integer? a) 24 b) 39 c) 52 d) 68
Answer: b) 39 The answer choices are far apart, so we can save time by estimating. 41/60 is close to 40/60, which is 2/3. 240 × 2/3 = 160. 23% is close to 25%. To calculate 25% of a number, just divide by 4. 160/4 is 40. The best answer is b) 39.
Distribute: (b + 7)(b - 10)
Answer: b2 − 3b − 70 Use FOIL - First, Outer, Inner, Last (b)(b) + (b)(−10) + (7)(b) + (7)(−10) b2 - 10b + 7b - 70 b2 - 3b - 70
Which number is closest to 7% of 1,440? a) 50 b) 75 c) 100
Answer: c) 100 We can save time by estimating. 1,440 is approximately 1,400, which is 14 × 100. 7% of (14)(100) = (7/100)(14)(100) = 7 × 14 = 98. This is a slight underestimate, so answer choice c) must be correct.
If c < 4, what is the range of possible values of d for the equation 3c = −6d?
Answer: d > -2 We can actually replace c with its extreme value, which is "less than 4." The equation will read 3(less than 4) = −6d. So (less than 12) = −6d. Divide by −6, and remember to flip the sign, because we're dividing by a negative. Thus we have (greater than −2) = d.
Solve: (x - 4)2 = 49
Answer: x = 11 or -3 Do not multiply out (x - 4)2 if there is a perfect square on one side of the equation. Instead, take the square root of both sides, and remember to place the side of the equation containing the unknown in an absolute value. x 4 7 . Our two solutions to this equation are x - 4 = 7 and x - 4 = −7. Solving these two equations gives us x = 11 and −3.
What are all possible values of x? x^2 - 27x + 50 = 0
Answer: x = 2 or 25 Since the last sign is positive, set up 2 parentheses with the sign of the middle term. (x - ) (x - ) Find two numbers that multiply to 50 and add to 27 and place them in the parentheses. (x - 2) (x - 25) = 0.
The first few steps of a problem are shown. Finish the problem and answer the question: what is x? x 3 x 3 x 3 (x 3)2 x 3 x2 6x 9 0 x2 7x 6
Answer: x = 6 (x does NOT equal 1!) Although this equation can be simplified and factored into (x − 6)(x − 1)=0, you need to be careful. When you square an equation containing a variable, you may create extraneous solutions. Potential answers need to be plugged back in to the original equation and verified. 6 is a genuine solution, 1 is not. Try plugging 1 back into the original equation to verify that x cannot equal 1.
f x is even and y is odd, is x2 + y2 even or odd?
Answer: x2 + y2 is odd An even number can be represented as 2m, and an odd number can be represented as 2n + 1, where m and n are integers. Squaring the even number yields 4m2; the odd, 4n2 + 4n + 1. Adding these together yields 4m2 + 4n2 + 4n + 1. The 1st 3 terms all have 4 as a factor, so their sum is even, and an even number plus 1 is odd. More simply, we could just recall: an even number squared is always even, an odd number squared is always odd, and an even plus an odd is always odd. When in doubt, try it out! Pick numbers to test properties.
If x is odd and y is even, is xy odd or even
Answer: xy is even An odd number can be represented as 2m + 1, and an even number can be represented as 2n, where m and n are integers. (Think about why this is.) Multiplying 2m + 1 and 2n would yield 4mn +2n, which is always even, since 2 is a factor of both terms. (Factor out the 2 to get 2(2mn + n), which shows that this number will be even.) More simply, we could just recall: an odd number times an even number is always even. When in doubt, try it out! Pick numbers to test properties.
Given that y7 < y6, describe all of the possible values for y.
Answer: y < 1, but not equal to 0 (alternatively, 0 < y < 1 or y < 0) Think about various categories of numbers: if y were negative, then y7 would also be negative, while y6 would be positive; then y7 < y6. If y = 0 or 1, then y7 = y6, which is not acceptable. When y is between 0 and 1, y7 < y6, since y7 would equal y6 times some fraction between 0 and 1. Finally, when y > 1, y7 > y6.
Solve for y: y2 + 7y - 60 = 0
Answer: y = 12, 5 Since the last sign is negative, set up 2 parentheses with opposite signs. (y + )(y - ) Find two numbers that multiply to 60 and subtract to 7: 12 × 5 = 60 12 - 5 = 7 Place the larger number in the parentheses with the same sign as the middle term (+7y): (y + 12)(y - 5) = 0 If y + 12 = 0, then y = 12. If y - 5 = 0, then y = 5.
Simplify: a b a b
Anytime there is a square root term in the denominator that is added to or subtracted from another term we can multiply by the conjugate (the same expression, but with the sign on the 2nd term flipped) to simplify: a b a b a b a b a b a b a b a b a b a b a b a b Alternatively, you could use the special product a2 - b2 = (a + b)(a - b) to solve. In this case, a b a b a b , and so the term a b would cancel from the top and bottom, leaving a bv