Chapter 12: Math Fundamentals on the GRE
12 - ((6/3)-4×3) -8×3 = A) -46 B) 30 C) -18 D) -6 E) -2
E) -2
The sum of the least and the greatest of a set of three consecutive integers is 24. What is the sum of the three integers? A) 34 B) 35 C) 36 D) 37 E) 38
C) 36 While this is obvious, you can also represent the least of the integers with x. Then, since the integers are continuous you can represent the sum by writing x + x+2 = 24 Solving for x yields x=11 The least integer is 11 then 12 then 13 11+12+13= 36
<−r³−−s−−t−−0−−u²-> Based on the information in the number line above, which of the following statements must be true? Indicate ALL such statements A) rs> 0 B) stu > 0 C) r÷ t < 0
Choice A is correct because negative x negative = positive Choice C is correct because negative/negative = positive Choice B is wrong because while u² is bigger than zero, u by itself can be smaller than zero (negative) For example, the value of stu is greater than zero when u is positive. However, if it is negative, the value of stu <0 Answers: A) rs> 0 C) r÷ t < 0
-2,3,-5,-2,3,-5,-2,3,-5,... In the sequence above, the first 3 terms repeat without end. What is the product of the 81st term through the 85th term?
300 The 9th term of the sequence is -5 9x10 =81 so the 81th term is -5 just assign the rest of the values and multiply
A basin of water has two leaks. One leaks at a rate of 1/6 of a gallon every half hour. The other leaks at a rate of 2/7 of a gallon every half hour. How many gallons of water will have leaked from the basin in 5 hours?
95/21
Quantity A 0.25% of x Quantity B ¼x A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
B) Quantity B is greater You may think they are equal. But notice it says 0.25%. If it was 25% then they would've been equal
If a prime number, p, is squared and the result is added to the next prime number greater than p, which of the following integers could be the resulting sum? Indicate ALL such integers. A) 3 B) 4 C) 7 D) 14 E) 58 F) 60 G) 65 H) 69
C) 7 D) 14 F) 60
If the product of two distinct integers is 91, then which of the following could be the sum of the two integers? Indicate ALL such sums. A) -92 B) -91 C) 7 D) 13 E) 20
REMEMBER that integers can also be negative 91 is the result of 7 × 13. So, their sum is 20. HOWEVER, if you also multiply -1 × -91 = 91 Therefore, their sum is 92. ANSWERS: A) -92 E) 20
If a and b are integers such that -8 ≤ a ≤ 3 and -3 < b <10, then what is the least possible value of b - a? A) -11 B) -7 C) -6 D) -5 E) 0
The least possible value for b is 2, and the least possible value of a is -8. However, in order to get the least possible value of b -a, it is necessary to choose the greatest possible value for a. If a = 3 and b = -2, then -2 -3 = -5 Answer: D) -5
If 4xy[(3x³y²) -(8x⁴y³-2x³y²)] = ax⁴y³ - bx⁵y⁴, what is the value of a? A) 4 B) 5 C) 8 D) 20 E) 32
Use PEMDAS First distribute the negative sign before the second set of parenthesis 4xy[(3x³y²) -(8x⁴y³-2x³y²)] → 4xy[(3x³y²) -8x⁴y³+2x³y²)]→ combine like terms 4xy[5x³y²-8x⁴y³] Distribute 20x⁴y³ - 32x⁵y⁴ 20x⁴y³ - 32x⁵y⁴ = ax⁴y³ - bx⁵y⁴ If you compare both equations, you can see a=20 D) 20
If 1/20³ is expressed as a terminating decimal, how many nonzero digits does the decimal have? A) 2 B) 3 C) 4 D) 5 E) 6
When looking at a fraction's denominator, if the number can be reduced to the prime numbers 2, 5 or both (factors of 10), then the fraction has a terminating decimal. Anything else just goes on forever. So, let's reduce this one to 2's and 5's 1/20³ =1/(4³)(5³) =1/(2⁶)(5³) = 1/(2³)(5³)(2³) 1/(2³)(5³)(2³) = 1/(10³)(2³) = 1/(2³) ×1/(10³) rewrite 1/(2³) to 0.5³ and 1/(10³) to 10⁻³ (0.5³)(10-³) 0.5= 5×10⁻¹ so ((5)(10⁻¹)³(10⁻³) = (5³)(10⁻⁶) =(125)(10⁻⁶) Therefore, the number of nonzero digits in the fraction is 3. OR instead of using the stupid way just flip the fraction and divide 1 by 8000. You'll get the decimals easier. B) 3
If r, s, and t are odd integers, which of the following expressions must be an even integer? Indicate all such expressions A) t(r+s) B) r + t + s C) rs + (t -1) D) s(r+1) +t E) rs + t
If r =3, s=5, and t=7, then only choices A and E are true.
|4s-6| > 10 Quantity A s Quantity B 2 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
If s is POSITIVE, then 4s - 6 > 10 is equivalent to 4s> 16 and s>4. If s is NEGATIVE, then solve for s by flipping the inequality sign and making the number to the right of the inequality sign negative. The results are: 4s-6<-10 which yields 4s<-4 and s<-1 if s>4 then A is greater if s<-1 then B is greater So... D) The relationship cannot be determined from the information given.
n is an integer. n>0 Quantity A The units digit of 37ⁿ Quantity B The units digits of 84ⁿ A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
One way to solve this problem is to plug in different values for n and look for a consistent answer. Alternatively, determine the pattern for the units digits for each quantity. The units digit patter for Quantity A for all values n>0 is 7,9,3,1. The units digit pattern for Quantity B for all values n>0 is 4,6,4,6. Therefore, when n=1, Quantity A is greater. But, when n=4, Quantity B is greater. Because, each quantity is greater than the other under different circumstances, the correct answer is D) D) The relationship cannot be determined from the information given.
When the integer n is divided by 16, the quotient is x and the remainder is 7. When integer n is divided by 23, the quotient is y and the remainder is 11. A) 16x -23y = 18 B) 16x-23y = 4 C) 16x +23y =28 D) 16x -23y =141 E) 16x + 23y =365
dividend = (integer quotient)*(divisor) + remainder n=(x)(16)+7 n=(y)(23)+11 since both equal n, you can set them up so they equal each other 16x+7=23y+11 isolate x and y → 16x -23y=4 B) 16x-23y = 4
m= 14²⁸ + 1 When m is divided by 6, the remainder is r. Quantity A r Quantity B 4 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
14²⁸ would take a long time to solve. So, begin by looking for patterns using a couple examples of 14 raised to a smaller exponent plus 1 divided by 6 For example, the remainder when 14¹ +1 =15. Then, when you divide 15 by 6, the remainder is 3 When14² +1 = 197 and you divide by 6, the remainder is 5 14³ +1 =2,745; remainder is 3 14⁴ +1 = 38,417; remainder is 5 So, when the exponent is odd, the remainder is 3 when the exponent is even, the remainder is 5 14²⁸ is even so the remainder is 5 A) Quantity A is greater
What is the least positive integer that is not a factor of 8! and is not a prime number? A) 9 B) 11 C) 18 D) 22 E) 26
8! is the product of all integers from 1-8 8!=1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 Then, rewrite any of these numbers that are not prime as the product of their primes. 8!= 1×2×3×2²×5×(2*3)×7×2³ Use the multiply add exponent rule 8!=1×2⁷×3²×5×7 Choice A is a factor of 8! because 9=3² Choice B is not a factor of 8! but 11 is prime Choice C is a factor of 8! because 18=2×3² Choice D is not a factor of 8! because 22=2×11 and 11 is not a prime factor of 8! D) 22
Which of the following are multiples of both 2 and 3? A) 102 B) 416 C) 522 D) 534 E) 654 F) 918
A number is divisible by 2 if the units digit of the number is even, and a number is divisible by 3 if the sum of all the number's digits is divisible by 3. A) 102 C) 522 D) 534 E) 654 F) 918
5 is r percent of 25 s is 25 percent of 60 Quantity A r Quantity B s A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
A) Quantity A is greater
During a certain three-month period, Vandelay Industries reported a $3,500 profit. If over the next three-month period, Vandelay Industries reported $6,000 profit for those months, by approximately what percent did Vandelay Industries'' profit increase? A) 25% B) 32% C) 42% D) 55% E) 70%
Subtract the difference between 6000 and 3500. The difference is 2500 Then divide 2500 by the original number 2500/3500 ≈ 70% E) 70%
If |z| < 1, then which of the following statements must be true? Indicate ALL such statements. A) z<z² B) z³<z² C) z² < 1
C) z² < 1 If you choose ½ then both B and C are true. However, if you choose 0, then only C remains true.
Quantity A The number of integers between 100 and 500 that are multiple of 12 Quantity B 34 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
You could do this manually but it will take too long Because 12 ×× 10 = 120, there are 10 integers that are multiples of 12 for every 120 integers. So, between 120 and 240, there are 10 integers. Between 240 and 360 is another 10. Between 360 and 480 and 500 is another 10. Between 100 and 120, there are 2 integers and between 480 and 500 there is 1 integer. So, 10 + 10 + 10 + 2 +1 =33 Or do 100/12 = 8 500/12 = 41 41-8=33 OR The first number to be divided by 12 (that is greater than 100) is 108. Divide 108 and you get number 9. Now divide 500/12= 41. From 41-9 +1= 33 BEST METHOD Between 100 and 500, minimum multiple of 12 is 120 and Maximum multiple of 12 is 492. Therefore, we have (492-108)/12 +1 = 33 RULE FOR INCLUSIVE LISTS: A nice rule says: the number of integers from x to y inclusive equals y - x + 1
When positive integer r is divided by 6, the remainder is 2. When positive integer r is divided by 82, the remainder is 50. Quantity A The least possible value of r Quantity B 296 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
dividend = (integer quotient)*(divisor) + remainder r = 6p + 2 (Also when 2₊6, the remainder is 2 so that's why it's included) 2,8,14,20,26,32,38,44,50,56, etc. r = 82q + 50 The first number on the second list is 50 so there's no need to do any more calculations. 50 is also on the first list. Quantity A =50 Answer: B) Quantity B is greater
1/3 +2/5 =x y=3 Quantity A y/x Quantity B 4 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
1/3 +2/5 =45/11= 4 1/11 A is slightly bigger than 4 A) Quantity A is greater
(2+¼)/((4/5)-4) - 3/(4+4/7) = A) -87/64 B) -45/64 C) 3/64 D) 42/64 E) 87/64
A) -87/64
Which of the following inequalities is NOT true? Indicate ALL such inequalities A) 1/2<5/8<7/12 B) 4/9<5/11<6/13 C) 6/7<11/12<15/16 D) 1/5<2/9<3/14
A) 1/2<5/8<7/12 D) 1/5<2/9<3/14 Use cross multiplying to solve and save time
Quantity A The greatest number of consecutive nonnegative integers which have a sum less than 22 Quantity B 6 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
A) Quantity A is greater
If x⊗(y⊗z) =(x ⊗ y) ⊗ z and x ⊗ y = y ⊗ x then which of the following statements must be true? Indicate all such statements A) ⊗ could represent multiplication B) ⊗ could represent addition C) ⊗ could represent subtraction D) ⊗ could represent division
A) ⊗ could represent multiplication B) ⊗ could represent addition
A grocery store received a shipment of produce, 40 percent of which were apples, with the remainder consisting of equal numbers of bananas, carrots, and dates. By the next day, 60 percent of the apples, 25 percent of the bananas, 40 percent of the carrots, and 80 percent of the dates were sold. What percent of the produce in the shipment was sold? A) 36% B) 47% C) 53% D) 64% E) 72%
Apples = 40% (take 60% of that) = 24% Bananas = 20% (take 25% of that) = 5% Carrots = 20% (take 40% of that) = 8% Dates = 20% (take 80%percent of that) = 16% 24 + 5 + 8 + 16 =53 OR Assume the shipment contained 100 items to make things easier C) 53%
During a sale, a store decreases the prices on all of its scarves by 25 to 50 percent. If all of the scarves in the store were originally priced at $20, which of the following prices could be the sale price of a scarf? Indicate ALL such prices. A) $8 B) $10 C) $12 D) $14 E) $16
B) $10 C) $12 D) $14
4 1/3 + 3 2/5 =x (mixed numbers) y=3 Quantity A y/x Quantity B 7 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
B) Quantity B is greater
If M = (r-s) + t and N = r-(s + t), then what is the value of M-N? A) 2r B) 2s C) 2t D) 2s +2t E) 2r +2s
C) 2t Just set M-N and eliminate like terms
Which of the following statements is true? A) 3/8 < 2/9 < 4/11 B) 2/5 < 3/7 < 4/13 C) 4/14 <2/5 < 3/7 D) 3/7 < 3/8 < 2/5 E) 2/9 < 3/7 < 3/8
C) 4/14 <2/5 < 3/7
Quantity A 12/x + 7/y Quantity B (7x+12y)/xy A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
C) The two quantities are equal
Quantity A 4(1/2x +2y) Quantity B 2x+8y A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
C) The two quantities are equal
A bookstore will only order books that come in complete cases. Each case has 150 books and costs $1,757. Quantity A The number of books that can be ordered for $ 10,550 Quantity B The number of books that can be ordered for $12,290 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
C) The two quantities are equal ez math bb
The total amount that Jamal paid for cake ingredients was equal to the price of the ingredients plus a sales tax of 6 percent of the price of the ingredients. Jamal paid for the ingredients with a $20 bill and received the correct change, which was between $1.00 and $3.00. Which of the following statements must be true? Indicate ALL such statements A) The price of the ingredients was less than $17.75 B) The price of the ingredients was greater than $16.00 C) The sales tax was less than $1.07
Choice A is wrong because $17.75 × 1.06 = $18.82. This fulfills the $1-$3 change requirement. However, If the price is $17.80 the result becomes $18.87. So, the number CAN be greater than 17.75. This answer is wrong Choice C is also wrong. If the sales tax is $1.07, you can calculate the price of the ingredients by setting the following equation: $1.07= .06(price of ingredients) The price is $17.83 Then add $17.83 + $1.07 =$18.89 This fulfills the $1-$3 change requirement. If the tax was $1.08 the result would be $19.08 which would not fulfill the requirement. HOWEVER Answer C specifies that the sales tax was LESS than $1.07 We proved that $1.07 works fine so this is false Choice B = 16 × 0.06 = 0.96 16+ 0.96 =16.96 The change would be greater than 3. So... Answer: B) The price of the ingredients was greater than $16.00
Which of the following is the units digit for the sum of all of the distinct prime integers less than 20? A) 4 B) 5 C) 6 D) 7 E) 13 F) 20
D) 7
In the equation b(12+a) -3ab=4b², what is the value of a in terms of b? A) 6-2b B) 2b+6 C) 2b/4b²+12b D) 2b/ 12b-4b² E) 2b-6
Distribute the parenthesis 12b+ab-3ab=4b² then subtract 12b from both sides -2ab=4b² -12b then divide both sides by -2b a=-2b+6 A) 6-2b You can also plug in the answers into the equation. For example if you plug in 2 for b, try to make sure that the equations yield a=2
If Q is the product of all the positive multiples of 9 less than 100, what is the sum of the distinct prime factors of Q? A) 37 B) 29 C) 28 D) 21 E) 17
Each positive multiple of 9 less than 100 is the product of 9 multiplied by an integer. Because 9 × 11 = 99, the positive multiples of 9 less than 100 are the products of 9 multiplied by every integer 1 through 11. The question asks for the distinct prime factors of Q, which is simply all of the prime factors between 1 and 11 The prime factors between1 and 11 are 2,3,5,7, and 11. The sum of these numbers is 28 C) 28
rs ≠ 0 r < s x² - 100 = 0 Quantity A The distance between r and x on a number line Quantity B The distance between s and x on a number line A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
First, solve the quadratic equation by factoring. x² - 100 = 0 → (x-10)(x+10) = 0 So, x =±10 Then choose numbers to assign to r and s. Since s is bigger than r, we'll choose s=3 and r=2. If x=+10, quantity A = 8 and quantity B =7 if x= -10, quantity A = 12 and quantity B = 13 Answer: D) The relationship cannot be determined from the information given.
If f(x) = 2(5+x) -4-x AND g(x,y) = y(x-3)² +xy, then what is the value of f(g(5,2))? A) 15 B) 18 C) 24 D) 42 E) 48
First, substitute (5 and 2) into g(x,y) = y(x-x)² +xy 2(5-3)² +5(2) this will yield x=18 plug in 18 into the f(x) equation 2(5+18) -4-18 C) 24
The Smith family spends an average (arithmetic mean) of $430 a month on groceries. This amount is 20% of their monthly income. Quantity A The amount of the Smith family's annual income not used to buy groceries Quantity B $20,600 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
If $430 is 20% of their monthly income, just multiply 430 by 5 to find their true monthly income 430 × 5 = $2,150 per month multiply that by 12 and you get $25,800 HOWEVER that is their total earnings, you must subtract the 20% they spend on groceries. You can find that by multiplying 25,800 by 0.2 OR multiplying 430 by 12 $25,800 - $5,160 = $20,640 A) Quantity A is greater
Set A consists of 10 members, set B consists of 12 members, and set C consists of all the distinct members of sets A and B. If at least 3 of the members in set A are also in set B, then which of the following could be the numbers of member of set C? Indicate ALL such number of members A) 10 B) 12 C) 14 D) 16 E) 18 F) 20
If exactly 3 members of set A and B are the same, then there are 10-3 =7 members of set A that are distinct from the members of set B and 12-3 =9 members of set B that are distinct from the members of set A. In this case, set C is 7 + 9 +16. HOWEVER, for every additional member in set A that is in set B, the total members in set C decrease by 2. Because the problem states that there are AT LEAST three members in set A that are also in set B, the maximum number of members in set C is 16. If all the members of set A were in set B, then the total members of set C is 0+2 =2 Therefore, any even number between 2 and 16 is a possible number of members of set C Answers: A) 10 B) 12 C) 14 D) 16
n is an odd, negative integer. Quantity A (¼)ⁿ Quantity B (-4)ⁿ A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
If n=-3 Quantity A (¼)⁻³ = 4³ = 64 Quantity B (-4)⁻³ = 1/64 You can also choose n=-1 to verify your results Answer: A) Quantity A is greater
If 0 < a < b, which of the following inequalities must be true? Indicate ALL such inequalities A) a - b < b -a B) b < ab C) a + b < a² +b²
If you choose a = 2 and b =3, then all statements are true. However, the problem did not state that a or b must be integers. if you choose a =¼ and b = ½, then only statement A) remains true Answer: A) a - b < b -a
N is the set of all distinct integers between 0 and 200, inclusive. Quantity A The total number of even integers in set N Quantity B The total number of odd integers in set N A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Instead of listing out all even and odd integers in the set, you can search for a pattern instead. For example, if you start with a smaller number like 10, you can see that the even integers between 0 and 10 are 0,2,4,6,8, and 10. The odd integers are 1,3,5,7,9. There are more even integers between 0-10 than odd integers. Since 10 is a factor of 200, it is also true that there are more even integers than odd integers between 0 and 200 A) Quantity A is greater
Model Original Price Sale Price A $ 12,000 $ 9,500 B $ 16,000 $13,000 C $ 10,000 $ 7,500 D $ 17,500 $ 13,000 E $ 20,000 $ 15,500 F $ 22,000 $16,000 The table above shows the original price and the sale price for six different models of cars. For which car models is the percent decrease at least 25%
Just try to divide the original numbers by 4 to see what 1/4 looks for each number. Then subtract that number from the original price to see the minimum number required to meet the criteria. C,D,F
Which of the following are multiples of both 2 and 3? A) 102 B) 416 C) 522 D) 534 E) 654 F) 918
One way to solve this problem is to realize that any number that is a multiple of 2 and 3 is also a multiple of 6, so divide each of the answer choices by 6 and look for ones that divide evenly. Another way to solve this problem is to find the prime factors of each answer choice. If both 2 and 3 are prime factors of the answer choice, then the choice is divisible by 2 and 3 All but B) work A) 102 C) 522 D) 534 E) 654 F) 918
a<b<c<d<3 a, b, c, and d are odd integers Quantity A d × b Quantity B a÷ c A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Out of all the variables, the only one that can be positive is d. If d is positive then quantity B is greater If d is negative then quantity A is greater So, the only logical answer is: D) The relationship cannot be determined from the information given.
Quantity A 52¹⁰ × 52/51 Quantity B 52¹⁰ +52¹⁰/51 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Quantity A can be rewritten as (52¹⁰ × 52)/51 which equals 52¹¹/51 Use the bow tie method (cross multiply add the results and place them on top of a common denominator) for quantity B (52¹⁰(51)+52¹⁰(1))/51 C) The two quantities are equal
For integers x and y xy =10 x = y -3 Quantity A y Quantity B 0 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Rewrite formula to y² -3y-10=0, then find the values of y by factoring or quadratic formula. y= 5; y=-2 When y = 5, quantity A is greater than quantity B When y =-2, quantity A is lesser than quantity B So, D) The relationship cannot be determined from the information given.
If |x² + 4x +3| = 0 and |y+4| = y² +3y -4, what is the least possible value of xy? A) -8 B) -6 C) -2 D) 0 E) 4
Start with x Because the absolute value expression that contains x is set equal to 0, there is no positive and negative value of the expression. Therefore, you only need to solve for one possible equation: x² + 4x +3 = 0. Factor this quadratic to find that (x+3)(x+1)=0. Therefore, the possible values of x are x=-1 and x=-3 Now solve for y The absolute value expression that contains y is set equal to the expression y² +3y -4. Because of this, you need to consider the positive and negative value of the expression. Set up two equations. The firs equation is y+4 = y² +3y -4 The second equation is y+4 = -(y² +3y -4) The first equation yields y² +2y-8=0. So, y =-4 and y = 2 Now check that both solutions make the equation true (plug in). Both are true The second equation yields y²+4y=0. So, y=0 and y=-4 Check that both solutions make the equation true. y=0 is an extraneous solution (it doesn't work) Finally, find the least possible value of xy. Both values of x are negative, so the least possible value of xy results when y is positive. The least possible value of xy is when x =-3 and y =2 B) -6
If r is the greatest common factor of 32 and 24 and s is the least common factor of 18 and 45, then what is the value of r+s? A) 17 B) 11 C) 9 D) 7 E) 6
The factors of 32 are 1,2,4,8, 16, and 32. The factors of 24 are 1,2,3,4,6,8,12, and 24 Their greatest common factor is 8 The factors of 18 are 1,2,3,6,9, and 18 The factors of 45 are 1,3,5,9,15,and 45 Their least common factor is 1 So, 8+1=9 C) 9
-5 ≤ n ≤ 5 Set X contains all distinct integer values of n such that n = |n| Quantity A The least possible product of any two values in set X Quantity B 2 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
The first statement indicates that n is any value between -5 and 5. The second statement establishes that set X is all the distinct integer values of n where n = |n|. Therefore, the possible values of set X are 0,1,2,3,4 and 5. Quantity A is the least possible product of two values in set X. Since it has a zero the answer is zero. B) Quantity B is greater
If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest possible value of x+y? A) 2 B) 3 C) 5 D) 6 E) 9
The greatest value of x is when 4 is divided by 6 so the remainder is 4. The greatest value of y is when 2 is divided by 3 so the remainder is 2 So the answer is 6 D) 6
For all integers n > 0, the sequence xₙ is defined by xₙ = xⁿ. If x₁ = 13, then what is the units digits of x₃₂?
The problem states that x₁ =13, so place this value into the sequence as 13 = x¹. So, x= 13. For each subsequent value of the sequence, raise 13 to that power. For instance, x²= 13²=169. Calculating 13³² would take a lot of time. Instead, look for a pattern in the units digit of the sequence by writing out the first few values in the sequence x₁=13 x₂= 169 x₃ =2,197 x₄ = 28,561 x₅ =371,293 At this point, the pattern of the units digit repeats itself. The pattern is: 3,9,7,1,3,9,7,1... Every fourth value in the sequence has a units digit of 1. Of course, 32 is a multiple of 4. Therefore, the value of the units digit for x₃₂ falls on the fourth item in the pattern, the number 1 Answer: 1
k is a digit in the number 1.8k4 1.8k4 < 1.833 Quantity A k Quantity B 1 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
The top statement says that k has to be less than 3 otherwise the statement would be false. The possible k values are: 0,1, and 2 If k=2, the quantity A is greater than Quantity B If k=0 then quantity B is greater than quantity A D) The relationship cannot be determined from the information given.
Two distinct integers x and y are selected from the odd integers that are less than 10 and greater than 1. Which of the following integers could be divisible by both x and y? Indicate all such integers. A) 48 B) 83 C) 126 D) 150 E) 189
The values of x and y can only be 3,5,7, or 9. Choice A is divisible by 3 but not by the other odd numbers Choice B is a prime number so it is not divisible by anything but 1 and itself The answers are: C) 126 D) 150 E) 189
Of the 180 employees at a company, 2/3 are female. Of the employees, 3/5 of the the female and 7/15 of the male employees are entry level. What fraction of the employees at the company are entry level employees?
Total is 180. 2/3 are female so 120 the 60 left are male 3/5 of 120 is 72 7/15 of 60 is 28 28+72=100 so, 100/180 = 5/9 Answer: 5/9
If x>0, then which of the following is equal to 1.75 percent of x? A) x/400 B) 7x/400 C) x/200 D) 3x/4 E) 7x/4
Translate the problem into an equation. 1.75% of x can be rewritten as (1.75/100)xy multiply by 100 to yield 175/10,000 =7/400 7/400 =0.0175 So, 7x/400 is the answer B) 7x/400 NOTE if you set up x/400= 0.0175 you will get x=7 and choose answer A. That is wrong
S = {1,2,3,4,5,7} x and y are distinct numbers from set S. Quantity A The number of different possible values of xy Quantity B 30 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Two ways to answer this. You can either list all the options out. Or, realize that there are 6 elements in this set, which means that there are 6 choices for the first number and 5 choices for the second number. Therefore, 6 *5 = 30. However, note that xy = yx for multiplication. So, it's necessary to divide 30 by 2. Thus, quantity A is 15. Answer: B) Quantity B is greater
Quantity A The number of distinct prime factors of 6,208 Quantity B 2 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Use a factor tree 6,208 ↓ 2 and 3,104 ↓ 2 and 1,552 ↓ 2 and 776 ↓ 2 and 388 ↓ 2 and 194 ↓ 2 and 97 The DISTINCT and PRIME factors of 6,208 are 2 and 97. So, 2 in total. C) The two quantities are equal
Each of the 150 students in the sophomore class at a certain high school must take at least one of five math classes. If 67 students take algebra and 45 students take statistics, then what is the least number of sophomore students who take neither algebra nor statistics?
Use the group formula Total = group A + group B - both + neither 150 = 67 +45 - 0 + neither neither = 38 Answer= 38
The decimal number D is the fraction 7/22 rounded to 34 decimal places to the right of the decimal point. Quantity A The 34th digit to the right of the decimal point in D Quantity B 1 A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
When you divide 7/22 the answer is 0.3181818 etc. The pattern indicates that very even number will be a one. Therefore, the 34th number is a one HOWEVER The question states that the number is rounded. The 35th number is an 8 so the 34th number is rounded to a 2 A) Quantity A is greater
What is the greatest integer value of n such that 3ⁿ is a factor of 270⁸?
Whenever a question asks you to compare a very large number to a smaller prime number, the first thing to consider is prime factors. The prime factorization of 270 is 2 × 3³ ×5 Therefore, 270⁸ = (2 × 3³ ×5)⁸ = 2⁸ ×3²⁴ ×5⁸ The maximum value of n is 24
The sum of 8 consecutive odd integers is 256. What is the third number in the set? A) 25 B) 29 C) 31 D) 35 E) 43
You can plug in the offered answers and find the answer manually or you can do it algebraically. n is the least integer, so the remaining odd integers can be represented by n + n+2+n+4+n+6+ n+8 + n+10+n+12+n+14 8n + 56 = 256 n= 25 (least integer) n + 4 is the third integer so substitute n 25+4=29 Answer: B) 29
What is the greatest common factor of 280 and 5,100? A) 40 B) 24 C) 22 D) 20 E) 15
You could do it manually but that would take a lot of time. Instead, break these numbers down into their prime factors. The number 280 is the same as 28 × 10, which can be broken down further to (7 × 4)(2 × 5). This is the same as (7 × 2²)(2 × 5) = 2³ × 5 × 7 The number 5,100 is the same as 50 × 102, which can be simplified to (25 × 2)(2 × 51). This is the same as (5² × 2)(2 × 3 × 17) =2² × 3 × 5² × 17 The prime factors shared between both numbers (280 and 5,100) are 2² and 5. Therefore, the greatest common factor is 2² × 5 =20 D) 20
