Data Sufficiency

S is a set of integers such that i) if a is in S, then –a is in S, and ii) if each of a and b is in S, then ab is in S. Is –4 in S? (1) 1 is in S. (2) 2 is in S.

What is the value of n? (1) n(n - 1)(n - 2 ) = 0 (2) n^2 + n - 6 = 0

If n = p + r, where n, p, and r are positive integers and n is odd, does p equal 2? (1) p and r are prime numbers. (2) r ≠ 2

The participants in a race consisted of 3 teams with 3 runners on each team. A team was awarded 6 –n points if one of its runners finished in nth place, where 1 <= n <= 5. If all of the runners finished the race and if there were no ties, was each team awarded at least one point? (1) No team was awarded more than a total of 6 points. (2) No pair of teammates finished in consecutive places among the top five places. Show Spoiler

If n is a positive integer, is the value of b - a at least twice the value of 3 n − 2 n 3 𝑛 − 2 𝑛 ? (1) a= 2 (n+1) 𝑎 = 2 ( 𝑛 + 1 ) and b= 3 (n+1) 𝑏 = 3 ( 𝑛 + 1 ) (2) n=3 𝑛 = 3

Last year Luis invested x dollars for one year, half at 8 percent simple annual interest and the other half at 12 percent simple annual interest. Now he wants to reinvest the x dollars for one year in the same two types of investments, but the lower rate has decreased. If the higher rate is unchanged, what fraction of the x dollars must he reinvest at the 12 percent rate so that the total interest earned from the x dollars will be the same for both years ? (1) The lower rate is now 6 percent. (2) The total amount of interest earned from the two investments last year was $3,000.

What is the value of xy? (1) x + y = 10 (2) x - y = 6

What is the value of x yz 𝑥 𝑦 𝑧 ? (1) x= y 2 𝑥 = 𝑦 2 and z= 2x 5 𝑧 = 2 𝑥 5 (2) x z = 5 2 𝑥 𝑧 = 5 2 and 1 y = 1 10 1 𝑦 = 1 10 DS86602.01

The figure above represents a circle graph of Company H's total expenses broken down by the expenses for each of its five divisions. If O is the center of the circle and if Company H's total expenses are $5,400,000, what are the expenses for Division R ? (1) x = 94 (2) The total expenses for Divisions S and T are twice as much as the expenses for Division R. Show Spoiler

Did the population of Town C increase by at least 100 percent from the year 2000 to the year 2010 ? (1) The population of Town C in 2000 was 2/3 of the population in 2005. (2) The population of Town C increased by a greater number of people from 2005 to 2010 than it did from 2000 to 2005.

If p and q are positive integers, what is the value of q? (1) qp−1=1 (2) p=1

If the area of triangular region RST is 25, what is the perimeter of RST? (1) The length of one side of RST is 5 2 √ 5 2 . (2) RST is a right isosceles triangle.

If x and y are integers between 10 and 99, inclusive, is (x - y)/9 an integer? (1) x and y have the same two digits, but in reverse order. (2) The tens’ digit of x is 2 more than the units digit, and the tens digit of y is 2 less than the units digit. DS00502.01

If y= 2 (x+1) 𝑦 = 2 ( 𝑥 + 1 ) , what is the value of y – x? (1) 𝑦 = 2 ( 𝑥 + 1 ) , what is the value of y – x? (1) 2 (2x+ (2) —

If x+y+z>0 𝑥 + 𝑦 + 𝑧 > 0 , is z>1 𝑧 > 1 ? (1) z>x+y+1 𝑧 > 𝑥 + 𝑦 + 1 (2) x+y+1<0 𝑥 + 𝑦 + 1 < 0 DS18602.01

The price per share of Stock X increased by 10 percent over the same time period that the price per share of Stock Y decreased by 10 percent. The reduced price per share of Stock Y was what percent of the original price per share of Stock X ? (1) The increased price per share of Stock X was equal to the original price per share of Stock Y. (2) The increase in the price per share of Stock X was 10/11 the decrease in the price per share of Stock Y.

Is the integer n a multiple of 140? (1) n is a multiple of 10. (2) n is a multiple of 14.

Elena receives a salary plus a commission that is equal to a fixed percentage of her sales revenue. What was the total of Elena's salary and commission last month? (1) Elena's monthly salary is $1 ,000. (2) Elena's commission is 5 percent of her sales revenue.

An infinite sequence of positive integers is called an "alpha sequence" if the number of even integers in the sequence is finite. If S is an infinite sequence of positive integers, is S an alpha sequence? (1) The first ten integers in S are even. (2) An infinite number of integers in S are odd.

What is the value of 1 k + 1 r ? (1) k+r=20 (2) kr=64