Correct Answer: 90
Explanation: In the 100's, the numbers with 5 as the tens digit are 150, 151, 152, ..., 159 (ten numbers). Also, there will be 10 such numbers in the 200's, in the 300's, etc. Therefore, there are 10*9=90 three-digit numbers between 100 and 1,000 that have 5 as the tens digit.

question 4 of 7

The measures of the lengths of the three sides of a triangle are prime numbers. If two of the sides are 5 and 23, what is one possible value for the length of the third side?
(from the October 17, 1998 test)

Correct Answer: 19 or 23
Explanation: Since two of the sides of the triangle are 5 and 23, let y  represent the third side. By the triangle inequality

5 + y  > 23  or  y  > 18
5 + 23 > y   or  y  < 28
23 + y  > 5  or  y  > -18
The lengths of the three sides of the triangle are prime numbers so that y  must be a prime number between 18 and 28. Thus, the two possible values for the length of the third side of the triangle are 19 and 23. Either answer may be gridded as the correct answer to the problem.

question 5 of 7

The sum of the digits is 6.
Each digit is different.
The number is odd.
What is the greatest 4-digit number that has all of the characteristics listed above?

(from the October 20, 1998 test)

Correct Answer: 3,201
Explanation: Since the sum of the digits of the four-digit number is 6, none of the digits can be greater than 6. The greatest four-digit number whose digits sum to 6 is 6,000. However, each digit must be different and the number must be odd. The greatest four-digit number having the given characteristics will have the largest digit in the thousands place. To maximize the number in the thousands place, let the units digit be 1. The thousands place cannot be 5 since 5,001 does not have four different digits. The thousands place cannot be 4 since 4,101 still does not have four different digits. If the thousands place is 3, then the number could be 3,201 and this is the greatest four-digit number that satisfies all three given conditions.

question 6 of 7

The sum of r and p is equal to twice s, and p is 36 less than twice the sum of r and s. What is the value of r ?
(from the October 12, 1999 test)

Correct Answer: 12
Explanation: The phrase "the sum of r and p is equal to twice s" can be written as r + p = 2s. The phrase "p is 36 less than twice the sum of r and s" can be written as p = 2(r + s) - 36. From the first equation, p = 2s - r. Substituting this expression for p into the second equation yields

2s - r = 2(r + s) - 36

2s - r = 2r + 2s - 36

-r = 2r - 36

-3r = -36

r = 12

The value of r is 12.

question 7 of 7

RESULTS OF AN ELECTION  Candidate   Number of Votes  
X 7,400
Y 2,375
Z 5,250

The results for three candidates in an election are shown in the table above. If each voter voted for exactly one candidate, what is the fewest number of voters who would have had to vote differently in order for Candidate Z to have received more votes than Candidate X?

(from the October 16, 1999 test)

Correct Answer: 1076
Explanation: For this question, let k be the number of voters who changed their vote. Since you want to make k as small as possible, the k voters should come from those who voted for Candidate X. To determine an answer to the problem, you would need to solve the inequality 5,250 + k > 7,400 - k. Solving this inequality yields 2k > 2150 or k > 1075. Therefore, 1,076 voters who had voted for Candidate X would have to change their vote and vote for Candidate Z in order for Candidate Z to receive more votes than Candidate X. The correct answer to this question is 1076.

Grid-ins (student-produced response questions) require you to solve the problem and enter your answer by marking the ovals in the special grid.