Solutions to Set 1

• Problem #1: 7/18
• Problem #2: 12
• Problem #3: 225
• Problem #4: 13.5
• Problem #5: 36
• Problem #6: 912
• Problem #7: 12421
• Problem #8: 11/28
• Problem #9: 30
• Problem #10: 15

Explanations

• Problem #1: You can reach Chamber D either by going to B and then D (let's call that Case 1), or by going to C and then D (Case 2). The probability of Case 1 is:
  (1/3)*(1/2) = 1/6.
  The probability of Case 2 is:
  (2/3)*(1/3) = 2/9.
  The probability of either Case 1 or Case 2 is:
  1/6 + 2/9 = 7/18.

  Read more about Combining Probabilities.

• Problem #2: Set up and solve an algebraic equation:

  (1 + 7 + 9 + 20 + x)/5 = x - 11
  1 + 7 + 9 + 20 + x = 5x - 55
  37 + x = 5x - 55
  92 = 4x
  23 = x

  Therefore, the arithmetic mean is
  (1 + 7 + 9 + 20 + 23)/5
  = 12.

  Read more about The Arithmetic Mean in Equations.

• Problem #3: The two perfect square integers surrounding 240 are 15²=225 and 16²=256. Test both of those perfect squares:
  240 - 225 = 15
  256 - 240 = 16

  The first difference is smaller, so 225 is the closest to 240.

  Read more about Perfect Squares.

• Problem #4: 12, x, y, z, 15 is an arithmetic sequence, so the difference between consecutive numbers stays the same. The total difference between 15 and 12 is 3. This increment of 3 is reached in 4 steps (12 to x, then x to y, then y to z, then z to 15). Therefore, the difference in each of these steps is 3/4 = 0.75.

  x = 12 + 0.75 = 12.75
  y = x + 0.75 = 12.75 + 0.75 = 13.5.

  Also notice that y is the average (arithmetic mean) of 12 and 15. This is true because it is an arithmetic sequence and because y falls halfway between 12 and 15.

  Read more about Completing Arithmetic Sequences.

• Problem #5: Let angle A measure 9x degrees and angle B measure 21x degrees. Because angles A and B are complementary,

  9x + 21x = 90
  30x = 90
  x = 3

  Therefore, angle A measures 9*3=27 degrees, and angle B measures 21*3=63 degrees. The measure of angle B minus the measure of angle A is 63-27=36 degrees.

  Read more about going From Ratios to Algebraic Equations and about Adding Angles.

• Problem #6: Let the variables x, y, and z represent the hundreds, tens, and ones digits, respectively. Based on the zombie's information, you can write the following equations:
  x + y + z = 12
  x + y = 5z
  y = 1.

  From here, you just have to solve the system of equations. Use the third equation, y = 1, to substitute 1 for y in the second equation:
  x + y = 5z
  x + 1 = 5z
  x = 5z - 1

  Now take the first equation, substituting 1 for y and 5z - 1 for x:
  x + y + z = 12
  5z - 1 + 1 + z = 12
  6z = 12
  z = 2.

  Now that you know the value of z, you can also find x:
  x = 5z - 1 = 5*2 - 1 = 9

  Since x = 9, y=1, and z = 2, the zombie's number is 912.

  Read more about going From Digits to Algebraic Expressions.

• Problem #7: In a five-digit palindrome, the first three digits automatically determine the rest of the number. So, just focus on the first three digits of the number. Since you are given 12345, which starts with the digits 123-, try the palindrome starting with 123-. This would be 12321. However, this is too small, so try the next smallest three-digit-start: 124-. This gives 12421, which is the answer.
• Problem #8: There are two cases to consider, both of which result in Johnny sleeping at your chosen time:

  Case 1: You choose Monday, Tuesday, Wednesday, or Thursday; the probability of this is 4/7. Within that day, you also choose one of the 9 hours in which he's asleep; the probability of this is 9/24, which is equivalent to 3/8. The probability of this case (choosing Monday, Tuesday, Wednesday, or Thursday, and choosing an hour when Johnny is asleep) is:
  (4/7) * (3/8) = 12/56 = 3/14.

  Case 2: You choose Friday, Saturday, or Sunday; the probability of this is 3/7. Within that day, you also choose one of the 10 hours in which he's asleep; the probability of this is 10/24, which is equivalent to 5/12. The probability of this case (choosing Friday, Saturday, or Sunday, and choosing an hour when Johnny is asleep) is:
  (3/7) * (10/24) = 30/168 = 5/28.

  The total probability of choosing a time when Johnny is asleep (the probability of either Case 1 or Case 2 occurring) is:
  3/14 + 5/28 = 11/28.

  Read more about Combining Probabilities.

• Problem #9: M and M-N are two integers that are factors of 31. You know that M is positive, so M-N is positive too.

  Since 31 is a prime number, its only two positive factors are 1 and 31.

  First, let's try M=1. Then M-N=31, and N=-30. But this is wrong because N is supposed to be a positive number.

  Our only other choice is M=31. Then M-N=1, and N=30. This works, so our answer is 30.

  Read more about Prime Numbers.

• Problem #10: Work backward. At the end of 2004, there were 90 ninjas. At the end of 2003, there were 90-54=36 ninjas. At the end of 2002, there were 36/2=18 ninjas. At the beginning of 2002, there were 18-3=15 ninjas.

  Read more about Translating Word Problems.