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Note: These "mathematical reasoning" problems were taken from practice tests for a Computer Programmer's Aptitude Exam. For each problem, select from the five choices A-D....and document your reasoning.

  1. A taxi charges X cents for the first mile and Y cents for every mile thereafter. About how far can I travel in it for Z cents? (Assume Z > X.)
    (A) Z/Y - X/Y (B) 1 + (Z-X)/Y (C) Z/(X+Y) (D) Z/(X-Y)
  2. A man is X years old. His older brother, who is Y years older than X, is Z years younger than his father. How old is the father?)
    (A) X+Y+Z (B) X-Y+Z (C) X+Y-Z (D) X-Y-Z
  3. If Y = A/(A+B) + B/(A+B) and A increases at twice the rate B decreases, then Y
    (A) gets larger (B) gets smaller (C) remains the same (D) not determinable from the data
  4. The average height of three boys is X inches. If adding a fourth boy to the group reduces the average height by 2 inches,
    (A) all of the original boys are taller than the fourth boy
    (B) at least one of the original three boys is shorter than the fourth boy
    (C) at least two of the original three boys are taller than the fourth boy
    (D) none of the above conclusions are necessaruily true
  5. If X is between 0 and -1, the largest of the following is
    (A) 2X (B) -2X (C) X2 (D) -2 + X
  6. On the average, X percent of a doctor's operations are successful. Of the successful ones, Y percent live. Out of Z patients how many live although their operations were not successful?
    (A) [Z(X-Y)]/100 (B) [XYZ]/10000 (C) [Y/100][Z + (2X)/100] (D) [(XY)/100][1 - X/100]
  7. If X > T and Y > 2T, it is true that
    (A) X > Y (B) (X+Y) > 2T (C) (X-T) < (2T-Y) (D) Y/X > Z
  8. If S = (X-1)(X+2) and T = (2X+3)(X-1), for how many values of X are S and T equal?
    (A) 1 (B) 2 or less (C) 3 (D) 0
  9. John started from his home by car traveling on a certain road at X mph. Two hours later his brother started after him on the same road traveling at X+Y mph. In how many hours did his brother overtake John?
    (A) [2X]/[X+Y] (B) [X+Y]/X (C) [X+Y]/[2X] (D) [2X]/Y
  10. If X > Y and Y > Z, then it follows that
    (A) [X+Z]/2 > Y (B) (Y+Z) < 2X (C) (3Z+Y) < 4X (D) X/Y > Y/Z)
  11. If X/Y > Z, then
    (A) X > (Y+Z) (B) (X+Z) > Y (C) X > Y (D) X > YZ
  12. A barber can give X haircuts in an hour. he can shave Y men per hour. If all of his customers want both shaves and haircuts, how many men can he service in an hour?
    (A) 2/[X+Y] (B) [XY]/[X+Y] (C) [X+Y]/2 (D) 1/X + 1/Y
  13. The length of a rectangular box is X and the width is Y. If the rectangle is changed so that the area remains the same but the length becomes 2X+3, then the width must become
    (A) [XY]/[2X+3] (B) [X+Y]/{2X+3] (C) [2X+3]/X (D) Y + [2X+3]/X
Where you able to do these without substituting in numbers as a check...that is, could you "mathematically reason" with the variables and conditions as given?

Note: At least one of these problems should have made you scream "But none of these answers are right!" Any idea which one(s)?


Source: Adapted from John Jensen's How To Pass Computer Programmer Aptitude Tests, 1967

Hint: If you get stuck, sometimes substituting numbers in can be helpful...but, such can also lead you astray!


Solution Commentary: The text's given answers (in reverse order) is the title for this problem.

Now which problem(s) is impossible...for example, rethink through the third from the last problem: "If X/Y > Z ...etc." or ....