Authors: Dr. Gary R. Gruber

showed was “if two sides of a triangle are equal, then the base angles are equal.” The teacher drew a perpendicular line to start the proof—unfortunately the teacher probably did not tell you that the reason she did this was because when you draw something you get more information and are able to start solving the problem. A natural phenomenon! So in this problem, draw a line to make a rectangle in the figure and you will find that you will have a 3-4-5 triangle.

x = 5.

37. (c) 40 percent

Translate words to math. Percent can be translated to 1/100, of to X, what to x, is to =.

A = [250/100](B).

x /100 (A) = B.

Substitute: x /100[(250/100)]B = B.

Cancel B: x /100[250/100] = 1. 250 x /10000 = 1 so x = 10,000/250 = 40.

38. The average rate is less than the average of the rates.

Average rate = 2AB/(A + B).

Average of the rates = (A + B)/2.

Detailed solution:

From the formula of Rate x Time = Distance, calling the time for the car to go uphill, t, and the time for the car to go downhill, T, with D being the distance downhill and uphill,

(1) A x t = D and

(2) B x T = D.

Now, Average rate = Total distance/Total time.

So,

(3) Average rate = 2 D/( t + T).

From (1) we have

(4) t = D/A

and from (2) we have

(5) T = D/B

Substituting from (4) and (5) in (3), we get

(6) Average rate = 2D/(D/A + D/B)

This becomes

(7) Average rate = 2AB/(A + B)

But the average of the rates is just (A + B)/2. So we ask, is

2AB/(A+B) greater, equal to, or less than (A+B)/2?

So we compare

2AB/(A+B) with (A+B)/2.

Put 2AB/(A+B) under a Column A and (A+B)/2 under a Column B and let’s compare the Columns:

Now multiply both columns by 2 and then by (A + B). We get

This becomes

Now subtract 4AB from both columns.

We get:

But A x A - 2AB + B x B = (A - B)(A - B) so we get:

Since we are told that A and B are different rates, A - B cannot be 0, and so (A - B)(A - B) is always greater than 0 whether A is greater than B or less than B. So Column B is greater than Column A and the original quantity in Column B, the average of the rates, is greater than the original quantity in Column A, the average rate.

39. e, a, c, b, d

40. (e) 13

Almost everyone who tries this problem gets it wrong! To get the right answer, you have to visualize something that is not straightforward.

When this problem first appeared in the Washington Post and other newspapers throughout the country in 1995, forty percent of people answered (a) 9 because they just added the small triangles in the figure, ABC, BDE, BEC, etc. Twenty-five percent of people answered (b)10 because they added the small triangles in the figure and also added the big triangle AGJ. Eight percent of people answered (c) 11 because they added the small triangles, the big triangle AGJ, and triangle ADF. Sixteen percent chose (f) none of the above, many of whom merely guessed at an answer without really working it out. Seven percent of people chose the correct answer, (e) 13, because they added the small triangles, the big triangle, triangle ADF, and the triangles BGI and CHJ.

Note: Four percent of people chose (d) 12—they were almost correct. They added the small triangles, the big triangle, and triangle ADF but then only added the triangle BGI and not also CHJ.

41. When the bird flies, it pushes down on the air, which pushes down on the scale. The scale reads the same.

42. (e) cannot be determined because this is an ambiguous question

How can you overtake the person who is last?

43. (b) 202/962 is greater. Add 208/962 to both quantities and compare.

44. (e) termite : house

Create a sentence expressing a specific relationship between the capitalized words.

MOTH is a living thing that destroys CLOTHING, as termite is a living thing that destroys a house.

45. (a) No good person lives to an old age and (d) All bad people do not die young. “Only the good die young” means that those who are not good do not die young and no good

x = 5.

37. (c) 40 percent

Translate words to math. Percent can be translated to 1/100, of to X, what to x, is to =.

A = [250/100](B).

x /100 (A) = B.

Substitute: x /100[(250/100)]B = B.

Cancel B: x /100[250/100] = 1. 250 x /10000 = 1 so x = 10,000/250 = 40.

38. The average rate is less than the average of the rates.

Average rate = 2AB/(A + B).

Average of the rates = (A + B)/2.

Detailed solution:

From the formula of Rate x Time = Distance, calling the time for the car to go uphill, t, and the time for the car to go downhill, T, with D being the distance downhill and uphill,

(1) A x t = D and

(2) B x T = D.

Now, Average rate = Total distance/Total time.

So,

(3) Average rate = 2 D/( t + T).

From (1) we have

(4) t = D/A

and from (2) we have

(5) T = D/B

Substituting from (4) and (5) in (3), we get

(6) Average rate = 2D/(D/A + D/B)

This becomes

(7) Average rate = 2AB/(A + B)

But the average of the rates is just (A + B)/2. So we ask, is

2AB/(A+B) greater, equal to, or less than (A+B)/2?

So we compare

2AB/(A+B) with (A+B)/2.

Put 2AB/(A+B) under a Column A and (A+B)/2 under a Column B and let’s compare the Columns:

Now multiply both columns by 2 and then by (A + B). We get

This becomes

Now subtract 4AB from both columns.

We get:

But A x A - 2AB + B x B = (A - B)(A - B) so we get:

Since we are told that A and B are different rates, A - B cannot be 0, and so (A - B)(A - B) is always greater than 0 whether A is greater than B or less than B. So Column B is greater than Column A and the original quantity in Column B, the average of the rates, is greater than the original quantity in Column A, the average rate.

39. e, a, c, b, d

40. (e) 13

Almost everyone who tries this problem gets it wrong! To get the right answer, you have to visualize something that is not straightforward.

When this problem first appeared in the Washington Post and other newspapers throughout the country in 1995, forty percent of people answered (a) 9 because they just added the small triangles in the figure, ABC, BDE, BEC, etc. Twenty-five percent of people answered (b)10 because they added the small triangles in the figure and also added the big triangle AGJ. Eight percent of people answered (c) 11 because they added the small triangles, the big triangle AGJ, and triangle ADF. Sixteen percent chose (f) none of the above, many of whom merely guessed at an answer without really working it out. Seven percent of people chose the correct answer, (e) 13, because they added the small triangles, the big triangle, triangle ADF, and the triangles BGI and CHJ.

Note: Four percent of people chose (d) 12—they were almost correct. They added the small triangles, the big triangle, and triangle ADF but then only added the triangle BGI and not also CHJ.

41. When the bird flies, it pushes down on the air, which pushes down on the scale. The scale reads the same.

42. (e) cannot be determined because this is an ambiguous question

How can you overtake the person who is last?

43. (b) 202/962 is greater. Add 208/962 to both quantities and compare.

44. (e) termite : house

Create a sentence expressing a specific relationship between the capitalized words.

MOTH is a living thing that destroys CLOTHING, as termite is a living thing that destroys a house.

45. (a) No good person lives to an old age and (d) All bad people do not die young. “Only the good die young” means that those who are not good do not die young and no good

Louis Trimble

Debbie Macomber

David Ashton

Avery Flynn

Catt Ford

Dianne Nelson, Dianne Nelson Oberhansly

Jennifer Johnson

James R. Benn

Nick Oliver

Gregory Martin