E X A M P L E S

•   Begin with Choice (C)
•   Take A Guess
•   Pick A Number

Picking Smart
Numbers




Which of the following is equal to ?

(A)
(B)
(C)
(D)
(E)





Let’s choose a value for k, say k = 3. We first substitute a 3 in for k into the given expression and use our calculator.




We type in the following: (3 + 60)/12 and we get k = 5.25. Put a nice big, dark circle around this number so that you can find it easily later. We now substitute a 3 into each answer choice and use our calculator.

(A) 3/12 + 5 = 5.25
(B) 3 + 5 = 8
(C) 5*3 = 15
(D) (3 + 30)/6 = 5.5
(E) (3 + 5)/6 ~ 1.33 (~ means “is approximately”)





We now compare each of these numbers to the number that we put a nice big, dark circle around. Since (B), (C), (D) and (E) are incorrect we can eliminate them. Therefore the answer is choice (A).




Important note: (A) is not the correct answer simply because it is equal to 5.25. It is correct because all 4 of the other choices are not 5.25. You absolutely must check all five choices!




As an example of how things could go wrong with incorrect reasoning, suppose we choose k = 0. Then the given expression becomes 5, and the answer choices become
(A) 5
(B) 5
(C) 0
(D) 5
(E) 5/6





In this case we have eliminated (C) and (E), but (A), (B) and (D) are all potential solutions. A common error is to choose the first answer to come out correct.




This is certainly a good guessing strategy, especially if you’re running out of time, but it can potentially lead to the wrong answer. This is also why we generally try not to pick numbers to be too simple. It often leads to multiple answer choices coming out to the correct number.




A quick algebraic solution: A discussion of this problem wouldn’t be complete without showing how to solve this problem very quickly using an algebraic method.




Most students have no trouble at all adding two fractions with the same denominator. For example,




   k/12 + 60/12 = (k + 60)/12




But these same students have trouble reversing this process.




   (k + 60)/12 = k/12 + 60/12




Note that these two equations are identical except that the left and right hand sides have been switched. Note also that to break a fraction into two (or more) pieces, the original denominator is repeated for each piece.




* An algebraic solution to the above problem consists of the following quick computation




(k + 60)/12 = k/12 + 60/12 = k/12 + 5




This is choice (A).