Strategy 12: Adapt Solutions to Related Problems

When solving a new problem that seems similar to a problem that you know how to solve, you can try to solve the new problem by adapting the solution — both the strategies and the results — of the problem you know how to solve.

If the differences between the new problem and the problem you know how to solve are only surface features — for example, different numbers, different labels or different categories — that is, features that are not fundamental to the structure of the problem, then solve the new problem using the same strategy as you used before.

If the differences between the new problem and the problem you know how to solve are more than just surface features, try to modify the solution to the problem you know how to solve to fit the conditions given in the new problem.

• This strategy is used in the following two sample questions.

This is a Multiple-Choice – Select One or More Answer Choices Question.

1. Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is \$25,000 for the employees in Department X and \$35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?

Indicate all such amounts.

(A) \$26,000
(B) \$28,000
(C) \$29,000
(D) \$30,000
(E) \$31,000
(F) \$32,000
(G) \$34,000

Explanation

One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between \$25,000 and \$35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of \$25,000 and \$35,000, which is \$30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean, dollars

where the weight for \$25,000 is 2 and the weight for \$35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to \$25,000 because the weight for \$25,000 is greater than 2. This means that \$28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore \$26,000 and \$28,000. Thus, the correct answer consists of Choices A (\$26,000) and B (\$28,000).

Intuitively, you might expect that any amount between \$25,000 and \$28,333 is a possible value of the average salary. To see that \$26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that \$28,000 is possible, use the respective weights 7 and 3.

Note: This question and explanation also appear as an example of Strategy 8.

This is a Multiple-Choice – Select One or More Answer Choices Question.

1. Which of the following could be the units digit of where n is a positive integer?

Indicate all such digits.

 (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 (F) 5 (G) 6 (H) 7 (I) 8 (J) 9

Explanation

The units digit of is the same as the units digit of for all positive integers n. To see why this is true for compute by hand and observe how its units digit results from the units digit of Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with and proceeding consecutively, the units digits of 7,   and are 7, 9, 3, 1 and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of and therefore of The correct answer consists of Choices B (1), D (3), H (7) and J (9).

Note: This question and explanation also appear as an example of Strategy 7. 