Notice that in some problems, there is a hidden constraint on the possible quantities. This would apply, for instance, to the number of apples and bananas that Jane bought. Because each fruit is a physical, countable object, you can only have a whole number of each type. Whole numbers are the integers 0, 1, 2, and so on. So you can have 1 apple, 2 apples, 3 apples, and so on, and even 0 apples, but you cannot have fractional apples or negative apples.
As a result of this implied “whole number” constraint, you often have more information than you might think, and you might be able to answer a question with fewer facts.
Consider the following example:
If Kelly received 1/3 more votes than Micah in a student election, which of the following could have been the total number of votes cast for the two candidates?
- 12
- 13
- 14
- 15
- 16
Let M be the number of votes cast for Micah. Then Kelly received M + (1/3)M, or (4/3)M votes. The total number of votes cast was therefore “votes for Micah” plus “votes for Kelly,” or M + (4/3)M. This quantity equals (7/3)M, or 7M/3.
Because M is a number of votes, it cannot be a fraction—specifically, not a fraction with a 7 in the denominator. Therefore, no matter what M is, the 7 in the expression 7M/3 will not be canceled out. As a result, the total number of votes cast must be a multiple of 7. Among the answer choices, the only multiple of 7 is 14, so the correct answer is (C).
Another way to solve this problem is this: the number of votes cast for Micah (M) must be a multiple of 3, since the total number of votes is a whole number. So M = 3, 6, 9, etc. Kelly received 1/3 more votes, so the number of votes she received is 4, 8, 12, etc. Thus, the total number of votes is 7, 14, 21, etc.
Not every unknown value related to “real-world” objects is restricted to whole numbers. Physical measurements such as weights, times, or speeds have to be positive numbers, but do not have to be integers. A few physical measurements can even be negative (e.g., temperatures, x- or y-coordinates). Think about what is being measured or counted, and you will recognize whether a hidden constraint applies.
Translate the following statements.
In a certain word, the number of consonants is 1/4 more than the number of vowels. Which of the following is a possibility for the number of letters in the word?