mathschallenge.net

Problem

A teacher, attempting to avoid revealing his real age, made the following statement:

"I'm only about twenty-three years old if you don't count weekends."

Can you work out how old the teacher really is?

Solution

If weekends are not counted, the given age only represents ⁵/₇ of his actual age.

Using this information:

⁵/₇ of 31 = 22.142... 22
⁵/₇ of 32 = 22.857... 23
⁵/₇ of 33 = 23.571... 24

So we can deduce that the teacher is about 32 years old.

Be careful, though. You may be thinking, "Why can't I just work out ²³/₅ 7 = 32.2?"

Suppose that the teacher claimed to be about 24 years old; ²⁴/₅ 7 = 33.6, and this would suggest that he was really 34 years old. However, consider what happens if we work the other way:

⁵/₇ of 33 = 23.571... 24
⁵/₇ of 34 = 24.285... 24

In other words, the ages 33 and 34 both map onto the reduced age of 24, therefore it is impossible to solve the problem in this particular case.

By investigating real ages mapped onto "reduced" ages, can you discover which ages produce unique values and which ages have more than one value?