You have a group of eighty-two men. Some of the men are tall, the number of the medium-height men is double that of the tall men and the number of the short men is triple that of the medium-height men and one man is very tall.
You know that all tall men are bald but not all of the short men are bald, and that some of the medium-height men are bald but the number of the medium-height men with hair is double of that of the bald medium-height men. You also know that none of the tall men does not have moustache while none of the short men with hair has moustache, but half of the bald medium-height men have moustaches and nine of the medium height men with hair have moustaches too. If the number of the bald short men with moustache is less by three than the rest of other men that have moustaches and the percentage of total men with moustaches is 50%, then the question is: does the very tall man have moustache or not (do not guess) and how can you prove that?
Answer
Yes, the very tall man has moustache.
Let the number of tall men = X then the number of medium-height men is 2X and the number of short men is 6X (3*2X).
Total number of men = X+2X+6X+1(very tall man) = 82
Solving 9X = 81 gives:
Number of tall men is 9
Number of medium-height men is 18
Number of short men is 54
And 1 very tall man
Grouping facts:
- All tall men are bald and none of the tall men does not have moustache (in other words, all of them have moustaches). That means you have 9 tall bald people with moustaches.
- The number of medium-height men with hair is double of the number of bald medium-height men (assume it Y) gives: Y+2Y = 18, and so Y = 6 which is the number of bald medium-height men and the number of medium height-men with hair = 12.
9 medium-height men with hair have moustaches and 3 bald medium-height men (half of them) have moustaches too. - Adding up the number of men that have moustaches up till this point from 1 and 2 above:
Number of men with moustaches = 9(tall) + 9(medium-height men with hair) + 3(bald medium- height men = 21
If the very tall man does not have a moustache, then the number of bald short men have moustache = 18 (less by 3 than the above number.) If the very tall man does have moustache, then the number of bald short men is 19 (less by 3 than the above number plus one for the very tall man.)
Percentage of the total number of men with moustache = 0.5
= 0.5 * 82 = 41 men
There must be 21 + (21 + W) – 3 + W = 41 men with moustaches (where W is the one very tall man if having moustache, and either equals 1 if he has moustache or = 0 if he does not have one.)
Solving the equation: 42 + 2W – 3 = 41 gives that W = 1 and thus the very tall man has moustache.
