A couple is taking their two daughters to a local fair. They are celebrating a birthday of one of the girls. It also marks the fathers’ favorite day of the year because it reminds him how lucky he is to not be barred from such festivities. Use the clues below to determine the ages each family member; which daughter is adopted; which daughter is the birthday girl; what is the difference in age of the two girls.

Bonus question: Why is it the fathers favorite day?

1. Dad is six times older than his youngest daughter.
2. The adopted daughters' age in days is 100 times her adopted mother's age in years.
3. In three years the dad will be exactly three times the difference in age between him and his younger wife.
4. The age of the two parents together is one less than ten times the eldest daughter's age.
5. The combined age of all four family members in whole years is seventy-one.

Note: Ages are to be taken in years unless otherwise stated and assume there is no leap year.

first off, let the family members ages be represented by:
Father - F
Mother - M
Adopder Daughter - A
Deaughter - D
aslo
older daughter: O
younger daughter: Y

Firstly looking at clue 4.
The parents ages + 1 can be no greather then 70 as according to clue 5 this wouldnt leave any room for the younger daughter

so Obviously 10*O cannot be greater then70 as there is not enough years for the younger daughter.   (it is possible for 10*O to be 70 while satisfying both clues 4 and 5, but thie leaves both daughters being 1. And by lookign at clue 1 this makes the father 6. which is ... absurd so i feel we can rule that possibilty out already as well)
Also 10*O cannot be 50 or less. as this would mean the older daughter is 5, the parents add up to 49, leaving the younger daughter as 17 years old.

Thereofore O must equal 6, and the parents ages combined must be 59.
10 * 6 - 1 = 59

Now plugging this into clue 5 we get
59 + 6 + Y = 71
Therefore Y = 6.
Since it is one daughters birthday, one daughter must be exactly 6 and one must be 6 and x days. Therefore the youngest daughter is the birthday girl

We now know that both daughters are 6 years old. substituting this into clue 1 gives us.
F = 6 * 6
   = 36

Now going back to clue 5 again.
36 + M + 6 + 6 = 71
M = 23

So in years.
F - Father = 36
M - Mother = 23
A - Adopted Daughter = 6
D -Daughter = 6

Now we must determine how many days older is the oldest daughter.

If we look at clue 2 we can determine that  365A = 100M
If we assume the adopted daughter is the birthday girl we know that her age in days is 365A = 2190 (since A = 6)
Therefore 100M = 2190
                    M = 21.9
Since we know the mothers age is 23 this cannot be correct, which means the adopted daughter is not the birthday girl and is therefore the older daughter.

We now know all the ages as well as
which daughter is the birthday girl: The youngest
Which daugter is adopted: The oldest

all thats left is to determine the age difference.between the 2 girls.
Referring to clue 2 again we know that the daughters age in days is 2300 (100 times mothers age).
and since it isthe younger daughters birthday we know her age in days is 2190 (365*6)
which leaves the difference as
2300-2190 = 110 days.

and we've finally got the answers for everything except the bonus question.
From the wording of the problem it seems pretty obvious that he was in prison, or atleast in danger of being there.
Since the rest of the problem is all about ages, i would assume the reason for this would be something to do with the fact that the mother was underage, seeing as she gave birth at only 17. But i cant be sure about this part :)


Pretty sure i got all that right, and i hope it actuallly makes sense to other people :)

Posted by Urza on 2004-11-21 16:27:31