All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
An age conversation (Posted on 2013-01-22) Difficulty: 3 of 5
I was chatting with an older friend of mine when I happened to mention the ages of my two children. He said to me, "when your youngest was born your age was the square of the older one. Now your age is the product of both their ages."

I then thought about his age and the ages of his children and replied, "When your youngest was born your age was the product of the older two. Later on your age was the product of the oldest and youngest. Now your age is the product of the younger two."

What are our two ages?

(Assume we all have the same birthday.)

No Solution Yet Submitted by Jer    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution to the first part | Comment 1 of 3
Let the respective ages of the man, his youngest child and the oldest child be A, B and C.

Then, by the problem:

(i) A=B*C with C > B
                      2  
(ii) A-B = (C-B)  
                                                                          2
Let C-B=x, then from (ii) we must have: A-B = x  so that from (i), we have:

      2
B+x  = B(B+x)
                           2                  
                         x
or,   B+x = 1 + --- ...... (iii)
                         B
                 
                                            
If x=B, then, (iii) => 2B = B+1=> B = 1. Hence, x=1, so that: C=2, giving: A=1*2=2, which is inadmissible, since no father's age can be 2.


If, x< B then, B+x < 1+B, so x < 1. Accordingly, x=0, which yields A=B=C, which is a contradiction.

Consequently, we must have x > B ....(iv)

Then,
 
 2           2                   2                       2
x < B + x  = B(B+x) = B +Bx < (B + x/2)

or, x < B + x/2

or, B > x/2  ..... (v)
                                                               
                                                                       2
Combining (iii), (iv) and (v) we observe that B is a factor of x  with the restriction x/2 < B < x ----(vi)

                                                                               
The normal lifetime of a human being is at most 100 years.
                                    2
So, 100 >= A =B*C > B  (since C>B )=> B <10=> B<=9 ....(vii)

                                                                              2
If x is a prime number, then there is no factor of x  satisfying the restriction in (vi) unless x/2=1 or, x=2
which again violates the said restriction.
                                                     a
If x is a prime power, so that x= p where p is a prime number, then we must have x/2 > x/a, so that a<2, or a=1,
which is not a prime number and hence a contradiction.

The only positive integers less than or equal to 9 which are not a prime or a prime power are 1 and 6 so that x= 1, 6.
It has been shown earlier that x=1 is not valid.

For x=6, the only  factor of 36 which is greater than 3 but less than 6 is 4. Accordingly, B=4.

Hence, C=4+6=10, giving: A = 4*10=40

Consequently, the respective ages of the man, his youngest child and the oldest child are 40, 10 and  4.

Edited on January 22, 2013, 2:36 pm

Edited on January 22, 2013, 2:47 pm

Edited on January 22, 2013, 3:16 pm
  Posted by K Sengupta on 2013-01-22 14:25:15

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information