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Age of ages (Posted on 2003-07-16) Difficulty: 3 of 5
Thanks to Ravi for introducing me to these with his problems but here goes. A father, mother, son, daughter and grandmother are all compairing ages. They find out this:

1) When the mother was the daughter' current age, the grandmother was twice as old as the father.

2) In one year, the father will be twice as old as his son.

3) When the daughter was 2/3 her current age, the son was twice as old as she was.

4)When the mother was 10 years younger than the daughter will be when she is twice her current age, the grandmother was twice as old as the mother was.

What are all the ages in the family?

See The Solution Submitted by Jon    
Rating: 3.0000 (2 votes)

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Solution Solution | Comment 3 of 12 |
If the ages are: father, F; mother, M; daughter, D; son, S; grandmother, G, then:

1. This was M-D years ago so G-(M-D)=2(F-(M-D)) or G=2F-M+D

2. F+1=2(S+1) or F = 2S+1

3. S-D/3=2(2D/3) or S=5D/3

4. This was M-(2D-10) years ago so G-(M-(2D-10))=2(2D-10) or G = 2D+M-10

and from 1 & 4: M=(2F-D+10)/2

From 3, D must be a multiple of 3 and from the last equation, from 1&4, D must be even, thus D must be a multiple of 6. Using the possible values for D we get a table of the other values:

d s f m g
6 10 21 23 25
12 20 41 40 54
18 30 61 57 83
24 40 81 74 112
30 50 101 91 141

The first row has the grandmother to young to be a parent to either the father or the mother.

The second row is conceivable if the grandmother gave birth to the mother at age 14.

The third row is presumably the sought answer with the grandmother having given birth to either the mother at age 26 or the father at age 22, and the mother having given birth at ages 27 and 39.

The third row is also conceivable with the grandmother having given birth to the father at age 31 and the mother having given birth at ages 34 and 50.

The last row has an age of 141, which age I don't think anyone has reached, and the mother would have had to have given birth at age 61.
  Posted by Charlie on 2003-07-16 08:54:39
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