Hey! I have a math riddle/puzzle/problem that I don't understand. Any help?

Question

When Bert was just one year younger than Bill was half as old as Bill will be 3 years from now, Ben was twice as old as Bill was when Ben was 1/3 as old as Bert was 3 years ago. But, when Bill was twice as old as Bert, Ben was 1/4 as old as Bill was one year ago. What are the ages?

Any help would be greatly appreciated!

Answer 1

Let Bert be x years old, Bill be y years old and Ben be z years old.

Take one condition at a time. I'll start with the last one: "when Bill was twice as old as Bert, Ben was 1/4 as old as Bill was one year ago."
Take this to be at some point c years in the past; Bill was (y-c) years old, Bert (x-c) and Ben (z-c). Then
- Bill was twice as old as Bert, i.e. y-c = 2(x-c)
- Ben was 1/4 as old as Bill was one year ago, i.e. z-c = (y-1)/4
For the first equation we get c = y - 2x, and then putting this into the second equation we get
z - (y-2x) = (y-1)/4
<=> 4z - 4y + 8x = y - 1
<=> 8x - 5y + 4z = -1

I don't quite understand the earlier bit - it seems to be missing some words. ("One year younger than Bill was half as old" - there seems to be something missing between "than" and "Bill", or between "was" and "half". We only have two conditions and three variables, so I suspect there's a line missing or something and that we have the start of one condition tacked onto the end of another.) I'll show you how to set up what I can get from this, though.

At some point in the past, "Ben was twice as old as Bill was when Ben was 1/3 as old as Bert was 3 years ago."

"when Ben was 1/3 as old as Bert was 3 years ago" means when Ben's age was (x-3)/3 = x/3 - 1. Ben's age now is z, so this was (z - x/3 + 1) years ago.
At this point, Bill's age was y - z + x/3 - 1.

So, "when Bert was just one year younger than Bill was half as old as Bill will be 3 years from now", Ben's age was 2(y - z + x/3 - 1). Ben's age now is z, so this was z - 2(y - z + x/3 - 1) = (3z - 2y - 2x/3 + 2) years ago.
Bert's age then was x - (3z - 2y - 2x/3 + 2) = 5x/3 + 2y - 3z - 2 years.

We're up to the point now where I'm not sure of the interpretation, so I'll have to stop there. But continuing in this way you should be able to get another equation. As I said before, you need three equations (in general) to get a solution, so I think there's something missing from the problem statement above.

Answer 2

The question is really written improperly, so I'm going to make an attempt to decipher.

The first statement is misssing a word or two....
bert = bill - 1 when ?=(1/2)(bill +3)
ben = 2 *bill, when ben = 1/3 (bert-3)
bill = 2 bert when ben = (bill-1)/4
2 bill = 1/3 (bert-3)

ben = (1/2 bert - 1)/4
ben = 1/8 bert - 1/4
bert = 8 ben + 2

2 bill = 1/3 (8 ben + 2 - 3)
6 bill = 8 ben -1

unable to finish because of first statement

Answer 3

Bert is one year younger than Bill. He will ALWAYS be one year younger than Bill.
When Bill was twice as old as Bert (Bert was still a year younger than Bill) Bill was 2 years old and Bert was a year old.
I'm not sure about the rest.. Better grammar would have helped me understand the question...

Answer 4

I have NO clue what you are saying. You need to double check that you have the right wording!!!!

Answer 5

wtf