> This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.

How did you actually make that deduction? I can only see it by writing down an equation. I can't see anything in the problem that directly implies it.

▲ losvedir 4 hours ago | parent | next [-]

> How did you actually make that deduction?

Think of it on a number line. Mary is 24 now, Ann was 12 then. There's a point somewhere between the two, representing Mary's age at that time.

     ---*-------*---------------*--------
        12      ?               24

The problem states that time has passed such that Mary has aged to 24 and Ann has aged to that point. You can think of time passing as a line growing out of each of them:

     ---*===----*===------------*--------
        12      ?               24

Since time progresses equally for both of them, and Ann is now that mystery age, and Mary is now 24, you can see that the distance from "12" to "?" has to be the same distance as "?" to "24".

	▲	seemaze 4 hours ago \| parent [-]
		That's pretty much how I approached it; `A1 M1 A0 M0 +----+----+----+----+----+ 0 6 12 18 24`

▲ judofyr 6 hours ago | parent | prev | next [-]

I imagined that I'm 24 years old and I have a 20 years old brother. When I were his age, he would have been 16 years old.

I also realized that this can be expressed in terms of a two pairs of sibling:

> Mary and Ann are the same age difference as Jane and Claire. Mary is 24 years old. Mary is twice as old as Jane. Claire is as old as Ann. How old is Ann?

This highlights also why it's so confusing:

> Mary is 24 years old. She [Mary, today] is twice as old as Ann was [Ann, past] when Mary was [Mary, past] as old as Ann is [Ann, today] now. How old is Ann?

In one sentence we're comparing past and present ages.

▲ stavros 6 hours ago | parent | prev [-]

Mary is 24 years old. When Ann was 12, Mary was Ann’s current age.

Let's say Ann's current age is 13. Then, when Ann was 12, Mary must have been 13. Now that Ann is 13, Mary must be 14, but she's 24.

This means that the only way for Mary's-age-when-Ann-was-12 to have been Ann's-age-now is for the same amount of years to have passed between Ann being 12 to being Ann's-age-now than from Ann's-age-now/Mary's-age-then to Mary's 24, which is 18.