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Mary is 24, which is twice as old as Ann was some time ago. At that time, Mary was the age that Ann is now. What is that age?
Wait so how old is she?
> A man is twice the age his wife was when he was the age she is now. When she reaches his present age, their combined years will be 100. Find the age of each.
This seemed initially like something amenable to guessing and checking a few ratios. It was not.
My answer: The man is 44 4/9 years old, the wife 33 1/3 years.
Earlier quoted context omitted.
> 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.
> 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: ---*===----*===-…
Wild assumption. I saw absolutely nothing in the text to indicate their relative velocities.
My approach at parsing this: There's two point in time with two ages each: M, M0, and A, A0 Then the sentence can be expressed as: * M = M0 + X and A = A0 + X and (X is the time difference) * M = 24 and ("Mary is 24") * M = 2 * A0 and ("Mary is twice as old as Ann was") * M0 = A ("Mary was as old as Ann is now")
MO (mary old), MN (mary now), AO (ann old), AN (ann now)
MN = 24
MO = AN
2AO = MN
MN - AN = MO - AO
Rephrasing it makes it easier to grasp: Mary is 24 years old. When Mary was Ann’s current age, Ann was 12 years old (half Mary’s current age). This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.
I'm not sure why it needs rephrasing—it seems pretty straightforward? What am I missing?
16 works (as another pointed out). Ann is 16 and Mary would have been that age 8 years ago—when of course Ann would then have been 8 (Mary, at 16, was twice her age).
Rephrasing it makes it easier to grasp: Mary is 24 years old. When Mary was Ann’s current age, Ann was 12 years old (half Mary’s current age). This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.
Without rephrasing it… I determined that Ann is 16. I'm not sure why it needs rephrasing—it seems pretty straightforward? What am I missing? 16 works (as another pointed out). Ann is 16 and Mary would have been that age 8 years ago—when of course Ann would then have been 8 (Mary, at 16, was twice her age).
The problem states “She _is_ twice as old as Ann was”, not “She _was_ twice as old as Ann was”.
Ann is 18. I think what makes it confusing is the "variable overloading" referring to both the current ages of Mary and Ann (M & A), and their ages at some point in the past "when Mary was as old as Ann is now" (M' & A'). So, what we're given is: M = 24 M' = A M = 2A' => A' = M/2 = 12 Since the age gap between Mary and Ann is constant, we know: M - A = M'- A' So, substituting in the known values: 24 - A = A - 12 2A =…
Mary := 24 = 2(A - Δ)
A = Mary - Δ
And then it’s just substitution. But of course what appeals to one’s intuition is very personal