Mary me

Mary (now) = 24.

Let Ann be x years old now.

before p years, Mary was x years old.

implies, 24-p = x.

Before p years, Ann would be x - p = 24 - p - p = 24 - 2p.

Given , 24 - 2p = 24/2 = 12.

Thus, 2p = 12. implies, p = 6.

thus, x = 24 - p = 24 - 6 = 18 years old.

$M$ = age of Mary now = $24$

$M-x$ = age of Mary $x$ years ago

$A$ = age of Ana now

$A-x$ = age of Ana $x$ years ago

We have

$M=2(A-x)$

$24=2A-2x$ $\implies$ $\color{#D61F06}\boxed{1}$

$M-x=A$

$24-x=A$

$x=24-A$ $\implies$ $\color{#D61F06}\boxed{2}$

Substitute $\color{#D61F06}\boxed{2}$ in $\color{#D61F06}\boxed{1}$

$24=2A-2(24-A)$

$24=2A-48+2A$

$24+48=4A$

$72=4A$

$\boxed{A=18}$

• We know Mary is 24 years old, and Ann is " A " years old.
• First, lets state X as the difference between Mary and Ann. So, X = (24 - A) .
• So, according to the question, " Mary is twice as old as Ann was when Mary was as old as Ann is now "; so, we can set that Ann was (A - X) when Mary was (24 - X) . So, we can assume:
  $24 = 2\times[A - X]$
  $24 = 2\times[A - (24 - A)]$
  $24 = 2\times[2A - 24]$
  $24 = 4A - 48 => 4A = 72$
  
  
  
  
• So, after this whole process, we see that Ann is $A = \frac{72}{4} = 18$ years old.

The answer is 18. Let's check that in the words , to see just why 18 is correct: ,

Mary is 24 years old. Ann is 18 years old. Therefore , the difference in their ages is 24-18 or 6 years. , Mary, 24, is twice as old as Ann was when Mary was , as old as Ann now. Since the difference in their ages , is 6 years, when Mary was 18, Ann was 18-6 or 12, and , 24, which is Mary's age now, equals twice Ann's age , 6 years ago, which was 12. ,

Now let's let Ann's age be A instead of 18, and use , the same words: ,

Mary is 24 years old. Ann is A years old. Therefore , the difference in their ages is 24-A years. , Mary, 24, is twice as old as Ann was when Mary was , as old as Ann now. Since the difference in their ages , is 24-A years, when Mary was A, Ann was A-(24-A), and , 24, which is Mary's age now, equals twice Ann's age , , 24-A years ago, which was A-(24-A). ,

So we take those last words ,

>...24, which is Mary's age now, equals twice Ann's , age 24-A years ago, which was A-(24-A)...<< ,

which shortens to ,

>...24...equals twice A-(24-A)...<< ,

and becomes the equation ,

24 = 2[A-(24-A)] ,

which you can easily solve and get A = 18.