Ytukyg

Problem Given: Twelve years ago the age of $A$ was the double of age of $B$ and in the next twelve years, the age of $A$ would be $68$ years less than triple of $B$. Find the current ages.

What I've done is set up the equations

$$2x-12=3(x+12)-68+12$$

where $2x-12$ is the age of $A$, $3(x+12)$ is the triple of age of $B$ in the future $(+12)$. And the last $12$ because in the sentence says 'is in the next twelve years.'

But the answer doesn't match with the solution on the book $A=52$ and $B=32$.

Could someone tell how to solve?

Since for some reason no one felt the need to elaborate on where these equations came from, much less OP's error, I'll do it then.

Though a word of import -- To note for future reference, OP, showing your process might help because otherwise no one can really understand where your own errors come from without it. I happened on your error by simply working the problem the same by mere coincidence; these problems can be solved many different ways, so it's merely luck we coincided.

I'm going to start from the beginning.

Twelve years ago the age of A was the double of age of B and in the next twelve years, the age of A would be 68 years less than triple of B. Find the current ages.

Let $A,B$ denote the respective persons' ages. Then, since "Twelve years ago the age of A was the double of age of B," we know

$$A - 12 =2(B - 12) = 2B - 24 $$

Similarly, per "in the next twelve years, the age of A would be 68 years less than triple of B," we know

$$A + 12 = 3(B+12) - 68 = 3B + 36 - 68 = 3B - 32$$

This gives us a system of equations:

$$A - 12 = 2B - 24$$
$$A + 12 = 3B - 32$$

Equivalently, since $A$ seems easiest to solve for,

$$A = 2B - 12 = 3B - 44$$

Thus, we have the equation in $B$, for which we solve:

$$2B - 12 = 3B - 44 ;;; Rightarrow ;;; B = 32$$

Plug $B = 32$ into any equation of our system of equations, and you'll find $A = 52$.

As for where your error lies, OP, it's this:

What I've done is
$$2x-12=3(x+12)-68+12$$ where
$2x-12$ is the age of $A$, $3(x+12)$ is the triple of age of $B$ in the future $(+12)$. And the last $12$ because in the sentence says 'is in the next twelve years.'

That last $+12$ is unnecessary. Yes, it is in $12$ years from the present, but that $12$ years is meant to turn $x$ into $x+12$ (as you already have done in parentheses). Since, at that $12$-year-mark, the "$68$ less than the triple" relation is satisfied, you do not add a further $12$.

Twelve years ago the age of A was the double of age of B so .........$$A-12 = 2(B - 12)$$

$$2B - A = 12 tag{1}$$

In the next twelve years, the age of A would be 68 years less than triple of B so ......$$3(B + 12) - (A + 12) = 68$$

$$3B - A = 44 tag{2}$$

Subtract $(1)$ from $(2)$:

$$B = 32$$

Then $$A = 52$$

The problem with your attempted solution is you are using one unknown $x$ for two people. You must use two unknowns for two people.

You can organize all in table:
$$begin{array}{c|c|c}
&12 text{years ago}&Now& 12 text{year after}\
hline
Adam&A-12&A&A+12\
Beth&B-12&B&B+12\
hline
Conditions&A-12=2(B-12)&&A+12=3(B+12)-68end{array}.$$
Can you solve the system of two equations and find $A$ and $B$?