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**Hint:**In this question, we have to find the age of two persons from the given conditions. Thus, we will use the age-mathematical rule to get the solution. We start solving this problem by letting the age of A and B be some variables. Then, we will convert both the statement conditions into the mathematical equations. Thus, we will get two separate equations, so we will solve them by equating each other. After necessary calculations, we get the values of variables as the age of A and B, which is our required answer.

**Complete step by step answer:**

According to the question we have to find the age of two persons.

Thus, we will use the age mathematical rules to get the solution.

The two conditions given to us is “five years ago, A’s age was four times the age of B” ---- (1) and “five years hence, A’s age will be twice the age of B” ------- (2)

First, we will let the age of A be equal to $x$ and the age of B is equal to $y$ .

Now, we will convert the statement (1) into mathematical equations, that is in the statement (1), we see that the age of A is four time the age of B, but it was five years ago, therefore we will subtract 5 from the age of A and B, that is

$(x-5)=4(y-5)$

So, we will apply the distributive property $a(b-c)=ab-ac$ in the above equation, we get

$x-5=4y-20$

We will add 5 on both sides in the above equation, we get

$x-5+5=4y-20+5$

As we know, the same terms with opposite signs cancel out each other, thus we get

$x=4y-15$ ---------- (3)

Statement (2) states that the age of A is twice the age of B but it is after five years, hence we will add 5 in the age of A and B, that is

$(x+5)=2(y+5)$

So, we will apply the distributive property $a(b-c)=ab-ac$ in the above equation, we get

$x+5=2y+10$

We will subtract 5 on both sides in the above equation, we get

$x+5-5=2y+10-5$

As we know, the same terms with opposite signs cancel out each other, thus we get

$x=2y+5$ ---------- (4)

Thus, we see that the value of both the equation (3) and (4) are equal, thus we get

$4y-15=2y+5$

We will add 15 on both sides in the above equation, we get

$4y-15+15=2y+5+15$

As we know, the same terms with opposite signs cancel out each other, thus we get

$4y=2y+20$

So, we will again subtract 2y on both sides in the above equation, we get

$4y-2y=2y+20-2y$

As we know, the same terms with opposite signs cancel out each other, thus we get

$4y-2y=20$

On further solving, we get

$2y=20$

So, we will divide 2 on both sides in the above equation, we get

$\dfrac{2y}{2}=\dfrac{20}{2}$

Therefore, we get

$y=10$

In the last, we will substitute the value of the above equation in equation (3), we get

$x=4(10)-15$

On further simplification, we get

$x=40-15$

Therefore, we get

$x=25$

**Therefore, the age of A is equal to $x=25$ and the age of B is equal to $y=10$**

**Note:**While solving this problem, do mention all the formulas and the steps properly to avoid confusion and mathematical error. You can also solve the two equations using the cross multiplication method or the elimination method, to get an accurate answer.

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