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**Hint:**Firstly, we have to consider I and II and create an equation to find the age of Tanya. From I, we can write the age of Tanya as 3x and that of Rahul as 4x. From II, we can create an equation $\dfrac{3x+5}{4x+5}=\dfrac{4}{5}$ and solve for x and substitute in 3x to obtain Tanya’s age. Next, we have to consider I and II. Here, we will consider Tanya’s age as x. and Rahul’s age as $x+5$ (using III). Then, we will create an equation using I of the form $\dfrac{x}{x+5}=\dfrac{3}{4}$ and solve for x. Then, we will consider II and III. Here also Tanya’s age will be x. We can obtain an equation $\dfrac{x+5}{x+10}=\dfrac{4}{5}$ and find x. From these steps, we can infer any of the given options.

**Complete step by step answer:**

Let us consider I and II. We are given that the ratio between the present ages of Tanya and her brother Rahul is 3 : 4 respectively. Let the present ages of Tanya be 3x and Rahul be 4x. In II, we are given that after 5 years the ratio between the ages of Tanya and Rahul will be 4 : 5. We can write Tanya’s age after 5 years as 3We can write this mathematically as

$\Rightarrow \dfrac{3x+5}{4x+5}=\dfrac{4}{5}$

Let us find the value of x. We have to cross multiply.

$\Rightarrow 5\left( 3x+5 \right)=4\left( 4x+5 \right)$

Let us apply distributive property on both sides.

$\Rightarrow 15x+25=16x+20$

We have to collect variable terms on one side and constants on the other.

$\Rightarrow 15x-16x=20-25$

Now, we can add the like terms.

$\Rightarrow -x=-5$

Let us cancel the negative sign from both sides.

$\Rightarrow x=5$

Therefore, Tanya’s age $=3x=3\times 5=15$ years.

Now, let us consider I and III. Let Tanya’s age be x. In III, we are given that Rahul is 5 years older than Tanya. We can write this mathematically as $x+5$ . Therefore, we can write the ratio in I as

$\dfrac{x}{x+5}=\dfrac{3}{4}$

Let us cross multiply.

$\Rightarrow 4x=3\left( x+5 \right)$

We have to apply distributive property on the RHS.

$\Rightarrow 4x=3x+15$

Let us collect variable terms on the LHS.

$\Rightarrow 4x-3x=15$

We have to add the like terms.

$\Rightarrow x=15$

Therefore, Tanya’s age is 15 years.

Now, let us consider II and III. Let Tanya’s age be x. From III, we can write Rahul’s age as $x+5$ . Tanya’s age after 5 years will be $x+5$ and Rahul’s age after 5 years will be $\left( x+5 \right)+5=x+10$ . Therefore, we can form an equation from II as

$\Rightarrow \dfrac{x+5}{x+10}=\dfrac{4}{5}$

Let us cross multiply.

$\Rightarrow 5\left( x+5 \right)=4\left( x+10 \right)$

We have to apply distributive property on both sides.

$\Rightarrow 5x+25=4x+40$

We have to collect variable terms on one side and constants on the other.

$\Rightarrow 5x-4x=40-25$

We have to add the like terms.

$\Rightarrow x=15$

Therefore, Tanya’s age is 15 years.

Hence, we have used either I and II, or I and III or II and III. We need not use all the three together.

**So, the correct answer is “Option e”.**

**Note:**Students must be able to create algebraic expressions and equations using the given situations. They must deeply learn to solve algebraic equations and the rules associated with it. They must always consider the unknown quantity as a variable.

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