Question

Choose the correct answer from the alternatives given:
A dog after seeing a cat finds that the cat is 25 leaps ahead of the dog. Cat after seeing the dog starts running and the dog chases the cat. If in every minute, dog takes 5 leaps while cat takes 6 leaps and 1 leap of dog is equal to 2 leaps of cat, in how many minutes will dog catch the cat?
A. $12$ minutes
B. $15$ minutes
C. $12.5$ minutes
D. $14$ minutes

Answer 1

Hint:For this question first we will assume a variable for the dog leap and then we will apply the condition given in the question and find the distance covered by the cat leap in terms of dog leap that is we will divide the variable into half. After that, we will write down the speed of the dog and the cat as given in the question in terms of the variable and finally, we will find out the time by using the formula: Time is taken by dog $=\dfrac{\text{Distance between the dog and the cat}}{\text{Relative speed of dog and cat}}$

Complete step by step answer:
Now it is given that the cat is 25 dog leaps ahead of the dog. Let one dog leap be equal to $x$ units of distance, now it is given that: one leap of dog = two leaps of a cat.
Therefore, $x=$ two leaps of cat $\Rightarrow $ one leap of cat$=\dfrac{x}{2}$ .
Now, we know that the formula for finding the speed is: $\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$
Now, the dog covers 5 leaps that is $5x$ units of distance in one minute, therefore the speed of dog will be: $5x/\min $
Similarly, the cat covers 6 leaps that is $6\times \dfrac{x}{2}=3x$ units of distance in one minute, therefore the speed of cat will be: $3x/\min $.
Now, we need to find the time in which the dog catches the cat, so:
Time taken by dog $=\dfrac{\text{Distance between the dog and the cat}}{\text{Relative speed of dog and cat}}$
Now it is given that the distance between the dog and the cat is 25 dog leaps $\Rightarrow 25x$ and
the relative speed of the dog and the cat = $\text{Speed of dog}-\text{Speed of cat}=5x-3x$ . Now we will put these values in the formula for finding out the time written above, therefore:
Time taken in which dog catches the Cat will be: $\dfrac{25x}{\left( 5x-3x \right)/\min }=\dfrac{25}{2/\min }=12.5\text{ min}$

Hence, option C will be the correct answer.

Note:
Remember that when two objects or subjects run in the same direction then the relative speed of both the objects will be the difference between them let’s say an object is running at the speed of ${{s}_{1}}$ and the second object is running at the speed of ${{s}_{2}}$ then the relative speed of both the objects will be: $\left( {{s}_{1}}-{{s}_{2}} \right)$ in case they are running in the same direction where $\left( {{s}_{1}}>{{s}_{2}} \right)$ and the relative speed of both the objects will be: $\left( {{s}_{1}}+{{s}_{2}} \right)$ in case they are running in the opposite direction.