In covering a certain distance, the speeds of $A$ and $B$ are in the ratio of 3 : 4. $A$ takes 30 minutes more than $B$ to reach the destination. The time taken by $A$ to reach the destination is

### Concept & Inverse Proportion of Speed and Time For a constant distance, the time taken is inversely proportional to the speed. If the ratio of speeds of two moving bodies is $a : b$, the ratio of the time taken by them to cover the same distance is $b : a$. ### Step-by-Step Solution 1. We are given the ratio of speeds of $A$ and $B$ as $S_A : S_B = 3 : 4$. 2. Therefore, the ratio of time taken by them will be $T_A : T_B = 4 : 3$. 3. Let the time taken by $A$ be $4x$ minutes and the time taken by $B$ be $3x$ minutes. 4. The problem states that $A$ takes $30$ minutes more than $B$: $$4x - 3x = 30$$ $$x = 30 \text{ minutes}$$ 5. We need to find the time taken by $A$, which is $4x$: $$T_A = 4 \times 30 = 120 \text{ minutes}$$ 6. Convert minutes to hours: $$120 \text{ minutes} = 2 \text{ hours}$$ ### Exam Strategy & Shortcut The time ratio is $4 : 3$. The difference in ratio units is $4 - 3 = 1$ unit. This $1$ unit difference corresponds to the $30$ minutes given in the problem. $A$ takes $4$ units of time, so $A$'s time is $4 \times 30 = 120$ minutes $= 2$ hours. Very fast and requires no complex algebra! ### Common Pitfall A common mistake is applying the ratio directly without taking its inverse, incorrectly assuming the faster person ($B$) takes more time. ### Final Answer Therefore, the correct answer is **2 hours**.