A and B are two towns. A car goes from A to B at a speed of 64 km / hr and returns to A at a sl

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A and B are two towns. A car goes from A to B at a speed of...

A and $B$ are two towns. A car goes from $A$ to $B$ at a speed of $64 \mathrm{~km} / \mathrm{hr}$ and returns to $A$ at a slower speed. If its average speed for the whole joumey is $56 \mathrm{~km} / \mathrm{hr}$, it returned with speed (a) $52.54 \mathrm{~km} / \mathrm{hr}$ (b) $47.74 \mathrm{~km} / \mathrm{hr}$ (c) $49.78 \mathrm{~km} / \mathrm{hr}$ (d) None of these

A and $B$ are two towns. A car goes from $A$ to $B$ at a speed of $64 \mathrm{~km} / \mathrm{hr}$ and returns to $A$ at a slower speed. If its average speed for the whole joumey is $56 \mathrm{~km} / \mathrm{hr}$, it returned with speed
(a) $52.54 \mathrm{~km} / \mathrm{hr}$
(b) $47.74 \mathrm{~km} / \mathrm{hr}$
(c) $49.78 \mathrm{~km} / \mathrm{hr}$
(d) None of these

Key Concepts

Algebraic Equation Solving

Solving the problem involves setting up an algebraic equation that relates the known values—such as one speed, the overall average speed, and the distance traveled—to the unknown variable. This requires forming an equation based on the definition of average speed and then solving for the unknown quantity. Such algebraic manipulation is a fundamental technique in many physics and mathematics problems involving rates, distances, and times.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. It is a measure that takes into account the variability of speeds over different segments of a journey, rather than simply averaging the speeds. In problems where consecutive segments of travel occur at different speeds, determining the overall average speed requires a careful consideration of the time spent at each speed.

Harmonic Mean

The harmonic mean is particularly useful when calculating the average speed over equal-distance segments traveled at different speeds. Unlike the arithmetic mean, the harmonic mean gives a more accurate measure of the central tendency for rates, as it appropriately weights the speeds relative to the distances covered in each segment. In the context of round-trip problems where the same distance is traveled at different speeds, the harmonic mean formula, (2ab)/(a+b), is commonly applied.

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