Train Journey Speed Ratio Calculation

The problem requires finding the ratio between the average speed and the lowest speed of a train covering three equal distances at different speeds.

Defining Variables

• Let the distance covered in each part of the journey be d.
• The speeds for the three equal distances are in the ratio 4:5:6. Let the speeds be $4x$, $5x$, and $6x$ respectively, where $x$ is a common factor.

Calculating Time for Each Segment

The time taken ($t$) is calculated using the formula: $t = \frac{\text{Distance}}{\text{Speed}}$.

• Time for the first distance ($t_1$): $t_1 = \frac{d}{4x}$
• Time for the second distance ($t_2$): $t_2 = \frac{d}{5x}$
• Time for the third distance ($t_3$): $t_3 = \frac{d}{6x}$

Calculating Total Distance and Total Time

• Total Distance = $d + d + d = 3d$.
• Total Time = $t_1 + t_2 + t_3 = \frac{d}{4x} + \frac{d}{5x} + \frac{d}{6x}$.
• To add these fractions, find the least common multiple (LCM) of the denominators (4, 5, 6), which is 60.
• Total Time = $\frac{15d}{60x} + \frac{12d}{60x} + \frac{10d}{60x} = \frac{(15 + 12 + 10)d}{60x} = \frac{37d}{60x}$.

Calculating Average Speed

Average speed is defined as Total Distance divided by Total Time.

• Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$
• Average Speed = $\frac{3d}{\frac{37d}{60x}}$
• Average Speed = $3d \times \frac{60x}{37d}$
• Average Speed = $\frac{180x}{37}$.

Identifying the Lowest Speed

The speeds are $4x$, $5x$, and $6x$. The lowest speed is $4x$.

Determining the Required Ratio

The question asks for the ratio of the average speed to the lowest speed.

• Ratio = $\frac{\text{Average Speed}}{\text{Lowest Speed}}$
• Ratio = $\frac{\frac{180x}{37}}{4x}$
• Ratio = $\frac{180x}{37 \times 4x}$
• Ratio = $\frac{180}{37 \times 4}$
• Ratio = $\frac{45}{37}$.

Therefore, the ratio of the average speed to the lowest speed is 45:37.