Question : The ratio of the distance between two places A and B to the distance between places B and C is 3 : 5. A man travels from A to B at a speed of $x$ km/hr and from B to C at a speed of 50 km/hr. If his average speed for the entire journey is 40 km/hr, then what is the value of $(x - 10) : (x + 1)$?
Option 1: 11 : 10
Option 2: 20 : 31
Option 3: 31 : 20
Option 4: 10 : 11
Correct Answer: 20 : 31
Solution :
According to the question
Let the distance from A to B is $3a$ and from B to C is $5a$
Time Taken by man to travel from A and B at a speed of $x$ km/hr = $\ \frac {3a}{x}$
Time Taken by man to travel from B and C at a speed of 50 km/hr = $\frac{5a}{50}$ = $\frac{a}{10}$
Average speed = $\frac{\text{Total distance}}{\text{Total time}}$
⇒ $\frac{3a + 5a}{\frac{3a}{x} + \frac{a}{10}}$ = 40
⇒ $\frac{8a}{\frac{3a}{x} + \frac{a}{10}}$ = 40
⇒ $8a = 40a(\frac{3}{x} + \frac{1}{10}$)
⇒ $\frac{1}{5}$ = $\frac{(30 + x)}{10x}$
⇒ 2$x$ = 30 + $x$
⇒ $x$ = 30 km/h
Value of ($x - 10) : (x + 1) = (30 – 10) : (30 + 1) = 20: 31$
Hence, the correct answer is 20 : 31.
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