Question : X and Y travel a distance of 90 km each such that the speed of Y is greater than that of X. The sum of their speeds is 100 km/hr and the total time taken by both is 3 hours 45 minutes. The ratio of the speed of X to that of Y is:
Option 1: 2 : 3
Option 2: 1 : 3
Option 3: 2 : 4
Option 4: 1 : 4
Correct Answer: 2 : 3
Solution :
Let the speed of Y be $a$
⇒ Speed of X = $100 - a$
Time = 3 hours 45 minutes = $\frac{15}{4}$ hours
According to the question,
$\frac{90}{a} + \frac{90}{100 - a} = \frac{15}{4}$
⇒ $\frac{9000 - 90a + 90a}{100a - a^2}= \frac{15}{4}$
⇒ $36000 = 1500a - 15a^2=0$
⇒ $15a^2 - 1500a + 36000 = 0$
⇒ $a^2 - 100a + 2400 = 0$
⇒ $a^2 - 60a - 40a + 2400 = 0$
⇒ $a(a - 60) - 40(a - 60) = 0$
⇒ $(a - 60) (a - 40) = 0$
⇒ $a = 60, 40$
As Y is faster than X
So, Y = 60 km/hr
And X = 40 km/hr
So, the required ratio = 40 : 60 = 2 : 3
Hence, the correct answer is 2 : 3.
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