Question

It takes eight hours for a $ 600 $ km journey, if $ 120 $ km is done by train and the rest by car. It takes $ 20 $ minutes more, if $ 200 $ km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is:
A.\[2:3\]
B.\[3:2\]
C.\[3:4\]
D.\[4:3\]

Answer 1

Hint: Identify the known and unknown ratios and set up the proportion and solve accordingly. In these ratio types of questions, take any variable as the reference number. Convert the word statements in the form of mathematical expressions and simplify for the required solution.

Complete step-by-step answer:
Let the speed of the train be $ = x{\text{ km/hr}} $
And the speed of the car be $ = y{\text{ km/hr}} $
It takes eight hours for a $ 600 $ km journey, if $ 120 $ km is done by train and the rest by car.
Convert the above word statement in mathematical form-
Then,
 $ \dfrac{{120}}{x} + \dfrac{{480}}{y} = 8 $
Take the number common from both the sides of the equations and so it is removed.
 $ $ $ \therefore \dfrac{1}{x} + \dfrac{4}{y} = \dfrac{1}{{15}}\,{\text{ }}.....{\text{(a)}} $
 Also, given that
It takes $ 20 $ minutes more, if $ 200 $ km is done by train and the rest by car.
Convert minutes in hour
 $ \therefore \dfrac{{200}}{x} + \dfrac{{400}}{y} = 8 + \dfrac{{20}}{{60}} $
Simplify the above equation –
 $
   \Rightarrow \dfrac{{200}}{x} + \dfrac{{400}}{y} = 8 + \dfrac{1}{3} \\
   \Rightarrow \dfrac{{200}}{x} + \dfrac{{400}}{y} = \dfrac{{25}}{3} \\
  $
Take common from both the sides of the equation and remove
 $ $ $ \therefore \dfrac{1}{x} + \dfrac{2}{y} = \dfrac{1}{{24}}{\text{ }}.......{\text{(b)}} $
Take equations $ (a){\text{ and (b)}} $ and find the value of unknown
By using elimination method- subtract equation (b) from the equation (a)
 $ \Rightarrow \dfrac{4}{y} - \dfrac{2}{y} = \dfrac{1}{{15}} - \dfrac{1}{{24}} $
( $ \dfrac{1}{x} $ is removed in subtraction)
 $
   \Rightarrow \dfrac{2}{y} = \dfrac{{24 - 15}}{{(24)(15)}} \\
   \Rightarrow \dfrac{2}{y} = \dfrac{9}{{24 \times 15}} \\
  $
Do cross-multiplication and make unknown “y” as the subject-
 $
  \therefore y = \dfrac{{24 \times 15 \times 2}}{9} \\
   \Rightarrow y = 80{\text{ km/hr}} \\
  $
Substitute value of “y” in equation (a)
 $
  \therefore \dfrac{1}{x} + \dfrac{4}{{80}} = \dfrac{1}{{15}}\,{\text{ }} \\
   \Rightarrow \dfrac{1}{x} + \dfrac{1}{{20}} = \dfrac{1}{{15}}{\text{ }} \\
  $
Make unknown “x” as the subject –
 $
  \therefore \dfrac{1}{x} = \dfrac{1}{{15}} - \dfrac{1}{{20}} \\
   \Rightarrow \dfrac{1}{x} = \dfrac{{20 - 15}}{{\left( {15} \right)\left( {20} \right)}} \\
   \Rightarrow \dfrac{1}{x} = \dfrac{5}{{300}} \\
   \Rightarrow \dfrac{1}{x} = \dfrac{1}{{60}} \\
   \Rightarrow x = 60{\text{ km/hr}} \\
  $
Now ratio of the speed of car and train is
 $
   = \dfrac{x}{y} \\
   = \dfrac{{60}}{{80}} \\
   = \dfrac{3}{4} \\
  $
Therefore, if it takes eight hours for a $ 600 $ km is done by train and the rest by car. It takes $ 20 $ minutes more, if $ 200 $ km is done by train and the rest by car then the ratio of the speed of the train to that of the cars is $ 3:4 $
Hence, from the given multiple choices- Option C is the correct answer.

So, the correct answer is “Option C”.

Note: Always convert the given word statement in the correct mathematical form and simplify using basic mathematical operations. Ratio is the comparison between two numbers without any units. Whereas, when two ratios are set equal to each other are called proportion. Always check the units of the given terms and change all the units of the quantities to the same unit. Write the given quantities in the ratio as a fraction. The ratio can also be expressed as a decimal or the percentage. The words such as proportion, rate, relationship, relation are all the same and are the synonym for the word “ratio”.