Question

A bike travels a distance of 200 km at a constant speed. If the speed of the bike is increased by 5 kmph the journey would have taken 2 hours less. What is the speed of the bike?
A) 30 kmph
B) 25 kmph
C) 20 kmph
D) 15 kmph

Answer 1

Hint: Using the fact that a product is zero if any of its factors is zero we follow these steps:
* Bring all terms to the left and simplify, leaving zero on the right side.
* Factorise the quadratic expression
* Set each factor equal to zero
* Solve the resulting linear equations
* Check the solutions in the original equation

Complete step-by-step answer:
The total distance travelled = 200 km
Let the speed of bike be x km \hr
Then the time taken to complete 200 km = $\dfrac{{200}}{x}$hr
If the speed of the bike is increase by 5 kmph = $x + 5$km/hr
The journey would have taken 2 hours less =$\left( {\dfrac{{200}}{x} - 2} \right)$hr……………(1)
Then time taken in 200 km with speed (x+5) km\hr = $\dfrac{{200}}{{x + 5}}$hr……………(2)
Now we can equate the equation (1) and equation (2)
$\left( {\dfrac{{200}}{x} - 2} \right)$=$\dfrac{{200}}{{x + 5}}$
$ \Rightarrow (200 - 2x)(x + 5) = 200x$
$ \Rightarrow 200x + 1000 - 2{x^2} - 10x = 200x$
$ \Rightarrow 2{x^2} + 10x - 1000 = 0$
$ \Rightarrow {x^2} + 5x - 500 = 0$
So the first thing with the above equation, I have to do is factor:
$ \Rightarrow {x^2} + 25x - 20x - 500 = 0$
Now I can restate the original equation in terms of a product of factors, with this product being equal to zero:
$ \Rightarrow (x + 25)(x - 20) = 0$
Now I can solve each factor by setting each one equal to zero and solving the resulting linear equations.
These two values are the solution to the original quadratic equation. So my answer is:
$ \Rightarrow x = 20, - 25$
But for speed we need a positive number.
So, the speed of the bike is 20 kmph.

Option C is the correct answer.

Note: Time and speed are inversely proportional. For a constant distance if speed increases then the time decreases and if the speed decreases then the time increases. DIstance and speed on the other hand are directly proportional. For a constant time with the increase in speed distance increases and if speed decreases then distance also decreases.