Case study:
Malik and Rakesh are best friends who are also neighbours living in Gurugram. Their families are also on excellent terms.

During summer, they decide to go for a vacation together along with their families to Ahmedabad by their own vehicles.

Malik travels by a Jeep and Rakesh by a car.

Malik’s jeep travels at a speed of x km/h and Rakesh’s car travels 10 km/h slower than Malik’s jeep. Rakesh took 2 hours more than Malik to reach Ahmedabad. The distance between Gurugram and Ahmedabad (by road) is approximately 975 km.

(i) What will be the distance covered by Rakesh’s car in five hours?

(ii) Form a quadratic equation to describe the speed of Malik’s jeep.

(iii) What is the speed of Malik’s jeep?

(iv) How much time is taken by Rakesh to reach Ahmedabad?

(i) Given,

The speed of Malik's jeep = x km/h.

Speed of Rakesh's jeep = (x - 10) km/h

Distance covered by Rakesh in 5 hours

= 5(x − 10) km

Hence, distance covered by Rakesh in 5 hours = 5(x − 10) km.

(ii) The distance between Gurugram and Ahmedabad = 975 km

Time taken by Malik = $\dfrac{975}{x}$

Time taken by Rakesh = $\dfrac{975}{x - 10}$

Given, that Rakesh took 2 hours more than Malik to reach Ahmedabad,:

$\Rightarrow \dfrac{975}{x - 10} = \dfrac{975}{x}$ + 2

$\Rightarrow \dfrac{975}{x - 10} = \dfrac{975 + 2x}{x}$

⇒ 975x = 975(x − 10) + 2x(x − 10)

⇒ 975x = 975x − 9750 + 2x² − 20x

⇒ 0 = 2x² − 20x − 9750

⇒ 0 = 2(x² − 10x − 4875)

⇒ x² − 10x − 4875 = 0

Hence, quadratic equation is x² − 10x − 4875 = 0.

(iii) From part (ii),

⇒ x² − 75x + 65x − 4875 = 0

⇒ x(x - 75) + 65(x − 75) = 0

⇒ (x − 75)(x + 65) = 0

⇒ (x − 75) = 0 or (x + 65) = 0

⇒ x = 75 or −65

Since speed cannot be negative,

x = 75 km/h

Hence, speed of Malik's jeep = 75 km/h.

(iv) Speed of Rakesh's jeep = x - 10

= 75 - 10 = 65 km/h.

Time taken by Rakesh to reach Ahmedabad =

= $\dfrac{975}{65}$

= 15 hours.

Hence, Rakesh takes 15 hours to reach Ahmedabad.