Question

While covering a distance of 30 km. Ajeet takes 2 hours more than Amit. If Ajeet doubles his speed, he would take 1 hour less than Amit. Find their speeds of walking.

Answer 1

Given:

Distance = 30 km.

Let Amit’s speed = x km/h and Ajeet’s speed = y km/h.

Conditions:

Ajeet takes 2 hours more than Amit: `30/y = 30/x + 2`.

If Ajeet doubles his speed, he takes 1 hour less than Amit: `30/(2y) = 30/x - 1`.

Step-wise calculation:

1. From (1): `30/y - 30/x = 2`

⇒ Multiply by xy: 30x – 30y = 2xy

⇒ xy = 15x – 15y (A)

2. From (2): `30/(2y) = 30/x - 1`

⇒ `15/y = 30/x - 1` 

⇒ `15/y + 1 = 30/x`

⇒ Multiply by xy: 15x + xy = 30y

⇒ xy + 15x = 30y (B)

3. Substitute (A) into (B): 

(15x – 15y) + 15x = 30y 

⇒ 30x – 15y = 30y

⇒ 30x = 45y

⇒ `x = (3/2)y`

4. Substitute `x = (3/2)y` into (A):

xy = 15x – 15y

⇒ `(3/2)y xx y = 15 xx (3/2)y - 15y` 

⇒ `(3/2)y^2 = (45/2)y - 15y = (15/2)y` 

⇒ `(3/2) y^2 = (15/2)y` 

⇒ 3y² = 15y 

⇒ y² – 5y = 0 

⇒ y(y – 5) = 0

⇒ y = 5   ...(Reject y = 0)

5. Then `x = (3/2) xx 5 = 7.5`

Check:

Amit time = `30/7.5` = 4 h.

Ajeet time = `30/5 = 6` h (2 h more). If Ajeet doubles speed → `30/10` = 3 h (1 h less than Amit). 

Ajeet’s walking speed = 5 km/h.

Amit’s walking speed = 7.5 km/h.