A fast train takes 1 hour less than a slow train for a journey of 96 km. If the speed of the slow train is 8 km/h less than that of the fast train, then find the speeds of the fast train and slow train respectively
- ((a))
26 km/h, 34 km/h
- ((b))
32 km/h, 24 km/h
- ((c))
34 km/h, 28 km/h
- ((d))
12 km/h, 14 km/h
Show Answer
32 km/h, 24 km/h

Shortcut Trick
Use the direct formula for speed difference: Distance = [Product of Speeds ÷ Difference of Speeds] × Time Difference.
96 = [S1 × S2 ÷ 8] × 1 ⇒ S1 × S2 = 96 × 8 = 768.
From the given options, only 32 × 24 = 768 and the difference 32 − 24 = 8 km/h satisfies the condition.
Hence, the speeds are 32 km/h and 24 km/h.
∴ The correct answer is 32 km/h, 24 km/h.

Alternate Method
Given:
Total Distance = 96 km
Time Difference (ΔT) = 1 hour
Speed Difference = 8 km/h

Formula Used:
Time = Distance ÷ Speed
Calculations:
Let the speed of the fast train be x km/h.
Then, the speed of the slow train is (x − 8) km/h.
⇒ 96/(x − 8) − 96/x = 1
⇒ 96[ (x − (x − 8)) / (x(x − 8)) ] = 1
⇒ 96 × 8 = x2 − 8x
⇒ x2 − 8x − 768 = 0
⇒ x2 − 32x + 24x − 768 = 0
⇒ x(x − 32) + 24(x − 32) = 0
⇒ (x − 32)(x + 24) = 0
⇒ x = 32 (Speed cannot be negative, so x ≠ −24)
Speed of fast train = 32 km/h
Speed of slow train = 32 − 8 = 24 km/h
∴ The correct answer is 32 km/h, 24 km/h.

Additional Information
Speed-Time Relation
For a constant distance, speed is inversely proportional to time (S ∝ 1/T). If speed increases, time decreases.
Quadratic Shortcut
In equations like x2 − Sx + P = 0, the sum of roots is S and the product is P. In speed problems, the positive root is the answer.
Average Speed
If a journey is covered at two different speeds x and y for equal distances, the average speed is (2xy) ÷ (x + y).


























































