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CAT 2024 Slot 3 QA Question 4

Multiple choice (+3 / −1) · Arithmetic · Time, Speed & Distance · Try it, then check the answer and solution below.

CAT 2024 Slot 3QAArithmetic • Time, Speed & DistanceModerate
A train travelled a certain distance at a uniform speed. Had the speed been 6 km per hour more, it would have needed 4 hours less. Had the speed been 6 km per hour less, it would have needed 6 hours more. The distance, in km, travelled by the train is
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Answer and solution

Answer: B) 720

📌 Core Concept
The problem involves the relationship between speed, time, and distance. The key formula is:
Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}
🔢 Step-by-Step Solution
1
Define Variables:
Let the original speed be vv km/h.
Let the original time be tt hours.
Distance d=v×td = v \times t.
2
Set Up Equations:
With increased speed: d=(v+6)(t−4)d = (v + 6)(t - 4).
With decreased speed: d=(v−6)(t+6)d = (v - 6)(t + 6).
3
Equation 1:
v×t=(v+6)(t−4)v \times t = (v + 6)(t - 4)
Expanding:
vt=vt−4v+6t−24vt = vt - 4v + 6t - 24
Simplify:
0=−4v+6t−24  ⟹  2v−3t+12=0(Equation A)0 = -4v + 6t - 24 \implies 2v - 3t + 12 = 0 \quad \text{(Equation A)}
4
Equation 2:
v×t=(v−6)(t+6)v \times t = (v - 6)(t + 6)
Expanding:
vt=vt+6v−6t−36vt = vt + 6v - 6t - 36
Simplify:
0=6v−6t−36  ⟹  v−t−6=0  ⟹  v=t+6(Equation B)0 = 6v - 6t - 36 \implies v - t - 6 = 0 \implies v = t + 6 \quad \text{(Equation B)}
5
Substitute Equation B into Equation A:
2(t+6)−3t+12=02(t + 6) - 3t + 12 = 0
Simplify:
2t+12−3t+12=0  ⟹  −t+24=0  ⟹  t=24 hours2t + 12 - 3t + 12 = 0 \implies -t + 24 = 0 \implies t = 24 \text{ hours}
6
Find Speed vv:
v=t+6=24+6=30 km/hv = t + 6 = 24 + 6 = 30 \text{ km/h}
7
Calculate Distance:
d=v×t=30×24=720 kmd = v \times t = 30 \times 24 = 720 \text{ km}
⚡ 30-Second Shortcut
Recognize the problem involves two scenarios altering speed and time.
Set up two equations based on the given conditions.
Solve the system of equations to find speed and time.
Multiply speed and time to get the distance.
🎯 Final Answer
The distance traveled by the train is 720 km.
Correct Answer: Option B

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