If a person covers $60 \text{ km}$ at a speed of $40 \text{ km... | Mathematics MCQ

MathematicsPrevious Year Paper 2024

If a person covers $60 \text{ km}$ at a speed of $40 \text{ km/h}$ and another distance of $120 \text{ km}$ at a speed of $80 \text{ km/h}$ , then his average speed for the whole journey is:

60 km/h

50 km/h

48 km/h

55 km/h

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option A

60 km/h

Quick Summary:

Calculate individual times: $t_1 = 60/40 = 1.5$ hrs, $t_2 = 120/80 = 1.5$ hrs. Since times are equal, $V_{avg} = (v_1 + v_2) / 2 = (40 + 80) / 2 = 60$ km/h.

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📐MAMath SolutionDirect Formula

📋 Given

Distance 1 (d1) = 60 km, Speed 1 (v1) = 40 km/h. Distance 2 (d2) = 120 km, Speed 2 (v2) = 80 km/h.

🔢 Formula Used

Vavg=Total DistanceTotal Time=d1+d2d1v1+d2v2V_{avg} = \frac{\text{Total} \ \text{Distance}}{\text{Total} \ \text{Time}} = \frac{d_1 + d_2}{\frac{d_1}{v_1} + \frac{d_2}{v_2}}Vavg​=Total TimeTotal Distance​=v1​d1​​+v2​d2​​d1​+d2​​

⚡ Exam Hall Shortcut / Speed Trick

Calculate individual times: $t_1 = 60/40 = 1.5$ hrs, $t_2 = 120/80 = 1.5$ hrs. Since times are equal, $V_{avg} = (v_1 + v_2) / 2 = (40 + 80) / 2 = 60$ km/h.

⚠️ Common Student Trap / Pitfall

Students often wrongly calculate average speed as the simple arithmetic mean of the two speeds: (40 + 80) / 2 is coincidentally correct only because the time spent on each segment is identical, but the formula is specifically weighted by time.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate time for the first part

Using the formula $t = d/v$ , the time for the first 60 km at 40 km/h is $t_1 = 60/40$ .

t1=1.5 hourst_1 = 1.5 \text{ hours}t1​=1.5 hours

Calculate time for the second part

The time for the second 120 km at 80 km/h is $t_2 = 120/80$ .

t2=1.5 hourst_2 = 1.5 \text{ hours}t2​=1.5 hours

Calculate total distance and total time

Sum the distances: $d_{total} = 60 + 120 = 180$ km. Sum the times: $t_{total} = 1.5 + 1.5 = 3$ hours.

dtotal=180 km,ttotal=3 hoursd_{total} = 180 \text{ km}, t_{total} = 3 \text{ hours}dtotal​=180 km,ttotal​=3 hours

Calculate Average Speed

Divide total distance by total time to get the average speed.

Vavg=1803=60 km/hV_{avg} = \frac{180}{3} = 60 \text{ km/h}Vavg​=3180​=60 km/h

✅

C is correct because the total distance of 180 km covered in a total time of 3 hours results in an average speed of 60 km/h.

Core Concepts Used

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Average Speed Time-Distance Relationship Total Motion Kinematics

💡 EXAM TIP

This concept is fundamental for time-speed-distance problems in exams like RRB and SSC, where understanding the weighted average vs simple arithmetic mean is key.

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