A car travels 120 km at an unknown speed x km/h and then retu... | Mathematics MCQ | Examoogle

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MathematicsTime, Speed and Distance

A car travels 120 km at an unknown speed x km/h and then returns at (x + 20) km/h. The average speed for the whole journey is 60 km/h. Find the value of x. (Round to one decimal place.)

54.5

51.6

50.2

52.4

Correct Answer

📐 MA • Math Harmonic Mean FormulaMathematicsTime, Speed and Distance

Option B

51.6

Quick Summary:

Use the Harmonic Mean formula directly: $60 = \frac{2x(x+20)}{2x+20}$ . Simplify to $6x + 60 = x^2 + 20x$ , leading to the quadratic equation $x^2 + 14x - 600 = 0$ . Solve by testing options or quadratic formula.

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📐MAMath SolutionHarmonic Mean Formula

📋 Given

Distance for each trip = 120 km, forward speed = x km/h, return speed = x + 20 km/h, average speed = 60 km/h.

🔢 Formula Used

Vavg=2v1v2v1+v2V_{avg} = \frac{2v_{1}v_{2}}{v_{1} + v_{2}}Vavg​=v1​+v2​2v1​v2​​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often incorrectly calculate average speed as the arithmetic mean (x + x + 20) / 2, which ignores the time spent traveling at each speed.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Set up the equation

Since the distance is constant for both legs, the average speed is the harmonic mean of the two speeds.

60=2×x×(x+20)x+(x+20)60 = \frac{2 \times x \times (x + 20)}{x + (x + 20)}60=x+(x+20)2×x×(x+20)​

Simplify the equation

Cross-multiply to remove the fraction: $60(2x + 20) = 2x(x + 20)$ , which simplifies to $120x + 1200 = 2x^2 + 40x$ .

x2−40x−600=0x^2 - 40x - 600 = 0x2−40x−600=0

Solve the quadratic equation

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1, b=-40, c=-600$ : $x = \frac{40 \pm \sqrt{1600 + 2400}}{2} = \frac{40 \pm \sqrt{4000}}{2} = 20 \pm 10\sqrt{10}$ .

x=20+10(3.162)=51.62x = 20 + 10(3.162) = 51.62x=20+10(3.162)=51.62

✅

B is correct because the quadratic solution yields approximately 51.6 km/h as the value for x.

Core Concepts Used

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Average Speed Harmonic Mean Quadratic Equations

💡 EXAM TIP

This concept of Harmonic Mean is identical to finding effective resistance in parallel circuits ( $R_{eq} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}$ ); always use this when dealing with rates over the same distance.

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