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ICSE Class 10 Mathematics Question 15 of 21

Solved 2023 Question Paper ICSE Class 10 Mathematics — Question 11

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11
Question

Question 7(ii)

A man covers a distance of 100 km, travelling with a uniform speed of x km/hr. Had the speed been 5 km/hr more it would have taken 1 hour less. Find x the original speed.

Answer

In first case :

Distance = 100 km

Speed = x km/hr

Time = DistanceSpeed=100x\dfrac{\text{Distance}}{\text{Speed}} = \dfrac{100}{x}

In second case :

Distance = 100 km

Speed = (x + 5) km/hr

Time = DistanceSpeed=100x+5\dfrac{\text{Distance}}{\text{Speed}} = \dfrac{100}{x + 5}

According to question,

The difference between the time taken between first and second case is 1 hour.

∴100x−100x+5=1⇒100(x+5)−100xx(x+5)=1⇒100x+500−100xx2+5x=1⇒500x2+5x=1⇒500=x2+5x⇒x2+5x−500=0⇒x2+25x−20x−500=0⇒x(x+25)−20(x+25)=0⇒(x−20)(x+25)=0⇒x−20=0 or x+25=0⇒x=20 or x=−25.\therefore \dfrac{100}{x} - \dfrac{100}{x + 5} = 1 \\[1em] \Rightarrow \dfrac{100(x + 5) - 100x}{x(x + 5)} = 1 \\[1em] \Rightarrow \dfrac{100x + 500 - 100x}{x^2 + 5x} = 1 \\[1em] \Rightarrow \dfrac{500}{x^2 + 5x} = 1 \\[1em] \Rightarrow 500 = x^2 + 5x \\[1em] \Rightarrow x^2 + 5x - 500 = 0 \\[1em] \Rightarrow x^2 + 25x - 20x - 500 = 0 \\[1em] \Rightarrow x(x + 25) - 20(x + 25) = 0 \\[1em] \Rightarrow (x - 20)(x + 25) = 0 \\[1em] \Rightarrow x - 20 = 0 \text{ or } x + 25 = 0 \\[1em] \Rightarrow x = 20 \text{ or } x = -25.

Since, speed cannot be negative.

∴ x = 20 km/hr.

Hence, original speed = 20 km/hr.