Arithmetic

Time, Speed & Distance

Hard

Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours. Her speeds in these four parts follow an arithmetic progression, and the corresponding time taken to cover these four parts follow another arithmetic progression. If she travels at a speed of 960 meters per minute for 30 minutes to cover the first part, then the distance, in meters, she travels in the fourth part is

112000

76800

86400

96000

✅ Correct Option: 3

Converting units to meters and minutes:

Total distance $= 224 \text{ km} = 224,000 \text{ m}$

Total time $= 3 \text{ hours} = 180 \text{ minutes}$

First part: speed $= 960 \text{ m/min}$ , time $= 30 \text{ min}$

Let the four times be $t_1, t_2, t_3, t_4$ in AP, with $t_1 = 30$ min.

Sum of all four times $= 180$ min

Average time $= \dfrac{180}{4} = 45$ min

In an AP, the average of all terms equals the average of the first and last terms:

$\dfrac{t_1 + t_4}{2} = 45$

$\dfrac{30 + t_4}{2} = 45$

$t_4 = 60$ min

Common difference $= \dfrac{60 - 30}{3} = 10$ min

So the four times are: $30, 40, 50, 60$ minutes.

Let the four speeds be $960, 960+d, 960+2d, 960+3d$ (in m/min).

Since Distance $=$ Speed $\times$ Time, the four distances are:

Part 1: $960 \times 30 = 28,800$

Part 2: $(960 + d) \times 40 = 38,400 + 40d$

Part 3: $(960 + 2d) \times 50 = 48,000 + 100d$

Part 4: $(960 + 3d) \times 60 = 57,600 + 180d$

Adding all four distances and setting equal to $224,000$ :

$28,800 + 38,400 + 48,000 + 57,600 + (40 + 100 + 180)d = 224,000$

$172,800 + 320d = 224,000$

$320d = 51,200$

$d = 160$

Speed in the 4th part $= 960 + 3(160) = 1,440$ m/min

Time in the 4th part $= 60$ min

Distance in the 4th part $= 1,440 \times 60 = 86,400$ meters

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