Leaving home at the same time, Amal reaches the office at $10:15 am$ if he travels at $8 km/hr$ , and at $9:40 am$ if he travels at $15 km/hr$ . Leaving home at 9:10, at what speed, in $km/hr,$ must be travel so as to reach office exactly at $10 am?$

✅ Correct Option: 4

We need to find Amal's required speed when leaving at 9:10 AM to reach by 10:00 AM.

Amal leaves home at the same time but:

Reaches office at 10:15 AM traveling at $8$ km/hr
Reaches office at 9:40 AM traveling at $15$ km/hr

The time difference is $10:15$ AM $- 9:40$ AM $= 35$ minutes.

Since distance is the same in both cases:

Let $t_1 =$ time taken at $8$ km/hr and $t_2 =$ time taken at $15$ km/hr

We know: $t_1 - t_2 = 35$ minutes $= \dfrac{35}{60}$ hours

Since distance is same: $8 \times t_1 = 15 \times t_2$

From $t_1 = t_2 + \dfrac{35}{60}$ , substitute into the distance equation:

$8 \times \left(t_2 + \dfrac{35}{60}\right) = 15 \times t_2$

$8t_2 + 8 \times \dfrac{35}{60} = 15t_2$

$8 \times \dfrac{35}{60} = 7t_2$

$t_2 = \dfrac{8 \times 35}{60 \times 7} = \dfrac{280}{420} = \dfrac{2}{3}$ hours $= 40$ minutes

Therefore: $t_1 = 40 + 35 = 75$ minutes

Distance $= 8 \times \dfrac{75}{60} = 10$ km

Since he reaches at 9:40 AM after traveling 40 minutes:

Original departure time $= 9:40$ AM $- 40$ minutes $= 9:00$ AM

Now Amal leaves at 9:10 AM and wants to reach at 10:00 AM:

Time available $= 10:00$ AM $- 9:10$ AM $= 50$ minutes $= \dfrac{50}{60}$ hours

Distance $= 10$ km

Required Speed $= \dfrac{\text{Distance}}{\text{Time}} = \dfrac{10}{\dfrac{50}{60}} = 10 \times \dfrac{60}{50} = 12$ km/hr

Answer: $12$ km/hr

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