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Leaving home at the same time, Amal reaches 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 $9:10$ am, at what speed, in km/hr, must he travel so as to reach office exactly at $10$ am?

1. $13$
2. $14$
3. $12$
4. $11$

Let the distance between home and office be $d\text{’} \; \text{km}.$

Time is taken to reach office.

• At speed of $8 \; \text{km/hr} \rightarrow 10 : 15 \; \text{AM}$
• At speed of $15 \; \text{km /hr} \rightarrow 9 : 40 \; \text{AM}$

Time difference $= \frac{35}{60} \; \text{hr}$

$\Rightarrow \frac{d}{8} – \frac{d}{15} = \frac{35}{60}$

$\Rightarrow \frac{15d – 8d}{120} = \frac{35}{60}$

$\Rightarrow 7d = 70$

$\Rightarrow \boxed{d= 10 \; \text{km}}$

Now,

Time $= 50 \; \text{minutes} = \frac{50}{60} = \frac{5}{6} \; \text{hr}$

Let the speed be $S\text{’} \;\text{km/hr}$

$\Rightarrow S = \frac{10}{\frac{5}{6}}$

$\Rightarrow S = \frac{10 \times 6}{5}$

$\Rightarrow \boxed{ S = 12 \; \text{km/hr}}$

$\therefore$ The required speed $= 12 \; \text{km/hr}$

Correct Answer $: \text{C}$

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