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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 $: \text{C}$