A person covers 11 km with a speed of 7 km/h, 25 km with a speed of 10 km/h and 30 km with a speed of 20 km/h. What is the average speed of the person?

The correct answer is Option 3: $11\frac{11}{13}\text{km/h}$

To find the average speed, we use the fundamental formula:

$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$

Step 1: Calculate the Total Distance

The person travels across three segments:

$\text{Total Distance}=11+25+30=66\text{km}$

Step 2: Calculate the Time Taken for Each Segment

Using $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

- Time for the 1st segment (11 km at 7 km/h):

$t_1=\frac{11}{7}\text{hours}$

- Time for the 2nd segment (25 km at 10 km/h):

$t_2=\frac{25}{10}=\frac{5}{2}\text{hours}$

- Time for the 3rd segment (30 km at 20 km/h):

$t_3=\frac{30}{20}=\frac{3}{2}\text{hours}$

Step 3: Calculate the Total Time

The LCM of 7, 2, and 2 is 14. Converting each fraction:

$\text{Total Time}=\frac{11}{7}+\frac{5}{2}+\frac{3}{2}=\frac{22}{14}+\frac{35}{14}+\frac{21}{14}=\frac{78}{14}=\frac{39}{7}\text{hours}$

Step 4: Calculate the Average Speed

$\text{Average Speed}=\frac{66}{\frac{39}{7}}=66\times \frac{7}{39}=\frac{462}{39}=\frac{154}{13}$

Step 5: Convert to Mixed Fraction

Dividing 154 by 13: 13 × 11 = 143, remainder = 154 - 143 = 11

$\frac{154}{13}=11\frac{11}{13}\text{km/h}$

Therefore, the average speed of the person is $11\frac{11}{13}\text{km/h}$.