A cyclist rides from her house to an office at 6 km/h and returns along the same route at 8 km/h. If the total time taken for the round trip is 3.5 hours, find the distance between her house and the office.

The correct option is 12 km.

Step 1: Understand the given information

Let the distance between the cyclist's house and the office be $d$ km.
Speed while going to the office = 6 km/h
Speed while returning along the same route = 8 km/h
Total time taken for the round trip = 3.5 hours

Step 2: Express time taken for each part of the journey

We know that time is calculated using the formula:

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

Time taken to travel from house to office ($t_1$):

$t_1=\frac{d}{6}\text{hours}$

Time taken to travel back from office to house ($t_2$):

$t_2=\frac{d}{8}\text{hours}$

Step 3: Set up the total time equation

The total time for the round trip is the sum of the time taken for going and returning ($t_1+t_2$):

$t_1+t_2=3.5$

Substituting the expressions for $t_1$ and $t_2$:

$\frac{d}{6}+\frac{d}{8}=3.5$

Step 4: Solve for $d$

Find a common denominator for the fractions on the left side (the Least Common Multiple of 6 and 8 is 24):

$\frac{4d+3d}{24}=3.5$

Combine the terms in the numerator:

$\frac{7d}{24}=3.5$

Multiply both sides of the equation by 24:

$7d=3.5\times 24$

$7d=84$

Divide both sides by 7 to solve for $d$:

$d=\frac{84}{7}=12$

Therefore, the distance between her house and the office is 12 km.