Ankita walks from A to C through B, and runs back through the same route at a speed that is 40% more than her walking speed. She takes exactly 3 hours 30 minutes to walk from B to C as well as to run from B to A. The total time, in minutes, she would take to walk from A to B and run from B to C, is

The correct answer is 444.

Let us break down the solution step-by-step to understand the logical reasoning and calculations behind it.

Step 1: Understand the relationship between speeds and times
Let Ankita's walking speed be denoted as $V_w$ and her running speed be denoted as $V_r$.
According to the problem, her running speed is 40% more than her walking speed. Therefore, we can express the running speed as:
$$V_r=V_w+0.40V_w=1.4V_w$$
Thus, the ratio of her walking speed to her running speed is:
$$\frac{V_w}{V_r}$$

For a constant distance, speed and time are inversely proportional. Thus, the ratio of the time taken to walk a distance ($T_w$) to the time taken to run the same distance ($T_r$) is:
$$\frac{T_w}{T_r}$$

Step 2: Calculate the times for individual segments
We are given that she takes exactly 3 hours 30 minutes to walk from B to C, and also to run from B to A.
First, convert this duration into minutes:
$$3\text{ hours }30\text{ minutes}=(3\times 60)+30=210\text{ minutes}$$

For segment AB:
The time taken to run from B to A (which is the distance AB) is 210 minutes:
$$T_r\left(\text{AB}\right)=210\text{ minutes}$$
Using the time ratio $\frac{T_w}{T_r}$, we can find the time she would take to walk from A to B:
$$T_w\left(\text{AB}\right)=T_r\left(\text{AB}\right)\times \frac{7}{5}=210\times \frac{7}{5}=42\times 7=294\text{ minutes}$$

For segment BC:
The time taken to walk from B to C (which is the distance BC) is 210 minutes:
$$T_w\left(\text{BC}\right)=210\text{ minutes}$$
Using the inverse ratio, we can find the time she would take to run from B to C:
$$T_r\left(\text{BC}\right)=T_w\left(\text{BC}\right)\times \frac{5}{7}=210\times \frac{5}{7}=30\times 5=150\text{ minutes}$$

Step 3: Calculate the total requested time
The total time she would take to walk from A to B and run from B to C is:
$$T_{\text{total}}=T_w\left(\text{AB}\right)+T_r\left(\text{BC}\right)$$
Substituting the calculated values:
$$T_{\text{total}}=294\text{ minutes}+150\text{ minutes}=444\text{ minutes}$$