Given that the total number of students $N$ is 125, we can use the relative frequencies and the total number of students to find the value of $w$ (the number of students who bike).

First, let's understand the known relative frequency for walking: $\text{Relative frequency for walking} = \frac{25}{125} = 0.2$

The sum of all relative frequencies should be 1. Therefore, we need to express the relative frequency of biking in terms of $w$:

$\text{Relative frequency for biking} = \frac{w}{125}$

Let's denote the unknown relative frequencies for public transport, car, and cycle as $rf_{\text{PT}}, rf_{\text{car}},$ and $rf_{\text{cycle}}$ respectively. So we have:

$rf_{\text{PT}} + rf_{\text{car}} + rf_{\text{cycle}} + \frac{w}{125} + 0.2 = 1$

Since we don't have the relative frequencies for public transport, car, and cycle, we will use the sum of all relative frequencies to isolate $w$:

$rf_{\text{PT}} + rf_{\text{car}} + rf_{\text{cycle}} + \frac{w}{125} + 0.2 = 1$

Rearranging to solve for $w$:

$rf_{\text{PT}} + rf_{\text{car}} + rf_{\text{cycle}} + \frac{w}{125} = 0.8$

Let's say the sum of relative frequencies for public transport, car, and cycle is $S$: $S + \frac{w}{125} = 0.8$

Since $S$ is unknown, we'll use the sum of the frequencies to isolate $w$:

$\frac{w}{125} = 0.8 - S$

To find $w$: $w = 125(0.8 - S)$

We need the actual relative frequencies for public transport, car, and cycle to determine $S$ and subsequently $w$. Assuming no additional information is given, the exact value of $w$ cannot be found without knowing the sum $S$. If we assume $S$ is 0, then:

$w = 125(0.8 - 0)$ $w = 100$

Thus, the value of $w$ would be 100 if there are no other students using public transport, car, or cycle. If you have the relative frequencies for public transport, car, or cycle, we can refine this calculation.

Would you like more details or have any questions?

Here are 5 questions you can ask:

1. How do you find the relative frequencies for public transport, car, and cycle?
2. How can we verify the consistency of relative frequencies in the table?
3. Can you explain the calculation of relative frequency in more detail?
4. What if the relative frequencies for public transport, car, and cycle are given? How would $w$ change?
5. How can we use relative frequency to analyze other similar problems?

Tip: When solving problems involving relative frequencies, ensure all given values add up correctly to maintain the balance of the total sum.