Understanding the Problem: Train or Bus Commuters

The question provides information about a group of 50 persons and how they commute to work, specifically focusing on those who use a train, a bus, or neither.

We are given the following data:

• Total number of persons = 50
• Number of persons who go by train = 18
• Number of persons who go by bus = 26
• Number of persons who go by neither train nor bus = 8

The question asks for the number of people who use either train or bus to go to work. This means the people who use the train, or the bus, or both train and bus. In terms of set theory, this is the number of people in the union of the set of train users and the set of bus users.

Calculating People Using Either Train or Bus

We know that the total number of people can be divided into two groups:

1. Those who use either train or bus (or both).
2. Those who use neither train nor bus.

Therefore, the total number of persons is the sum of these two groups.

Let T be the set of people who use the train and B be the set of people who use the bus.

The number of people who use either train or bus is represented by $|T \cup B|$.

The number of people who use neither train nor bus is given as 8.

The total number of people is 50.

We can write this relationship as:

Total persons = (Number using either train or bus) + (Number using neither train nor bus)

Using the given values:

$\text{50} = \text{(Number using either train or bus)} + \text{8}$

To find the number of people using either train or bus, we can rearrange the equation:

Number using either train or bus = Total persons - Number using neither train nor bus

Number using either train or bus = $50 - 8$

Number using either train or bus = $42$

So, 42 people use either train or bus to go to work.

Detailed Breakdown of Commuters

We can also find the number of people who use both train and bus using the Principle of Inclusion-Exclusion, although it wasn't directly asked. The formula is:

$|T \cup B| = |T| + |B| - |T \cap B|$

We know $|T \cup B| = 42$, $|T| = 18$, and $|B| = 26$. Plugging these values in:

$42 = 18 + 26 - |T \cap B|$

$42 = 44 - |T \cap B|$

$|T \cap B| = 44 - 42$

$|T \cap B| = 2$

So, 2 people use both train and bus.

Now we can break down the groups:

• Only Train: $|T| - |T \cap B| = 18 - 2 = 16$
• Only Bus: $|B| - |T \cap B| = 26 - 2 = 24$
• Both Train and Bus: $|T \cap B| = 2$
• Neither Train nor Bus: 8 (given)

Let's check the total:

Only Train + Only Bus + Both + Neither = $16 + 24 + 2 + 8 = 50$

This matches the total number of persons, confirming our calculation for "either train or bus".

Final Answer on Commute Type

The question asks for the number of people who use either train or bus. Based on our calculation:

Number of people using either train or bus = Total persons - Number using neither

Number of people using either train or bus = $50 - 8 = 42$

Thus, 42 people use either the train or the bus for their commute.

Revision Table: Commute Data Summary

Additional Information: Set Theory and Venn Diagrams

This problem is a classic example that can be solved using basic set theory concepts or visualised with a Venn diagram. The total group of 50 persons represents the universal set.

• The set of people who go by train (T) and the set of people who go by bus (B) are subsets of the universal set.
• The people who go by neither train nor bus are outside the union of sets T and B, but still within the universal set.
• The number of people using either train or bus corresponds to the number of elements in the union of sets T and B ($|T \cup B|$).
• The number of people using neither corresponds to the number of elements in the complement of the union of T and B, relative to the universal set.

A Venn diagram would show two overlapping circles for Train (T) and Bus (B) inside a rectangle representing the total group. The number 8 would be placed outside the circles but inside the rectangle. The areas within the circles represent people using train, bus, or both. The sum of the numbers in the 'Only Train', 'Only Bus', and 'Both' regions equals the number using 'Either Train or Bus'. This total must add up with the 'Neither' group to equal the 'Total Persons'.

This method is useful for understanding relationships between overlapping groups in a population.