Understanding the Problem: Train and Bus Commuters

The problem provides information about a group of 50 persons and their modes of transport to work (train, bus, or neither). We are asked to find the number of people who travel only by train and express this number as a percentage of the total group size.

Analyzing the Given Data

We are given the following information:

• 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

Calculating People Using Either Train or Bus

First, let's find out how many people use at least one of the given modes of transport (train or bus or both). This can be found by subtracting those who use neither from the total number of persons.

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

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

So, 42 people use either train or bus or both.

Finding People Using Both Train and Bus

We know the total number of people who use train ($18$) and the total number of people who use bus ($26$). The sum of these two numbers includes the people who use both modes twice. The number of people using at least one mode (calculated above as 42) represents those in the union of the two sets (Train users and Bus users).

We can use the principle of inclusion-exclusion, which states:

$\text{|A} \cup \text{B|} = \text{|A|} + \text{|B|} - \text{|A} \cap \text{B|}$

Where A is the set of train users and B is the set of bus users.

$\text{|Train} \cup \text{Bus|} = \text{|Train|} + \text{|Bus|} - \text{|Train} \cap \text{Bus|}$

We know $\text{|Train} \cup \text{Bus|} = 42$, $\text{|Train|} = 18$, and $\text{|Bus|} = 26$.

$42 = 18 + 26 - \text{|Train} \cap \text{Bus|}$

$42 = 44 - \text{|Train} \cap \text{Bus|}$

Rearranging the equation to find the number of people using both (Train $\cap$ Bus):

$\text{|Train} \cap \text{Bus|} = 44 - 42 = 2$

So, 2 people go by both train and bus.

Determining People Going by Train Only

The number of people who go by train only is the total number of people who go by train minus those who go by both train and bus.

Number of people going by train only = Number of people going by train - Number of people going by both

Number of people going by train only = $18 - 2 = 16$

Thus, 16 people go by train only.

Calculating Percentage of Train Only Commuters

We need to find what percentage the number of people going by train only (16) is of the total number of persons (50).

Percentage = $\left( \frac{\text{Number of people going by train only}}{\text{Total number of persons}} \right) \times 100\%$

Percentage = $\left( \frac{16}{50} \right) \times 100\%$

Percentage = $\left( \frac{16 \times 2}{50 \times 2} \right) \times 100\% = \left( \frac{32}{100} \right) \times 100\%$

Percentage = $0.32 \times 100\% = 32\%$

So, the number of people going by train only is 32% of the total number of persons.

Summary of Calculation Steps

Revision Table: Commuting Data Analysis

Additional Information: Set Theory and Venn Diagrams

This problem can also be visualized using a Venn diagram. We have two overlapping circles, one for Train users (T) and one for Bus users (B), inside a rectangle representing the total group.

• The area outside both circles represents those who use neither (8 persons).
• The area where the circles overlap ($\text{T} \cap \text{B}$) represents those who use both (calculated as 2 persons).
• The area in the Train circle only ($\text{T} \setminus \text{B}$) represents those who use train only (calculated as 16 persons).
• The area in the Bus circle only ($\text{B} \setminus \text{T}$) represents those who use bus only ($26 - 2 = 24$ persons).

The sum of all these disjoint areas must equal the total number of persons:

Train Only + Bus Only + Both + Neither = Total

$16 + 24 + 2 + 8 = 50$

This confirms our calculations are consistent with the total number of persons.