- Set up equations from the table constraints: \(a+b=47\), \(a+c=70\), and \(a+b+c+5=80\).
- Solve for \(c\) by substituting \(a+b=47\) into the total count: \(47+c+5=80 \implies c=28\).
- Find \(a\) using \(a+c=70 \implies a+28=70 \implies a=42\).
- Determine \(b, d, e\) using \(a+b=47\), \(d=c+5\), and \(e=b+5\).
- The values are \(\boxed{a=42, b=5, c=28, d=33, e=10}\).
Explanation
- Analysis of the Problem
We are given data about a bike club with 80 members and their preferences for touring bikes and mountain bikes. We need to determine the values of variables \(a\), \(b\), \(c\), \(d\), and \(e\) representing various subsets of members.
The provided information is:
- Total members: \(80\)
- Members who like touring bikes: \(70\)
- Members who like mountain bikes: \(47\)
- Members who like neither: \(5\)
The structure of the club preferences is organized in a contingency table. Let \(a\) be members who like both, \(b\) be members who like mountain but not touring, \(c\) be members who like touring but not mountain, and the value \(5\) represents members who like neither. From the table, the totals are:
- \(a + b = 47\) (Total who like mountain bikes)
- \(a + c = 70\) (Total who like touring bikes)
- \(c + 5 = d\) (Total who do not like mountain bikes)
- \(b + 5 = e\) (Total who do not like touring bikes)
- \(a + b + c + 5 = 80\) (Total members)
- Solving the System of Equations
We have a system of linear equations derived from the problem constraints.
From the grand total equation:
\[a + b + c + 5 = 80\]
We can simplify this to:
\[a + b + c = 75\]
We know that \(a + b = 47\). Substituting this into the simplified total equation:
\[47 + c = 75\]
\[c = 75 - 47 = 28\]
Now, using \(a + c = 70\), we substitute the value of \(c\):
\[a + 28 = 70\]
\[a = 70 - 28 = 42\]
With \(a\) known, we use \(a + b = 47\) to find \(b\):
\[42 + b = 47\]
\[b = 5\]
Finally, we calculate \(d\) and \(e\) using the relations from the table:
\[d = c + 5 = 28 + 5 = 33\]
\[e = b + 5 = 5 + 5 = 10\]
- Final Values and Verification
We have calculated all the unknown variables:
- \(a = 42\)
- \(b = 5\)
- \(c = 28\)
- \(d = 33\)
- \(e = 10\)
Verifying these results:
- Total who like mountain bikes: \(a + b = 42 + 5 = 47\). Correct.
- Total who like touring bikes: \(a + c = 42 + 28 = 70\). Correct.
- Total members: \(a + b + c + 5 = 42 + 5 + 28 + 5 = 80\). Correct.
- Column total for "Does Not Like Touring": \(b + 5 = 5 + 5 = 10 = e\). Correct.
- Row total for "Does Not Like Mountain": \(c + 5 = 28 + 5 = 33 = d\). Correct.
The values are \(a=42, b=5, c=28, d=33, e=10\).
Examples
This type of table is known as a contingency table, which is a fundamental tool in statistics for summarizing relationships between two categorical variables. In real life, market researchers use these tables to analyze customer behavior—for example, determining how many customers like both Product A and Product B versus those who prefer only one or neither. By solving for these segments, businesses can target their advertising more effectively to specific demographics.