Question

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what are the correct values of a, b, c, d, and e?
○ a = 42, b = 28, c = 33, d = 5, e = 10
○ a = 42, b = 5, c = 28, d = 33, e = 10
○ a = 5, b = 42, c = 28, d = 10, e = 33
○ a = 5, b = 10, c = 28, d = 33, e = 42

Explanation:

Step1: Find the value of \(e\)

The total number of people is \(80\). The number of people who like touring bikes is \(70\). Using the formula \(e=\text{Total}-\text{Likes Touring Bikes}\), we have \(e = 80 - 70\).
$$e=10$$

Step2: Find the value of \(d\)

The total number of people is \(80\). The number of people who like mountain bikes is \(47\). Using the formula \(d=\text{Total}-\text{Likes Mountain Bikes}\), we have \(d=80 - 47\).
$$d = 33$$

Step3: Find the value of \(c\)

The number of people who like touring bikes is \(70\). The number of people who like mountain bikes and touring bikes (\(a\)) and the number of people who do not like mountain bikes but like touring bikes (\(c\)) sum up to \(70\). Also, \(a=\text{Likes Mountain Bikes}-\text{Does not like touring bikes (mountain bike lovers)}\). Since \(d = 33\) (from step 2) and the number of people who do not like mountain bikes and do not like touring bikes is \(5\), and \(c=\text{Does not like mountain bikes}-\text{Does not like touring bikes (non - mountain bike lovers)}\). But another way: \(c=\text{Likes Touring Bikes}-\text{(Likes Mountain Bikes - b)}\). Also, using the row - column relations. Since \(d = 33\) (total of non - mountain bike lovers) and the number of non - mountain bike lovers who do not like touring bikes is \(5\), for the non - mountain bike lovers row: \(c + 5=d\). Substituting \(d = 33\), we get \(c=33 - 5=28\).

Step4: Find the value of \(a\)

Since \(c = 28\) (from step 3) and \(a + c=70\) (total of touring bike lovers), then \(a=70 - c\). Substituting \(c = 28\), we have \(a=70 - 28 = 42\).

Step5: Find the value of \(b\)

Since \(a + b = 47\) (total of mountain bike lovers) and \(a = 42\) (from step 4), then \(b=47 - a\). Substituting \(a = 42\), we get \(b=47 - 42=5\).

Answer:

\(a = 42\), \(b = 5\), \(c = 28\), \(d = 33\), \(e = 10\) (the second option: \(a = 42,b = 5,c = 28,d = 33,e = 10\))