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Chapter 12: Problem 7

Towns \(A, B, C,\) and \(D\) are all in the same voting district. Towns \(A\) and \(B\) have 3,000 people each who support referendum \(R\) and the referendum has an average (arithmetic mean) of 3,500 supporters in towns \(B\) and \(D\) and an average of 5,000 supporters in Towns \(A\) and \(C\). Quantity \(\mathbf{A}\) The average number of supporters of Referendum \(R\) in Towns \(C\) and \(D\) Quantity \(\mathbf{B}\) The average number of supporters of Referendum \(R\) in Towns \(B\) and \(C\) a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

Short Answer

Expert verified
a. Quantity A is greater.

Step by step solution

01

Find total supporters in towns \(B\), \(D\) and \(A\), \(C\)

Given that the average of supporters in towns \(B\) and \(D\) is 3,500, and the average of supporters in towns \(A\) and \(C\) is 5,000. Since average is calculated as \(\frac{Sum\, of\, Items}{Number\, of\, Items}\), the total supporters in \(B\) and \(D\) can be calculated as \(3,500 \times 2 = 7,000\), and the total supporters in \(A\) and \(C\) is \(5,000 \times 2 = 10,000\).
02

Determine number of supporters in Town \(C\)

Now we know that there are 10,000 supporters in Town \(A\) and \(C\) combined, and we know that there are 3,000 supporters in town \(A\). So, the number of supporters in Town \(C\) must be \(10,000 - 3,000 = 7,000\).
03

Determine number of supporters in Town \(D\)

Similar to the above step, we know that there are 7,000 supporters combined in Towns \(B\) and \(D\), and we know there are 3,000 supporters in Town \(B\). So the number of supporters in Town \(D\) must be \(7,000 - 3,000 = 4,000\).
04

Calculate Quantities \(A\) and \(B\)

Quantity \(A\) is the average number of supporters in \(C\) and \(D\), and this can be calculated as \(\frac{(C + D)}{2} = \frac{(7,000 + 4,000)}{2} = 5,500\). Quantity \(B\) is the average number of supporters in \(B\) and \(C\), and this can be calculated as \(\frac{(B + C)}{2} = \frac{(3,000 + 7,000)}{2} = 5,000\).
05

Determine which Quantity is Greater

Now we can compare the two quantities. \(Quantity\, A = 5,500\) is greater than \(Quantity\, B = 5,000\), so the answer is a. Quantity \(A\) is greater.

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