Question66 voters ranked cereals $A, B, C$ , and $D$ . Use the Borda count method to find the winner.

Studdy Solution

STEP 1

Assumptions1. The number of voters is66. . The voters are asked to rank four brands of cereal $A, B, C$ , and $$.
3. The votes are summarized in the given preference table.
4. We are using the Borda count method to determine the winner.

STEP 2

The Borda count method assigns points to each choice based on its rank. In this case, with four choices, the first choice gets points, the second choice gets2 points, the third choice gets1 point, and the fourth choice gets0 points. We can write this as $\text{First Choice} = \text{ points}$ $\text{Second Choice} =2 \text{ points}$ $\text{Third Choice} =1 \text{ point}$ $\text{Fourth Choice} =0 \text{ points}$

STEP 3

Now, we calculate the total points for each cereal brand. For each brand, we multiply the number of votes it got in each position by the points for that position, and then add up these products.
For brand $A$ , we have $\text{Total points for A} = (23 \times3) + (34 \times2) + (5 \times1) + ( \times0)$

STEP 4

Calculate the total points for brand $A$ .
$\text{Total points for A} = (23 \times3) + (34 \times2) + ( \times1) + (4 \times0) =69 +68 + +0 =142$

STEP 5

Repeat the process for brands $B, C$ , and $$.
For brand $B$ , we have $\text{Total points for B} = (5 \times3) + (23 \times2) + (34 \times1) + (4 \times0)$

STEP 6

Calculate the total points for brand $B$ .
$\text{Total points for B} = (5 \times3) + (23 \times2) + (34 \times1) + (4 \times0) =15 +46 +34 +0 =95$

STEP 7

For brand $C$ , we have $\text{Total points for C} = (4 \times3) + (5 \times2) + (23 \times1) + (34 \times0)$

STEP 8

Calculate the total points for brand $C$ .
$\text{Total points for C} = (4 \times3) + (5 \times2) + (23 \times1) + (34 \times0) =12 +10 +23 +0 =45$

STEP 9

For brand $, we have$ \text{Total points for D} = (34 \times3) + (4 \times2) + (5 \times) + (23 \times)$$

STEP 10

Calculate the total points for brand $$.
$\text{Total points for D} = (34 \times3) + (4 \times2) + (5 \times) + (23 \times0) =102 +8 +5 +0 =115$

STEP 11

Now, compare the total points for each brand. The brand with the highest total points is the winner.
$A142 \text{ points}$ $B95 \text{ points}$ $C45 \text{ points}$ $115 \text{ points}$ Brand $A$ has the highest total points, so it is the winner.
The winner is brand $A$ .

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