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Sagot :
To solve this problem, let's fill in the blanks one by one based on the provided total values and constraints.
First, we know:
1. The total number of campers is 32.
2. 22 campers swim.
3. 20 campers play softball.
4. 5 campers do not swim and do not play softball.
### Step 1: Define Known Values
Given:
- Total campers ([tex]\( T \)[/tex]) = 32
- Campers who swim ([tex]\( S \)[/tex]) = 22
- Campers who play softball ([tex]\( P \)[/tex]) = 20
- Campers who do neither ([tex]\( N \)[/tex]) = 5
### Step 2: Calculate Campers who do Either or Both
First, those who participate in either or both activities:
[tex]\[ T - N = 32 - 5 = 27 \][/tex]
### Step 3: Set up the Relationship
Let [tex]\( a \)[/tex] be the number of campers who both swim and play softball.
Let [tex]\( b \)[/tex] be the number of campers who swim but do not play softball.
Let [tex]\( c \)[/tex] be the number of campers who do not swim but play softball.
From the totals given, we know:
[tex]\[ S = a + b \][/tex]
[tex]\[ P = a + c \][/tex]
We also know that:
[tex]\[ a + b + c + N = T \][/tex]
Since [tex]\( N = 5 \)[/tex]:
[tex]\[ a + b + c + 5 = 32 \][/tex]
[tex]\[ a + b + c = 27 \][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
To find [tex]\( a \)[/tex]:
[tex]\[ a = (S + P - 27) = (22 + 20 - 27) = 15 \][/tex]
So we have [tex]\( a = 15 \)[/tex].
### Step 5: Solve for [tex]\( b \)[/tex] and [tex]\( c \)[/tex]
To find [tex]\( b \)[/tex]:
[tex]\[ b = S - a = 22 - 15 = 7 \][/tex]
To find [tex]\( c \)[/tex]:
[tex]\[ c = P - a = 20 - 15 = 5 \][/tex]
### Step 6: Calculate Total Numbers
[tex]\[ d = 32 - 22 = 10 \quad \text{(Total number of campers who do not swim)} \][/tex]
[tex]\[ e = 32 - 20 = 12 \quad \text{(Total number of campers who do not play softball)} \][/tex]
### Conclusion:
Thus, the table is completed as follows:
[tex]\[ \begin{array}{|c|c|c|c|} \cline{2-4} & \text{Softball} & \text{No Softball} & \text{Total} \\ \hline \text{Swim} & 15 & 7 & 22 \\ \hline \text{Do Not Swim} & 5 & 5 & 10 \\ \hline \text{Total} & 20 & 12 & 32 \\ \hline \end{array} \][/tex]
Therefore, the correct values are:
[tex]\[ \boxed{a=15, b=7, c=5, d=10, e=12} \][/tex]
Thus the correct values which complete the table are:
[tex]\[ a=15, b=7, c=5, d=10, e=12 \][/tex]
First, we know:
1. The total number of campers is 32.
2. 22 campers swim.
3. 20 campers play softball.
4. 5 campers do not swim and do not play softball.
### Step 1: Define Known Values
Given:
- Total campers ([tex]\( T \)[/tex]) = 32
- Campers who swim ([tex]\( S \)[/tex]) = 22
- Campers who play softball ([tex]\( P \)[/tex]) = 20
- Campers who do neither ([tex]\( N \)[/tex]) = 5
### Step 2: Calculate Campers who do Either or Both
First, those who participate in either or both activities:
[tex]\[ T - N = 32 - 5 = 27 \][/tex]
### Step 3: Set up the Relationship
Let [tex]\( a \)[/tex] be the number of campers who both swim and play softball.
Let [tex]\( b \)[/tex] be the number of campers who swim but do not play softball.
Let [tex]\( c \)[/tex] be the number of campers who do not swim but play softball.
From the totals given, we know:
[tex]\[ S = a + b \][/tex]
[tex]\[ P = a + c \][/tex]
We also know that:
[tex]\[ a + b + c + N = T \][/tex]
Since [tex]\( N = 5 \)[/tex]:
[tex]\[ a + b + c + 5 = 32 \][/tex]
[tex]\[ a + b + c = 27 \][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
To find [tex]\( a \)[/tex]:
[tex]\[ a = (S + P - 27) = (22 + 20 - 27) = 15 \][/tex]
So we have [tex]\( a = 15 \)[/tex].
### Step 5: Solve for [tex]\( b \)[/tex] and [tex]\( c \)[/tex]
To find [tex]\( b \)[/tex]:
[tex]\[ b = S - a = 22 - 15 = 7 \][/tex]
To find [tex]\( c \)[/tex]:
[tex]\[ c = P - a = 20 - 15 = 5 \][/tex]
### Step 6: Calculate Total Numbers
[tex]\[ d = 32 - 22 = 10 \quad \text{(Total number of campers who do not swim)} \][/tex]
[tex]\[ e = 32 - 20 = 12 \quad \text{(Total number of campers who do not play softball)} \][/tex]
### Conclusion:
Thus, the table is completed as follows:
[tex]\[ \begin{array}{|c|c|c|c|} \cline{2-4} & \text{Softball} & \text{No Softball} & \text{Total} \\ \hline \text{Swim} & 15 & 7 & 22 \\ \hline \text{Do Not Swim} & 5 & 5 & 10 \\ \hline \text{Total} & 20 & 12 & 32 \\ \hline \end{array} \][/tex]
Therefore, the correct values are:
[tex]\[ \boxed{a=15, b=7, c=5, d=10, e=12} \][/tex]
Thus the correct values which complete the table are:
[tex]\[ a=15, b=7, c=5, d=10, e=12 \][/tex]
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