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A summer camp has 32 campers. A total of 22 of them swim, 20 play softball, and 5 do not swim or play softball. Which values complete the table?

[tex]\[
\begin{tabular}{|c|c|c|c|}
\cline {2-4}
& Softball & No Softball & Total \\
\hline
Swim & $a$ & $b$ & 22 \\
\hline
Do Not Swim & $c$ & 5 & $d$ \\
\hline
Total & 20 & $e$ & 32 \\
\hline
\end{tabular}
\][/tex]

A. [tex]\(a=15, b=10, c=7, d=5, e=12\)[/tex]
B. [tex]\(a=15, b=7, c=5, d=10, e=12\)[/tex]
C. [tex]\(a=14, b=7, c=5, d=12, e=10\)[/tex]
D. [tex]\(a=14, b=12, c=7, d=5, e=10\)[/tex]



Answer :

GinnyAnswer
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]

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