If the sum of five consecutive integers is S, then the largest of those integers in terms of S is

Options

A. `(S-10)/(5)`
B. `(S+4)/(4)`
C. `(S+5)/(4)`
D. `(S+10)/(5)`

Answer

To find the largest of five consecutive integers whose sum is \( S \), we can follow these steps:

### Step 1: Define the integers
Let the five consecutive integers be represented as:
- \( x \) (the first integer)
- \( x + 1 \) (the second integer)
- \( x + 2 \) (the third integer)
- \( x + 3 \) (the fourth integer)
- \( x + 4 \) (the fifth integer)

### Step 2: Write the equation for the sum
The sum of these five integers can be expressed as:
\[ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = S \]

### Step 3: Simplify the equation
Combine like terms:
\[ 5x + (1 + 2 + 3 + 4) = S \]
This simplifies to:
\[ 5x + 10 = S \]

### Step 4: Solve for \( x \)
Rearranging the equation gives:
\[ 5x = S - 10 \]
Now, divide both sides by 5:
\[ x = \frac{S - 10}{5} \]

### Step 5: Find the largest integer
The largest of the five consecutive integers is:
\[ x + 4 \]
Substituting the value of \( x \) we found:
\[ x + 4 = \frac{S - 10}{5} + 4 \]
To combine the terms, convert 4 to a fraction with a denominator of 5:
\[ x + 4 = \frac{S - 10}{5} + \frac{20}{5} \]
This simplifies to:
\[ x + 4 = \frac{S - 10 + 20}{5} = \frac{S + 10}{5} \]

### Conclusion
Thus, the largest of the five consecutive integers in terms of \( S \) is:
\[ \frac{S + 10}{5} \]

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