`(8r+7)/(4s)=11`
If `(1)/(2)r+1=s+1`, what is the value of `r+s` for the equation above?

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. `1`
D. `(3)/(2)`

Answer

To solve the given problem step by step, we start with the two equations provided:

1. \(\frac{8r + 7}{4s} = 11\)
2. \(\frac{1}{2}r + 1 = s + 1\)

### Step 1: Simplify the first equation
We can start by simplifying the first equation. Multiply both sides by \(4s\) to eliminate the denominator:

\[ 8r + 7 = 44s \]

This is our first equation (Equation 1).

### Step 2: Simplify the second equation
Now let's simplify the second equation. We can subtract 1 from both sides:

\[ \frac{1}{2}r = s \]

To express \(r\) in terms of \(s\), multiply both sides by 2:

\[ r = 2s \]

This is our second equation (Equation 2).

### Step 3: Substitute Equation 2 into Equation 1
Now we will substitute \(r = 2s\) from Equation 2 into Equation 1:

\[ 8(2s) + 7 = 44s \]

This simplifies to:

\[ 16s + 7 = 44s \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to isolate \(s\):

\[ 16s + 7 - 44s = 0 \]

This simplifies to:

\[ -28s + 7 = 0 \]

### Step 5: Solve for \(s\)
Now, we can solve for \(s\):

\[ -28s = -7 \]

Dividing both sides by -28 gives:

\[ s = \frac{1}{4} \]

### Step 6: Substitute \(s\) back to find \(r\)
Now that we have the value of \(s\), we can substitute it back into Equation 2 to find \(r\):

\[ r = 2s = 2 \times \frac{1}{4} = \frac{1}{2} \]

### Step 7: Find \(r + s\)
Finally, we can find \(r + s\):

\[ r + s = \frac{1}{2} + \frac{1}{4} \]

To add these fractions, we need a common denominator:

\[ r + s = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]

### Final Answer
Thus, the value of \(r + s\) is:

\[ \boxed{\frac{3}{4}} \]