`(11+s)/(12r)=(1)/(6)+(1-3s)/(4r)`
In the equation above, what is r inn terms of s?

Options

A. `r=5s+4`
B. `r=(s-2)/(3)`
C. `r=4s+5`
D. `r=(s+3)/(2)`

Answer

To solve the equation \(\frac{11+s}{12r}=\frac{1}{6}+\frac{1-3s}{4r}\) for \(r\) in terms of \(s\), we will follow these steps:

### Step 1: Identify the Right-Hand Side (RHS)
The right-hand side of the equation is \(\frac{1}{6} + \frac{1-3s}{4r}\).

### Step 2: Find a Common Denominator
The common denominator for the terms on the RHS is \(12r\).

### Step 3: Rewrite Each Term with the Common Denominator
- For \(\frac{1}{6}\), we multiply the numerator and denominator by \(2r\):
\[ \frac{1}{6} = \frac{2r}{12r} \]

- For \(\frac{1-3s}{4r}\), we multiply the numerator and denominator by \(3\):
\[ \frac{1-3s}{4r} = \frac{3(1-3s)}{12r} = \frac{3 - 9s}{12r} \]

### Step 4: Combine the Terms on the RHS
Now we can combine the two fractions:
\[ \frac{2r + (3 - 9s)}{12r} = \frac{2r + 3 - 9s}{12r} \]

### Step 5: Set the Left-Hand Side (LHS) Equal to the RHS
Now we set the LHS equal to the RHS:
\[ \frac{11+s}{12r} = \frac{2r + 3 - 9s}{12r} \]

### Step 6: Cancel the Denominator
Since the denominators are the same, we can cancel them:
\[ 11 + s = 2r + 3 - 9s \]

### Step 7: Rearrange the Equation
Now, we will rearrange the equation to isolate \(r\):
1. Move the terms involving \(s\) to one side:
\[ 2r = 11 + s - 3 + 9s \]
Simplifying this gives:
\[ 2r = 8 + 10s \]

### Step 8: Solve for \(r\)
Now, divide both sides by 2 to solve for \(r\):
\[ r = \frac{8 + 10s}{2} = 4 + 5s \]

### Final Answer
Thus, \(r\) in terms of \(s\) is:
\[ \boxed{5s + 4} \]