To solve the equation:

\[
\frac{r}{3} + \frac{r - 1}{r} = \frac{7}{6}
\]

we need to find a common denominator and solve for \( r \).

### Step 1: Find a Common Denominator

The denominators are 3, \( r \), and 6. The least common multiple (LCM) of these is \( 6r \). We rewrite each term with this common denominator:

\[
\frac{r}{3} = \frac{2r}{6}, \quad \frac{r - 1}{r} = \frac{6(r - 1)}{6r}, \quad \frac{7}{6} = \frac{7r}{6r}
\]

Thus, the equation becomes:

\[
\frac{2r^2 + 6(r - 1)}{6r} = \frac{7r}{6r}
\]

### Step 2: Eliminate the Denominator

Multiply every term by \( 6r \) to eliminate the fraction:

\[
2r^2 + 6(r - 1) = 7r
\]

### Step 3: Expand and Simplify

Expand the terms:

\[
2r^2 + 6r - 6 = 7r
\]

Combine like terms:

\[
2r^2 + 6r - 7r - 6 = 0
\]

\[
2r^2 - r - 6 = 0
\]

### Step 4: Solve the Quadratic Equation

To solve the quadratic equation \( 2r^2 - r - 6 = 0 \), we can use the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 2 \), \( b = -1 \), and \( c = -6 \). Substituting these values in:

\[
r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}
\]

\[
r = \frac{1 \pm \sqrt{1 + 48}}{4}
\]

\[
r = \frac{1 \pm \sqrt{49}}{4}
\]

\[
r = \frac{1 \pm 7}{4}
\]

### Step 5: Find the Roots

1. \( r = \frac{1 + 7}{4} = \frac{8}{4} = 2 \)
2. \( r = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2} \)

### Conclusion

The solutions for the equation are:

\[
r = 2 \quad \text{and} \quad r = -\frac{3}{2}
\]

Would you like more details on any step or have any other questions?

Here are 5 related questions for further exploration:

1. How do you find the least common multiple (LCM) of multiple numbers?
2. What is the quadratic formula, and how is it derived?
3. How can you check if your solutions to a quadratic equation are correct?
4. What are some real-world applications of solving quadratic equations?
5. How do you simplify algebraic fractions with different denominators?

**Tip:** When solving equations with fractions, always look for a common denominator to simplify the equation.

Discover how to solve the equation r/3 + (r - 1)/r = 7/6 step-by-step. This tutorial covers algebraic concepts, including finding the least common multiple, simplifying fractions, and using the quadratic formula. Ideal for students in Grades 9-11.

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