Babylonian Equation 030F0E

1. **State the problem:** Solve the equation $$x + \frac{x}{7} + \frac{1}{11} \left(x + \frac{x}{7}\right) = 60$$ for $x$. 2. **Rewrite the equation:** Let’s simplify the terms inside the parentheses first. $$x + \frac{x}{7} = \frac{7x}{7} + \frac{x}{7} = \frac{8x}{7}$$ So the equation becomes: $$x + \frac{x}{7} + \frac{1}{11} \cdot \frac{8x}{7} = 60$$ 3. **Express all terms with a common denominator:** The terms are: - $x = \frac{7x}{7}$ - $\frac{x}{7}$ - $\frac{1}{11} \cdot \frac{8x}{7} = \frac{8x}{77}$ So the equation is: $$\frac{7x}{7} + \frac{x}{7} + \frac{8x}{77} = 60$$ 4. **Combine the first two terms:** $$\frac{7x}{7} + \frac{x}{7} = \frac{8x}{7}$$ So now: $$\frac{8x}{7} + \frac{8x}{77} = 60$$ 5. **Find a common denominator for the left side:** The denominators are $7$ and $77$. Since $77 = 7 \times 11$, the common denominator is $77$. Rewrite: $$\frac{8x}{7} = \frac{8x \times 11}{7 \times 11} = \frac{88x}{77}$$ So: $$\frac{88x}{77} + \frac{8x}{77} = 60$$ 6. **Combine the fractions:** $$\frac{88x + 8x}{77} = \frac{96x}{77} = 60$$ 7. **Solve for $x$:** Multiply both sides by $77$: $$\cancel{77} \times \frac{96x}{\cancel{77}} = 60 \times 77$$ $$96x = 4620$$ Divide both sides by $96$: $$x = \frac{4620}{96}$$ Simplify the fraction by dividing numerator and denominator by 12: $$x = \frac{\cancel{4620}^{385}}{\cancel{96}^{8}}$$ So: $$x = \frac{385}{8} = 48.125$$ **Final answer:** $x = 48.125$