1. The problem is to solve the equation $10 \div x = 45 \div 27$ for $x$. 2. First, rewrite the division as fractions: $\frac{10}{x} = \frac{45}{27}$. 3. Simplify the fraction on the right side: $\frac{45}{27} = \frac{45 \div 9}{27 \div 9} = \frac{5}{3}$. 4. Now the equation is $\frac{10}{x} = \frac{5}{3}$. 5. Cross multiply to solve for $x$: $10 \times 3 = 5 \times x$. 6. This gives $30 = 5x$. 7. Divide both sides by 5: $x = \frac{30}{5} = 6$. 8. Therefore, the solution is $x = 6$.