1. **State the problem:** Solve for $x$ in the equation $$\frac{0.853}{0.972} = 0.0750 \times \frac{0.642}{0.972} + \frac{x}{0.188}.$$\n\n2. **Rewrite the equation:** \n$$\frac{0.853}{0.972} = 0.0750 \times \frac{0.642}{0.972} + \frac{x}{0.188}.$$\n\n3. **Calculate the constants:** \nCalculate $$0.0750 \times \frac{0.642}{0.972} = 0.0750 \times 0.660493827 = 0.049537037.$$\nCalculate $$\frac{0.853}{0.972} = 0.877160494.$$\n\n4. **Substitute the values:** \n$$0.877160494 = 0.049537037 + \frac{x}{0.188}.$$\n\n5. **Isolate $\frac{x}{0.188}$:** \n$$\frac{x}{0.188} = 0.877160494 - 0.049537037 = 0.827623457.$$\n\n6. **Solve for $x$ by multiplying both sides by 0.188:** \n$$x = 0.188 \times 0.827623457.$$\nShow cancellation step: $$x = \cancel{0.188} \times 0.827623457.$$\n\n7. **Calculate $x$:** \n$$x = 0.155933.$$\n\n**Final answer:** $$x \approx 0.156.$$