$40.04 \div 0.4 = x \times 0.05$

## Concept & Logic The equation requires simplifying a decimal division on one side and then isolating the variable $x$ by dividing by a decimal fraction. $$ \frac{a}{b} = c \times d \implies c = \frac{a}{b \times d} $$ ## Step-by-Step Solution * **Calculation:** First, evaluate the left side of the equation: $\frac{40.04}{0.4}$ Multiply numerator and denominator by 10 to simplify: $\frac{400.4}{4} = 100.1$ Substitute this back into the main equation: $100.1 = x \times 0.05$ Solve for $x$: $x = \frac{100.1}{0.05}$ Multiply numerator and denominator by 100: $x = \frac{10010}{5}$ Perform the division: $x = 2002$ ## Exam Strategy & Shortcut Instead of dividing step-by-step, you can combine the denominators. $x = \frac{40.04}{0.4 \times 0.05}$ Calculate the denominator product first: $0.4 \times 0.05 = 0.02$ So, $x = \frac{40.04}{0.02} = \frac{4004}{2} = 2002$. This method heavily reduces calculation steps and prevents decimal shifting errors. ## Common Pitfall Students often mishandle the division by $0.05$, multiplying instead or misplacing the decimal, incorrectly landing on $20.02$ or $200.2$. Always double-check decimal shifts when dividing by values less than 1. ## Final Answer **Therefore, the correct answer is 2002.**