Which of the following fractions is nearest to $$\frac{2}{5}$$?

### Concept & Strategy To find which fraction is "nearest" to a target value, convert all fractions to decimals and find the minimum absolute difference. ### Step-by-Step Solution * The target fraction is $$\frac{2}{5}$$, which equals exactly $$0.4$$. * Let's convert the given options into their exact decimal formats: * (a) $$\frac{4}{5} = 0.8$$ * (b) $$\frac{21}{50} = \frac{42}{100} = 0.42$$ * (c) $$\frac{20001}{50000} = \frac{40002}{100000} = 0.40002$$ * (d) $$\frac{200001}{500000} = \frac{400002}{1000000} = 0.400002$$ * Now, calculate the distance (difference) from the target $$0.4$$: * Difference for (a) = $$0.8 - 0.4 = 0.4$$ * Difference for (b) = $$0.42 - 0.4 = 0.02$$ * Difference for (c) = $$0.40002 - 0.4 = 0.00002$$ * Difference for (d) = $$0.400002 - 0.4 = 0.000002$$ * The smallest difference is $$0.000002$$, meaning option (d) is incredibly close to $$0.4$$. ### Exam Strategy & Shortcut **Pattern Recognition:** Notice how options (b), (c), and (d) are built. They all represent a value slightly larger than $$\frac{2}{5}$$. Multiply the denominator of each by $$2$$ to make it a power of $$10$$, and multiply the numerator by $$2$$: * $$\frac{21}{50} \rightarrow \frac{42}{100}$$ * $$\frac{20001}{50000} \rightarrow \frac{40002}{100000}$$ * $$\frac{200001}{500000} \rightarrow \frac{400002}{1000000}$$ As you add zeros to the denominator and insert them between the $$4$$ and the $$2$$ in the numerator, the value pushes closer and closer to $$0.40000...$$ The fraction with the most leading zeros after the decimal point will naturally be the closest. ### Common Pitfall Trying to find a common denominator for $$5, 50, 50000$$, and $$500000$$ and manually subtracting numerators. Converting everything directly to decimals base-10 makes the comparison trivial. ### Final Answer **Therefore, the correct answer is option (d).**