Which of the following fractions lies between $$\frac{2}{3}$$ and $$\frac{3}{5}$$?

### Concept & Logic To determine which fraction falls between two boundaries, convert the boundary fractions and the given options into percentages or decimals for a direct, uniform comparison. ### Step-by-Step Solution * First, establish the numerical range of the given boundaries: * $$\frac{2}{3} \approx 0.666$$ * $$\frac{3}{5} = 0.600$$ * We are looking for a value strictly between $$0.600$$ and $$0.666$$. * Now, evaluate the decimal value of each option: * (a) $$\frac{2}{5} = 0.400$$ * (b) $$\frac{1}{3} \approx 0.333$$ * (c) $$\frac{1}{15} \approx 0.066$$ * (d) $$\frac{31}{50} = \frac{62}{100} = 0.620$$ * Comparing the options to our target range ($$0.600 < x < 0.666$$): $$0.620$$ is the only value that fits perfectly between the boundaries. ### Exam Strategy & Shortcut **Percentage Conversion:** Convert the bounds to percentages mentally. $$\frac{3}{5} = 60\%$$ and $$\frac{2}{3} = 66.66\%$$. Look at the options. Option (d) has a denominator of $$50$$. Multiply the numerator and denominator by $$2$$ to get a percentage instantly: $$\frac{31 \times 2}{50 \times 2} = \frac{62}{100} = 62\%$$. Since $$62\%$$ is between $$60\%$$ and $$66.66\%$$, mark it and move on without calculating the rest. ### Common Pitfall Getting confused by the phrasing because $$\frac{2}{3}$$ ($$0.666$$) is listed before $$\frac{3}{5}$$ ($$0.600$$), making it seem like a descending range. Always re-orient your bounds from smallest to largest mentally before checking options. ### Final Answer **Therefore, the correct answer is option (d).**