The rational numbers lying between $$\frac{1}{3}$$ and $$\frac{3}{4}$$ are

### Concept & Strategy When testing pairs of fractions against boundary conditions, always evaluate the simplest side of the boundary first. If one fraction in an option fails the test, you can instantly eliminate the entire option. ### Step-by-Step Solution * Let's establish our lower and upper decimal boundaries: * Lower bound: $$\frac{1}{3} \approx 0.333$$ * Upper bound: $$\frac{3}{4} = 0.750$$ * We need both fractions in a pair to fall within the range $$0.333 < x < 0.750$$. * Look at the first fraction in each option. They all have a denominator of $$300$$. Let's rewrite our lower bound ($$\frac{1}{3}$$) to match this denominator: $$\frac{1}{3} = \frac{100}{300}$$ * For a fraction to be greater than the lower bound, its numerator must be strictly greater than $$100$$. * (a) First fraction is $$\frac{117}{300}$$. Since $$117 > 100$$, it passes. * (b) First fraction is $$\frac{95}{300}$$. Since $$95 < 100$$, it fails. * (c) First fraction is $$\frac{99}{300}$$. Since $$99 < 100$$, it fails. * (d) First fraction is $$\frac{97}{300}$$. Since $$97 < 100$$, it fails. * Only option (a) has a valid first fraction. Let's quickly verify the second fraction in option (a) against the upper bound: $$\frac{287}{400} = \frac{287}{4} \div 100 = 71.75 \div 100 = 0.7175$$ Since $$0.7175 < 0.750$$, it successfully lies within the range. ### Exam Strategy & Shortcut **The Elimination Method:** Don't blindly calculate every decimal. Notice that all options start with a fraction over $$300$$. The lower limit is $$\frac{1}{3}$$, which translates identically to $$\frac{100}{300}$$. Instantly scan the numerators of the first fractions: $$117, 95, 99, 97$$. Only $$117$$ is greater than $$100$$. Options (b), (c), and (d) are eliminated in under $$5$$ seconds. ### Common Pitfall Checking the second fraction of every option first. Since the second fractions vary more in their denominators ($$400$$ and $$500$$), testing them takes longer. Always start by comparing against the boundary that offers a shared, easy denominator. ### Final Answer **Therefore, the correct answer is option (a).**