Which of the following pairs of fractions adds up to a number greater than $5$?

### Concept & Logic The task is to determine which pair of fractions has a sum strictly greater than $5$. Instead of finding the exact fractional sum for every option (which requires calculating Least Common Multiples), convert each fraction into its approximate decimal or mixed number equivalent to quickly estimate the sum. ### Step-by-Step Solution Let's evaluate the options one by one using decimal approximations: * **Option (a):** $\frac{5}{3}$ and $\frac{3}{4}$ $\frac{5}{3} \approx 1.66$ $\frac{3}{4} = 0.75$ Sum: $1.66 + 0.75 = 2.41$ (Less than $5$) * **Option (b):** $\frac{7}{3}$ and $\frac{11}{5}$ $\frac{7}{3} \approx 2.33$ $\frac{11}{5} = 2.2$ Sum: $2.33 + 2.2 = 4.53$ (Less than $5$) * **Option (c):** $\frac{11}{4}$ and $\frac{8}{3}$ $\frac{11}{4} = 2.75$ $\frac{8}{3} \approx 2.66$ Sum: $2.75 + 2.66 = 5.41$ (Greater than $5$) * **Option (d):** $\frac{13}{5}$ and $\frac{11}{6}$ $\frac{13}{5} = 2.6$ $\frac{11}{6} \approx 1.83$ Sum: $2.6 + 1.83 = 4.43$ (Less than $5$) ### Exam Strategy & Shortcut Use mixed fractions to instantly eliminate low-value pairs. For example, in option (b): $2\frac{1}{3} + 2\frac{1}{5}$. Since both fractional parts ($\frac{1}{3}$ and $\frac{1}{5}$) are less than a half, their sum is clearly less than $1$, meaning the total must be under $5$ ($2+2+<1$). In option (c): $2\frac{3}{4} + 2\frac{2}{3}$. Here, both fractional parts are greater than a half, guaranteeing the sum pushes past the integer $5$ ($2+2+>1$). ### Common Pitfall Students waste valuable exam time calculating exact common denominators for every single option. Always estimate first, and only calculate the exact LCM if two options are incredibly close to the target value. ### Final Answer **Therefore, the correct answer is 11/4, 8/3.**