\( \begin{array}{llll}\text { a } \frac{1}{4} \times 9=2 \frac{1}{4} & \text { T F } & \text { b } \frac{3}{5} \text { of } 25=15 \text { T F } & \text { C } \frac{2}{5} \text { of } 15=5 \frac{2}{5} \text { T F } \\ \text { d } 18 \times \frac{1}{5}=\frac{5}{18} & \text { T F } & \text { e } \frac{2}{6} \times 24=14 \text { T F } & \text { f } 17 \times \frac{1}{3}=\frac{17}{3}\end{array} \)

Let's break down the statements step by step: 1. **Statement a**: \( \frac{1}{4} \times 9 = 2 \frac{1}{4} \) is **False**. \( \frac{1}{4} \times 9 = \frac{9}{4} = 2 \frac{1}{4} \) actually holds true here, so this statement should be true instead. 2. **Statement b**: \( \frac{3}{5} \text{ of } 25 = 15 \) is **True**. \( \frac{3}{5} \times 25 = 15 \). 3. **Statement c**: \( \frac{2}{5} \text{ of } 15 = 5 \frac{2}{5} \) is **False**. \( \frac{2}{5} \times 15 = 6 \), so the statement about getting 5 and 2/5 is incorrect. 4. **Statement d**: \( 18 \times \frac{1}{5} = \frac{5}{18} \) is **False**. Actually, \( 18 \times \frac{1}{5} = \frac{18}{5} \). 5. **Statement e**: \( \frac{2}{6} \times 24 = 14 \) is **False**. Simplifying \( \frac{2}{6} = \frac{1}{3} \), we find \( \frac{1}{3} \times 24 = 8 \). 6. **Statement f**: \( 17 \times \frac{1}{3} = \frac{17}{3} \) is **True.** So, the true and false designations are: a: F, b: T, c: F, d: F, e: F, f: T. Now, let’s delve into some delightful insights: Fractions have been a crucial part of mathematics since ancient civilizations. The Egyptians used a system of unit fractions as far back as 1800 BC. Just imagine them using papyrus and ink to scribble down division problems! Understanding fractions not only makes math more manageable but also ties us to the rich history of human problem-solving! Real-world applications of fractions are everywhere! Whether you’re cooking and need to halve a recipe or you're sharing a pizza with friends, fractions help you divide things fairly. Knowing how to manipulate fractions is essential—not just for math class, but for everyday life!