Adaptive Practice Without multiplying, decide which symbol to use: \( >,< \), or \( = \). Explain how you decided. \[ 4 \frac{1}{7} \times \frac{1}{5}-4 \frac{1}{4} \]

To tackle the expression \( 4 \frac{1}{7} \times \frac{1}{5} - 4 \frac{1}{4} \) without directly multiplying, we can first convert the mixed numbers to improper fractions. So, \( 4 \frac{1}{7} \) becomes \( \frac{29}{7} \) and \( 4 \frac{1}{4} \) turns into \( \frac{17}{4} \). Now, compute \( \frac{29}{7} \times \frac{1}{5} \) which simplifies to \( \frac{29}{35} \). Next, we need to compare \( \frac{29}{35} \) with \( \frac{17}{4} \). By cross-multiplying (since direct division might be tricky), we see \( 29 \times 4 = 116 \) and \( 17 \times 35 = 595 \). Clearly, \( 116 < 595 \), determining that \( \frac{29}{35} < \frac{17}{4} \). Therefore, the final expression evaluates to \( 4 \frac{1}{7} \times \frac{1}{5} - 4 \frac{1}{4} < 0 \). In choosing the comparison symbols, we relied heavily on converting and scaling fractions, allowing us to avoid direct multiplication and keep our arithmetic brain cells intact! Additionally, if you often work with mixed numbers and operations, a handy tip is to always convert them to improper fractions first. It simplifies calculations and minimizes confusion! Keeping your fractions consistent can save you from mathematical chaos.