Solve the following problems by finding the LCM and comparing fractions. 22. A quality control inspector in an egg factory checks every forty-eighth egg for cracks and every fifty-fourth egg for weight. What is the number of the first egg each day that the inspector checks for both qualities? 3. A stock sold for \( 3 \frac{5}{8} \) one day and \( 3 \frac{1}{2} \) the next. Did the value of the stock go up or down? Explain. Marissa needs \( 2 \frac{2}{3} \) yards of ribbon for a wall-hanging she wants to mak She has \( 2 \frac{3}{4} \) yards. Does she have enough ribbon? Explain.

To find the first egg that the inspector checks for both qualities, we need to determine the least common multiple (LCM) of 48 and 54. The prime factorization gives us 48 = \( 2^4 \times 3^1 \) and 54 = \( 2^1 \times 3^3 \). To find the LCM, we take the highest power of each prime: \( 2^4 \) and \( 3^3 \). Thus, the LCM is \( 2^4 \times 3^3 = 16 \times 27 = 432 \). The inspector checks the first egg at number 432! Comparing the stock prices, start by converting them to improper fractions for better comparison. \( 3 \frac{5}{8} = \frac{29}{8} \) and \( 3 \frac{1}{2} = \frac{7}{2} = \frac{28}{8} \). Since \( \frac{29}{8} > \frac{28}{8} \), the value of the stock actually went up from the first day to the second. Going up for the win! Now about Marissa's ribbon! She needs \( 2 \frac{2}{3} \) yards, which is \( \frac{8}{3} \) in improper fractions. She has \( 2 \frac{3}{4} \), converting that gives \( \frac{11}{4} \). To compare, convert both to a common denominator (which is 12). Marissa needs \( \frac{32}{12} \) and has \( \frac{33}{12} \). Therefore, she has enough ribbon—just enough and a little extra for those crafty endeavors!