Perform the indicated operations and simplify, Express your answer as a single fraction, (a) \( \frac{2}{3}+\frac{9}{11} \) (b) \( \frac{5}{24}-\frac{3}{16} \)

Let's perform the indicated operations step by step and simplify each expression. ### Problem (a) **Add** \( \frac{2}{3} \) and \( \frac{9}{11} \). **Step 1: Find a Common Denominator** The denominators are 3 and 11. Since 3 and 11 are coprime (they have no common factors other than 1), the least common denominator (LCD) is: \[ \text{LCD} = 3 \times 11 = 33 \] **Step 2: Convert Each Fraction** Convert each fraction to have the denominator of 33: \[ \frac{2}{3} = \frac{2 \times 11}{3 \times 11} = \frac{22}{33} \] \[ \frac{9}{11} = \frac{9 \times 3}{11 \times 3} = \frac{27}{33} \] **Step 3: Add the Fractions** \[ \frac{22}{33} + \frac{27}{33} = \frac{22 + 27}{33} = \frac{49}{33} \] Since 49 and 33 have no common factors other than 1, the fraction is already in its simplest form. **Answer for (a):** \[ \frac{49}{33} \] ### Problem (b) **Subtract** \( \frac{3}{16} \) from \( \frac{5}{24} \). **Step 1: Find a Common Denominator** The denominators are 24 and 16. Let's find the LCD: - Prime factors of 24: \( 2^3 \times 3 \) - Prime factors of 16: \( 2^4 \) The LCD is the product of the highest powers of all primes present: \[ \text{LCD} = 2^4 \times 3 = 16 \times 3 = 48 \] **Step 2: Convert Each Fraction** Convert each fraction to have the denominator of 48: \[ \frac{5}{24} = \frac{5 \times 2}{24 \times 2} = \frac{10}{48} \] \[ \frac{3}{16} = \frac{3 \times 3}{16 \times 3} = \frac{9}{48} \] **Step 3: Subtract the Fractions** \[ \frac{10}{48} - \frac{9}{48} = \frac{10 - 9}{48} = \frac{1}{48} \] Since 1 and 48 have no common factors other than 1, the fraction is already in its simplest form. **Answer for (b):** \[ \frac{1}{48} \] ### Final Answers: - **(a)** \( \frac{49}{33} \) - **(b)** \( \frac{1}{48} \)