Perform the indicated operations and simplify, Express your answer as a single fraction,
(a)
(b)

Ask by Peterson Dawson. in the United States

Jan 24,2025

Question

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

UpStudy AI · Accepted Answer

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)** $$ \frac{1}{48} $$
(a) $$ \frac{49}{33} $$ (b) $$ \frac{1}{48} $$

Perform the indicated operations and simplify, Express your answer as a single fraction,
(a)
(b)

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