Simplify : g. `((3)/(8) xx (2)/(15)) + ((4)/(9) div (32)/(27))`

Answer

To simplify the expression \(\frac{3}{8} \times \frac{2}{15} + \frac{4}{9} \div \frac{32}{27}\), we will follow these steps:

### Step 1: Simplify the first term \(\frac{3}{8} \times \frac{2}{15}\)

1. Multiply the numerators: \(3 \times 2 = 6\)
2. Multiply the denominators: \(8 \times 15 = 120\)
3. So, \(\frac{3}{8} \times \frac{2}{15} = \frac{6}{120}\)

### Step 2: Simplify \(\frac{6}{120}\)

1. Find the greatest common divisor (GCD) of 6 and 120, which is 6.
2. Divide both the numerator and the denominator by 6:
- \(6 \div 6 = 1\)
- \(120 \div 6 = 20\)
3. Thus, \(\frac{6}{120} = \frac{1}{20}\)

### Step 3: Simplify the second term \(\frac{4}{9} \div \frac{32}{27}\)

1. To divide by a fraction, multiply by its reciprocal:
\[ \frac{4}{9} \div \frac{32}{27} = \frac{4}{9} \times \frac{27}{32} \]

### Step 4: Multiply the fractions

1. Multiply the numerators: \(4 \times 27 = 108\)
2. Multiply the denominators: \(9 \times 32 = 288\)
3. So, \(\frac{4}{9} \times \frac{27}{32} = \frac{108}{288}\)

### Step 5: Simplify \(\frac{108}{288}\)

1. Find the GCD of 108 and 288, which is 36.
2. Divide both the numerator and the denominator by 36:
- \(108 \div 36 = 3\)
- \(288 \div 36 = 8\)
3. Thus, \(\frac{108}{288} = \frac{3}{8}\)

### Step 6: Add the two simplified fractions

1. Now we have \(\frac{1}{20} + \frac{3}{8}\).
2. To add these fractions, we need a common denominator. The least common multiple (LCM) of 20 and 8 is 40.

### Step 7: Convert the fractions to have a common denominator

1. Convert \(\frac{1}{20}\) to have a denominator of 40:
\[ \frac{1}{20} = \frac{1 \times 2}{20 \times 2} = \frac{2}{40} \]
2. Convert \(\frac{3}{8}\) to have a denominator of 40:
\[ \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \]

### Step 8: Add the fractions

1. Now add \(\frac{2}{40} + \frac{15}{40} = \frac{2 + 15}{40} = \frac{17}{40}\)

### Final Answer

The simplified result of the expression is \(\frac{17}{40}\).

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