`(5 (9)/(14))/(5 + (3)/(3 + (1)/(3/5) )) ` is equal to

Options

A. 1
B. 1.5
C. 2
D. 2.5

Answer

To solve the expression \((5 \cdot \frac{9}{14}) / (5 + \frac{3}{3 + \frac{1}{\frac{3}{5}}})\), we will break it down step by step.

### Step 1: Simplify the numerator
The numerator is \(5 \cdot \frac{9}{14}\).
\[ 5 \cdot \frac{9}{14} = \frac{5 \cdot 9}{14} = \frac{45}{14} \]

**Hint:** Multiply the whole number by the fraction to get a single fraction.

### Step 2: Simplify the denominator
The denominator is \(5 + \frac{3}{3 + \frac{1}{\frac{3}{5}}}\). We will start from the innermost part.

First, simplify \(\frac{1}{\frac{3}{5}}\):
\[ \frac{1}{\frac{3}{5}} = \frac{5}{3} \]

Now substitute this back into the denominator:
\[ 5 + \frac{3}{3 + \frac{5}{3}} \]

### Step 3: Simplify the expression inside the denominator
Now simplify \(3 + \frac{5}{3}\):
\[ 3 + \frac{5}{3} = \frac{3 \cdot 3 + 5}{3} = \frac{9 + 5}{3} = \frac{14}{3} \]

Now substitute this back:
\[ 5 + \frac{3}{\frac{14}{3}} = 5 + \frac{3 \cdot 3}{14} = 5 + \frac{9}{14} \]

### Step 4: Combine the terms in the denominator
Convert \(5\) to a fraction with a denominator of \(14\):
\[ 5 = \frac{70}{14} \]
Now add:
\[ \frac{70}{14} + \frac{9}{14} = \frac{70 + 9}{14} = \frac{79}{14} \]

### Step 5: Substitute back into the overall expression
Now we have:
\[ \frac{\frac{45}{14}}{\frac{79}{14}} \]

### Step 6: Simplify the overall expression
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{45}{14} \cdot \frac{14}{79} = \frac{45 \cdot 14}{14 \cdot 79} = \frac{45}{79} \]

### Step 7: Final expression
Now we have:
\[ \frac{45}{79} \]

### Step 8: Evaluate the final expression
Since the numerator and denominator do not have any common factors, \(\frac{45}{79}\) is in its simplest form.

### Final Answer
The expression \((5 \cdot \frac{9}{14}) / (5 + \frac{3}{3 + \frac{1}{\frac{3}{5}}})\) simplifies to \(\frac{45}{79}\).

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