Simplify : `(19)/(43) + (1)/(2 + (1)/(3 + (1)/(1 + (1)/(4))))`

Options

A. 1
B. `(19)/(43)`
C. `(43)/(19)`
D. `(38)/(43)`

Answer

To simplify the expression \( \frac{19}{43} + \frac{1}{2 + \frac{1}{3 + \frac{1}{1 + \frac{1}{4}}}} \), we will follow these steps:

### Step 1: Simplify the innermost fraction
Start with the innermost fraction:
\[ 1 + \frac{1}{4} \]
This can be simplified as follows:
\[ 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]

**Hint:** Always start simplifying from the innermost part of the expression.

### Step 2: Substitute back into the next fraction
Now substitute \( \frac{5}{4} \) into the next fraction:
\[ 3 + \frac{5}{4} \]
This can be simplified as:
\[ 3 + \frac{5}{4} = \frac{12}{4} + \frac{5}{4} = \frac{17}{4} \]

**Hint:** Convert whole numbers to fractions with a common denominator for easier addition.

### Step 3: Substitute into the next fraction
Now substitute \( \frac{17}{4} \) into the next fraction:
\[ 2 + \frac{1}{\frac{17}{4}} \]
To simplify \( \frac{1}{\frac{17}{4}} \), we take the reciprocal:
\[ \frac{1}{\frac{17}{4}} = \frac{4}{17} \]
Now, we can add:
\[ 2 + \frac{4}{17} = \frac{34}{17} + \frac{4}{17} = \frac{38}{17} \]

**Hint:** Remember that adding fractions requires a common denominator.

### Step 4: Substitute into the final fraction
Now substitute \( \frac{38}{17} \) into the last fraction:
\[ \frac{1}{\frac{38}{17}} = \frac{17}{38} \]

**Hint:** Again, take the reciprocal when simplifying fractions.

### Step 5: Add the two fractions
Now we can add \( \frac{19}{43} \) and \( \frac{17}{38} \):
To add these fractions, we need a common denominator. The least common multiple of 43 and 38 is \( 43 \times 38 = 1634 \).

Convert each fraction:
\[ \frac{19}{43} = \frac{19 \times 38}{43 \times 38} = \frac{722}{1634} \]
\[ \frac{17}{38} = \frac{17 \times 43}{38 \times 43} = \frac{731}{1634} \]

Now, add them together:
\[ \frac{722}{1634} + \frac{731}{1634} = \frac{722 + 731}{1634} = \frac{1453}{1634} \]

### Final Step: Simplify the result
Now, we check if \( \frac{1453}{1634} \) can be simplified. Since both numbers do not have any common factors, this is our final answer.

Thus, the simplified expression is:
\[ \frac{1453}{1634} \]

### Summary of Steps:
1. Simplify the innermost fraction.
2. Substitute back and simplify the next fraction.
3. Continue substituting and simplifying until the outermost fraction.
4. Find a common denominator to add the two fractions.
5. Simplify the final result if possible.