If `97/19 = a+ 1/(b+1/c)` where a, b and c are positive integers, then what is the sum of a, b and c?

Options

A. 16
B. 20
C. 9
D. Cannot be determined

Answer

To solve the equation \( \frac{97}{19} = a + \frac{1}{b + \frac{1}{c} } \), we will follow these steps:

### Step 1: Divide 97 by 19
First, we need to perform the division of 97 by 19.

\[ 97 \div 19 = 5 \quad \text{(since } 19 \times 5 = 95 \text{)} \]

This gives us a quotient of 5 and a remainder.

### Step 2: Find the remainder
Now, we calculate the remainder:

\[ 97 - (19 \times 5) = 97 - 95 = 2 \]

So we can express \( \frac{97}{19} \) as:

\[ \frac{97}{19} = 5 + \frac{2}{19} \]

### Step 3: Rewrite the fraction
Next, we need to express \( \frac{2}{19} \) in the form \( \frac{1}{b + \frac{1}{c}} \). We can rewrite \( \frac{2}{19} \) as follows:

\[ \frac{2}{19} = \frac{1}{\frac{19}{2}} = \frac{1}{9.5} \]

However, we need to express it in terms of integers \( b \) and \( c \). To do this, we can convert \( 9.5 \) into a fraction:

\[ 9.5 = 9 + \frac{1}{2} \]

Thus, we can write:

\[ \frac{2}{19} = \frac{1}{9 + \frac{1}{2}} \]

### Step 4: Identify values of a, b, and c
From our expression, we can identify:

- \( a = 5 \)
- \( b = 9 \)
- \( c = 2 \)

### Step 5: Calculate the sum of a, b, and c
Now, we can find the sum of \( a \), \( b \), and \( c \):

\[ a + b + c = 5 + 9 + 2 = 16 \]

### Final Answer
Thus, the sum of \( a \), \( b \), and \( c \) is \( 16 \).

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