Question

If \[ A + \frac{1}{B + \frac{1}{C - 9}} = \frac{29}{5} \] then find the value of \( A + B + C \).

• A. 17
• B. 12
• C. 19
• D. 28

Hint:

In nested fractions, first convert the given value into a mixed number. Compare integer and fractional parts step-by-step to easily find unknowns.

Answer 1

The Correct Option is C

Step 1: Convert the given fraction into a mixed number.
\[ \frac{29}{5} = 5 + \frac{4}{5} \]
This means the integer part of the expression is 5.
Step 2: Compare integer parts.
Since the expression is \[ A + \frac{1}{B + \frac{1}{C - 9}} \]
and the integer part of the RHS is 5, we get:
\[ A = 5 \]
Step 3: Compare the fractional parts.
\[ \frac{1}{B + \frac{1}{C - 9}} = \frac{4}{5} \]
Step 4: Take reciprocal of both sides.
\[ B + \frac{1}{C - 9} = \frac{5}{4} \]
Step 5: Split the mixed fraction.
\[ \frac{5}{4} = 1 + \frac{1}{4} \]
Thus,
\[ B = 1 \]
Step 6: Compare remaining fractional parts.
\[ \frac{1}{C - 9} = \frac{1}{4} \]
Step 7: Take reciprocal again.
\[ C - 9 = 4 \]
\[ C = 13 \]
Step 8: Find the required sum.
\[ A + B + C = 5 + 1 + 13 \]
\[ = 19 \]
Step 9: Final conclusion.
Hence, the required value of \( A + B + C \) is 19.