Operations with Fractions $$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Operations with Fractions: Continued Fractions

In this lesson, we explore a more advanced operation with fractions by working with continued fractions. The goal is not only to simplify a fraction, but also to identify specific values hidden within its structure.

1. Problem Overview

We are given the following expression:

$$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Our task is to determine the values of A, B, C, and D, then find the value of A + B + C + D.

2. Step-by-Step Solution

Step 1: Convert the fraction into a mixed number.

$$ \frac{43}{30} = 1 + \frac{13}{30} $$

This works because a mixed number can be written as the sum of a whole number and a fraction.

Step 2: Convert the fraction into a continued fraction.

Recall the rule:

$$ \frac{a}{b} = \frac{1}{\frac{b}{a}} $$

Applying this rule to the fraction:

$$ \frac{13}{30} = \frac{1}{\frac{30}{13}} $$

Now simplify the denominator step by step:

$$ \frac{30}{13} = 2 + \frac{4}{13} $$ $$ \frac{4}{13} = \frac{1}{\frac{13}{4}} = \frac{1}{3 + \frac{1}{4}} $$

So the continued fraction becomes:

$$ \frac{13}{30} = \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

3. Identifying the Values

Substituting back into the original expression:

$$ \frac{43}{30} = 1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

From this form, we can identify:

• A = 1
• B = 2
• C = 3
• D = 4

4. Final Calculation

Now add all the values:

$$ A + B + C + D = 1 + 2 + 3 + 4 = 10 $$

Final Answer:

$$ A + B + C + D = 10 $$