Understanding the Problem: Solving a Continued Fraction Equation

The question asks us to find the value of the expression (4a - b + 3c), given a continued fraction equation where a, b, and c are positive integers. The equation provided is:

\(\frac{{45}}{{53}} = \frac{1}{{a + \frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}}}\)

To solve this, we need to manipulate the given fraction \(\frac{45}{53}\) to match the structure of the continued fraction on the right side of the equation. This process involves repeatedly inverting the fractional part of the expression.

Step-by-Step Solution for the Continued Fraction

We start by inverting the fraction \(\frac{45}{53}\) to match the \(\frac{1}{\text{something}}\) structure on the right side of the equation:

\(\frac{53}{45} = a + \frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}\)

Now, express \(\frac{53}{45}\) as a mixed number:

\(\frac{53}{45} = \frac{45 + 8}{45} = 1 + \frac{8}{45}\)

Comparing this with \(a + \frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}\), we can see that the integer part is \(a\).

\(a = 1\)

The remaining fractional part is \(\frac{8}{45}\), which must be equal to \(\frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}\). So, we invert \(\frac{8}{45}\):

\(\frac{45}{8} = b + \frac{1}{{c - \frac{2}{5}}}}\)

Next, express \(\frac{45}{8}\) as a mixed number:

\(\frac{45}{8} = \frac{40 + 5}{8} = 5 + \frac{5}{8}\)

Comparing this with \(b + \frac{1}{{c - \frac{2}{5}}}}\), the integer part is \(b\).

\(b = 5\)

The remaining fractional part is \(\frac{5}{8}\), which must be equal to \(\frac{1}{{c - \frac{2}{5}}}}\). So, we invert \(\frac{5}{8}\):

\(\frac{8}{5} = c - \frac{2}{5}\)

Now, solve for \(c\):

\(c = \frac{8}{5} + \frac{2}{5} = \frac{10}{5} = 2\)

So, we have found the values of a, b, and c:

• \(a = 1\)
• \(b = 5\)
• \(c = 2\)

We are given that a, b, and c are positive integers. Our calculated values 1, 5, and 2 are indeed positive integers, which matches the condition in the problem.

Calculating the Value of (4a - b + 3c)

Now that we have the values for a, b, and c, we can substitute them into the expression (4a - b + 3c):

Value \( = 4(a) - (b) + 3(c)\)

Substitute the values \(a=1\), \(b=5\), and \(c=2\):

Value \( = 4(1) - (5) + 3(2)\)

Perform the multiplication and subtraction:

Value \( = 4 - 5 + 6\)

Value \( = -1 + 6\)

Value \( = 5\)

Summary of Values

The value of the expression (4a - b + 3c) is 5.

Revision Table: Continued Fraction Solving

Additional Information: What are Continued Fractions?

A continued fraction is an expression obtained through an iterative process of representing a number as a sum of its integer part and the reciprocal of another number, which is then itself expressed as a sum of its integer part and the reciprocal of another number, and so on.

Finite continued fractions represent rational numbers (fractions like \(\frac{p}{q}\)). Infinite continued fractions can represent irrational numbers (like \(\sqrt{2}\) or \(\pi\)).

The general form of a simple finite continued fraction is:

\(a_0 + \frac{1}{{a_1 + \frac{1}{{a_2 + \frac{1}{{a_3 + \dots}}}}}}}\)

where \(a_0\) is an integer and \(a_1, a_2, a_3, \dots\) are positive integers.

Continued fractions are useful in various areas of mathematics, including number theory and approximation theory. The method used in this problem to find the values of a, b, and c is a standard algorithm for converting a rational number into a continued fraction representation.