If \(\frac{{36}}{{11}} = 3\; + \;\frac{1}{{x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}}}\) , where x, y and z are natural numbers then what is (x + y + z) equal to?
6
Solving Continued Fraction Equations
This problem asks us to find the value of \((x + y + z)\) given an equation involving a continued fraction. The equation is: \[\frac{{36}}{{11}} = 3\; + \;\frac{1}{{x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}}}\] We are told that x, y, and z are natural numbers (positive integers). To solve this, we need to express the fraction \(\frac{36}{11}\) in the form of the given continued fraction.
Step-by-Step Solution to find x, y, and z
We will systematically break down the fraction \(\frac{36}{11}\) to match the structure of the continued fraction on the right side of the equation.
Step 1: Convert the improper fraction to a mixed number.
Divide 36 by 11:
\[36 \div 11 = 3 \text{ with a remainder of } 3\]So, we can write \(\frac{36}{11}\) as a mixed number:
\[\frac{36}{11} = 3 + \frac{3}{11}\]Step 2: Compare the mixed number form with the given equation.
The equation is:
\[\frac{36}{11} = 3\; + \;\frac{1}{{x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}}}\]Substituting the mixed number form:
\[3 + \frac{3}{11} = 3\; + \;\frac{1}{{x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}}}\]Comparing both sides, we can see that:
\[\frac{3}{11} = \frac{1}{{x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}}}\]Step 3: Invert both sides of the equation.
To isolate the expression involving x, y, and z, we take the reciprocal of both sides:
\[\frac{11}{3} = x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}\]Step 4: Convert the new improper fraction \(\frac{11}{3}\) to a mixed number.
Divide 11 by 3:
\[11 \div 3 = 3 \text{ with a remainder of } 2\]So, \(\frac{11}{3}\) can be written as:
\[\frac{11}{3} = 3 + \frac{2}{3}\]Step 5: Compare this mixed number with the equation from Step 3.
\[3 + \frac{2}{3} = x\; + \;\frac{1}{{y\; + \;\frac{1}{z}}}\]By comparing the integer parts and the fractional parts, we find the value of x:
\[x = 3\]And the remaining fractional part is:
\[\frac{2}{3} = \frac{1}{{y\; + \;\frac{1}{z}}}\]Step 6: Invert both sides of the remaining equation.
Taking the reciprocal of both sides gives:
\[\frac{3}{2} = y\; + \;\frac{1}{z}\]Step 7: Convert the improper fraction \(\frac{3}{2}\) to a mixed number.
Divide 3 by 2:
\[3 \div 2 = 1 \text{ with a remainder of } 1\]So, \(\frac{3}{2}\) can be written as:
\[\frac{3}{2} = 1 + \frac{1}{2}\]Step 8: Compare this mixed number with the equation from Step 6.
\[1 + \frac{1}{2} = y\; + \;\frac{1}{z}\]By comparing the integer parts and the fractional parts, we find the value of y:
\[y = 1\]And the remaining fractional part is:
\[\frac{1}{2} = \frac{1}{z}\]Step 9: Solve for z from the final equation.
From \(\frac{1}{2} = \frac{1}{z}\), we can easily see that \(z = 2\).
Step 10: Verify that x, y, and z are natural numbers.
- \(x = 3\) (Natural number)
- \(y = 1\) (Natural number)
- \(z = 2\) (Natural number)
All the values satisfy the condition that x, y, and z are natural numbers.
Calculate (x + y + z)
Now that we have found the values of x, y, and z, we can calculate their sum:
\[x + y + z = 3 + 1 + 2 = 6\]Therefore, the value of \((x + y + z)\) is 6.
| Variable | Value |
| x | 3 |
| y | 1 |
| z | 2 |
Revision Table: Key Steps in Solving Continued Fractions
| Step No. | Action | Purpose |
| 1 | Convert the initial improper fraction to a mixed number. | Match the first integer part of the continued fraction. |
| 2 | Equate the fractional part to the remaining continued fraction. | Isolate the complex fraction part. |
| 3 | Invert both sides of the equation. | Bring the next part of the continued fraction structure to the numerator. |
| 4 | Convert the new improper fraction (if any) to a mixed number. | Identify the next integer part of the continued fraction. |
| 5 | Repeat steps 2-4 until all variables (x, y, z) are found. | Deconstruct the continued fraction layer by layer. |
| 6 | Verify the values meet any specified conditions (e.g., natural numbers). | Ensure the solution is valid. |
| 7 | Perform the final calculation (e.g., x + y + z). | Answer the specific question asked. |
Additional Information about 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, then writing this other number as a sum of its integer part and the reciprocal of another number, and so on.
They are particularly useful for representing real numbers, especially irrational numbers, which have infinite continued fraction representations. Rational numbers, like \(\frac{36}{11}\) in this problem, have finite continued fraction representations.
The general form of a simple 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. In our problem, \(a_0 = 3\), \(a_1 = x\), \(a_2 = y\), and \(a_3 = z\).
The method used to solve this problem, which involves repeatedly taking the reciprocal of the fractional part, is essentially the Euclidean algorithm applied to find the greatest common divisor (GCD) of the numerator and denominator, but expressed in terms of quotients.
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Match the following.
Column I
Column II
a.
Equivalent fraction of \(\frac{7}{12}\) is
i.
Proper fraction
b.
Equivalent fraction of \(\frac{9}{15}\) is
ii.
Improper fraction
c.
\(\frac{7}{11}\) is
iii.
\(\frac{21}{36}\)
d.
\(\frac{19}{5}\) is
iv.
\(\frac{3}{5}\)
