If \(\frac{1}{{x + \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}}} = \frac{{29}}{{79}}\) where x, y, and z are natural numbers, then the value of (2x + 3y - z) is:
2
Solving Continued Fraction Equations for Natural Numbers
The problem asks us to find the value of \(2x + 3y - z\) given a continued fraction equation involving natural numbers x, y, and z. The equation is:
\[ \frac{1}{{x + \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}}} = \frac{{29}}{{79}} \]We are given that x, y, and z are natural numbers. Natural numbers are positive integers (1, 2, 3, ...).
Step 1: Inverting the Equation
To make the equation easier to work with, we can take the reciprocal of both sides:
\[ x + \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}} = \frac{{79}}{{29}} \]Step 2: Simplifying the Right Side
We need to express the fraction \(\frac{79}{29}\) as a mixed number to match the structure of the left side. We divide 79 by 29:
\[ 79 \div 29 \]29 goes into 79 two times (\(29 \times 2 = 58\)) with a remainder of \(79 - 58 = 21\). So,
\[ \frac{{79}}{{29}} = 2 + \frac{{21}}{{29}} \]Now the equation is:
\[ x + \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}} = 2 + \frac{{21}}{{29}} \]Since x is a natural number and the structure matches, we can equate the integer parts and the fractional parts:
- \(x = 2\)
- \( \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}} = \frac{{21}}{{29}} \)
We have found the value of x.
Step 3: Solving for y and the Remaining Fraction
Now we work with the remaining equation:
\[ \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}} = \frac{{21}}{{29}} \]Again, take the reciprocal of both sides:
\[ y + \frac{2}{{z + \frac{1}{4}}}} = \frac{{29}}{{21}} \]Express \(\frac{29}{21}\) as a mixed number. Divide 29 by 21:
\[ 29 \div 21 \]21 goes into 29 one time (\(21 \times 1 = 21\)) with a remainder of \(29 - 21 = 8\). So,
\[ \frac{{29}}{{21}} = 1 + \frac{{8}}{{21}} \]Now the equation is:
\[ y + \frac{2}{{z + \frac{1}{4}}}} = 1 + \frac{{8}}{{21}} \]Since y is a natural number, we equate the integer parts and the fractional parts:
- \(y = 1\)
- \( \frac{2}{{z + \frac{1}{4}}}} = \frac{{8}}{{21}} \)
We have found the value of y.
Step 4: Solving for z
Now we solve the remaining equation for z:
\[ \frac{2}{{z + \frac{1}{4}}}} = \frac{{8}}{{21}} \]We can cross-multiply or first simplify the right side fraction \(\frac{8}{21}\). Let's cross-multiply:
\[ 2 \times 21 = 8 \times \left(z + \frac{1}{4}\right) \] \[ 42 = 8z + 8 \times \frac{1}{4} \] \[ 42 = 8z + 2 \]Now, subtract 2 from both sides to isolate the term with z:
\[ 42 - 2 = 8z \] \[ 40 = 8z \]Divide by 8 to find z:
\[ z = \frac{{40}}{{8}} \] \[ z = 5 \]We have found the value of z.
Summary of Found Values
We found the following values for x, y, and z:
- \(x = 2\)
- \(y = 1\)
- \(z = 5\)
These values are all natural numbers, as required by the problem statement.
Step 5: Calculating the Final Expression
The question asks for the value of the expression \(2x + 3y - z\). Substitute the values we found:
\[ 2x + 3y - z = 2(2) + 3(1) - 5 \] \[ = 4 + 3 - 5 \] \[ = 7 - 5 \] \[ = 2 \]Final Answer Calculation
The value of the expression \(2x + 3y - z\) is 2.
| Variable | Value | Calculation Part |
|---|---|---|
| x | 2 | \(2 \times x = 2 \times 2 = 4\) |
| y | 1 | \(3 \times y = 3 \times 1 = 3\) |
| z | 5 | \(-z = -5\) |
| Expression: \(2x + 3y - z\) | \(4 + 3 - 5 = 7 - 5 = 2\) | |
Revision Table: Key Steps in Solving Continued Fractions
| Step | Description | Action Taken in this Problem |
|---|---|---|
| 1 | If the fraction is \(1 / (...)\), invert both sides. | Inverted \( \frac{1}{{x + ...}} = \frac{29}{79} \) to get \( x + ... = \frac{79}{29} \). |
| 2 | Convert the resulting fraction to a mixed number. | Converted \( \frac{79}{29} \) to \( 2 + \frac{21}{29} \). |
| 3 | Equate the integer part to the first variable (if it's an integer). | Equated \( x \) to 2. |
| 4 | Take the fractional part and repeat the process (Step 1 onwards) for the remaining continued fraction expression. | Used \( \frac{1}{{y + ...}} = \frac{21}{29} \) and repeated steps. |
| 5 | Continue until all variables are determined. | Repeated for y and then solved directly for z. |
| 6 | Verify that the values meet the conditions (e.g., natural numbers). | Confirmed x=2, y=1, z=5 are natural numbers. |
| 7 | Calculate the final required expression. | Calculated \(2x + 3y - z\). |
Additional Information on Continued Fractions
A continued fraction is an expression obtained through an iterative process of representing a number as a sum of an integer and the reciprocal of another number, which is then represented as a sum of an integer and a reciprocal, and so on. They can be finite or infinite.
The general form of a simple continued fraction is:
\[ a_0 + \frac{1}{{a_1 + \frac{1}{{a_2 + \frac{1}{{a_3 + \ldots}}}}}} \]where \(a_0\) is typically an integer and \(a_1, a_2, a_3, \ldots\) are positive integers.
In this problem, the structure was slightly different because of the '2' in the numerator \( \frac{2}{{z + \frac{1}{4}}}} \). However, the core method of iteratively inverting and separating the integer part from the fractional part is fundamental to solving such problems, especially when the variables are restricted to be integers or natural numbers.
This method is closely related to the Euclidean algorithm for finding the greatest common divisor, as it involves repeated division with remainder.
Similar Questions
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Important Questions from Fractions
Simplify the expression 441 ÷ \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\)
Number 0.232323 can be written in rational form as:
Solve: \(\frac{1}{2}\) [{-2(2 + 3)*20}/2]
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.
\(\frac{3}{5}\)
Among the fractions \(\frac{7}{12}\) , \(\frac{8}{15}\) , \(\frac{17}{24}\) and \(\frac{13}{18}\) , which is the smallest?
