Find the value of k in `(26)/(21) : (24)/(9) :: k : (14)/(13)`

Options

A. a.`(1)/(3)`
B. b.`2`
C. `c.(1)/(2)`
D. d.`3`

Answer

To find the value of \( k \) in the proportion \( \frac{26}{21} : \frac{24}{9} :: k : \frac{14}{13} \), we can set up the equation based on the property of proportions.

### Step-by-Step Solution:

1. **Set Up the Proportion**:
We can express the proportion as:
\[ \frac{26}{21} : \frac{24}{9} = k : \frac{14}{13} \]
This can be rewritten as:
\[ \frac{26}{21} \cdot \frac{14}{13} = \frac{24}{9} \cdot k \]

2. **Cross Multiply**:
Cross multiplying gives us:
\[ 26 \cdot 14 = 21 \cdot 24 \cdot k \]

3. **Calculate the Left Side**:
Calculate \( 26 \cdot 14 \):
\[ 26 \cdot 14 = 364 \]

4. **Calculate the Right Side**:
Calculate \( 21 \cdot 24 \):
\[ 21 \cdot 24 = 504 \]
So we have:
\[ 364 = 504 \cdot k \]

5. **Solve for \( k \)**:
To find \( k \), divide both sides by 504:
\[ k = \frac{364}{504} \]

6. **Simplify \( k \)**:
We can simplify \( \frac{364}{504} \) by finding the greatest common divisor (GCD). The GCD of 364 and 504 is 252. Thus:
\[ k = \frac{364 \div 252}{504 \div 252} = \frac{1.44}{2} = \frac{1}{2} \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{2}} \]