Reduce the ratio in simplest form
`(12)/(17) : (9)/(34) : 3 (9)/(17)`

Answer

To reduce the ratio \( \frac{12}{17} : \frac{9}{34} : 3 \frac{9}{17} \) into its simplest form, we can follow these steps:

### Step 1: Convert Mixed Fraction to Improper Fraction
The third term in the ratio is a mixed fraction \( 3 \frac{9}{17} \). We need to convert it into an improper fraction.

To do this:
- Multiply the whole number (3) by the denominator (17): \( 3 \times 17 = 51 \)
- Add the numerator (9): \( 51 + 9 = 60 \)
- Therefore, \( 3 \frac{9}{17} = \frac{60}{17} \)

So, the ratio becomes:
\[ \frac{12}{17} : \frac{9}{34} : \frac{60}{17} \]

### Step 2: Make Denominators the Same
Next, we want to express all terms with a common denominator. The common denominator for \( 17 \) and \( 34 \) is \( 34 \).

We can rewrite the first and third fractions:
- For \( \frac{12}{17} \), multiply the numerator and denominator by 2:
\[ \frac{12 \times 2}{17 \times 2} = \frac{24}{34} \]
- The second term remains \( \frac{9}{34} \).
- For \( \frac{60}{17} \), multiply the numerator and denominator by 2:
\[ \frac{60 \times 2}{17 \times 2} = \frac{120}{34} \]

Now, the ratio is:
\[ \frac{24}{34} : \frac{9}{34} : \frac{120}{34} \]

### Step 3: Remove the Common Denominator
Since all terms have the same denominator (34), we can ignore it and focus on the numerators:
\[ 24 : 9 : 120 \]

### Step 4: Simplify the Ratio
Now we need to simplify the ratio \( 24 : 9 : 120 \). We can find the greatest common divisor (GCD) of the numbers.

- The GCD of \( 24, 9, \) and \( 120 \) is \( 3 \).

Now, divide each term by \( 3 \):
\[ \frac{24}{3} : \frac{9}{3} : \frac{120}{3} = 8 : 3 : 40 \]

### Final Answer
The simplest form of the ratio \( \frac{12}{17} : \frac{9}{34} : 3 \frac{9}{17} \) is:
\[ \boxed{8 : 3 : 40} \]