Simplify the ratio:
`5/6 : 3/.8 : 3 (3)/(4)`

Answer

To simplify the ratio \( \frac{5}{6} : \frac{3}{0.8} : 3 \frac{3}{4} \), we will follow these steps:

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we need to convert the mixed fraction \( 3 \frac{3}{4} \) into an improper fraction.

\[ 3 \frac{3}{4} = 3 \times 4 + 3 = 12 + 3 = \frac{15}{4} \]

So, the ratio now looks like this:
\[ \frac{5}{6} : \frac{3}{0.8} : \frac{15}{4} \]

### Step 2: Convert Decimal to Fraction
Next, we convert the decimal \( 0.8 \) into a fraction.

\[ 0.8 = \frac{8}{10} = \frac{4}{5} \]

Now, we can rewrite the ratio:
\[ \frac{5}{6} : \frac{3}{\frac{4}{5}} : \frac{15}{4} \]

### Step 3: Simplify the Fraction
To simplify \( \frac{3}{\frac{4}{5}} \), we multiply by the reciprocal:

\[ \frac{3}{\frac{4}{5}} = 3 \times \frac{5}{4} = \frac{15}{4} \]

Now the ratio is:
\[ \frac{5}{6} : \frac{15}{4} : \frac{15}{4} \]

### Step 4: Find the LCM of Denominators
Next, we need to find the least common multiple (LCM) of the denominators \( 6 \) and \( 4 \).

The LCM of \( 6 \) and \( 4 \) is \( 12 \).

### Step 5: Convert Each Fraction to Have the Same Denominator
Now we will convert each term to have a denominator of \( 12 \):

1. For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]

2. For \( \frac{15}{4} \):
\[ \frac{15}{4} = \frac{15 \times 3}{4 \times 3} = \frac{45}{12} \]

3. For the second \( \frac{15}{4} \):
\[ \frac{15}{4} = \frac{45}{12} \]

### Step 6: Write the Ratio
Now we can write the ratio:
\[ \frac{10}{12} : \frac{45}{12} : \frac{45}{12} \]

### Step 7: Remove the Common Denominator
Since all terms have the same denominator, we can ignore it and simplify the ratio:
\[ 10 : 45 : 45 \]

### Step 8: Simplify Further
Now we can simplify this ratio by dividing each term by \( 5 \):
\[ \frac{10}{5} : \frac{45}{5} : \frac{45}{5} = 2 : 9 : 9 \]

### Final Answer
Thus, the simplified ratio is:
\[ 2 : 9 : 9 \]

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