Express each of the following ratios in simplest form :
`2 (1)/(3) : 3 (1)/( 4) : 1 (1)/(6)`

Answer

To express the ratios \(2 \frac{1}{3} : 3 \frac{1}{4} : 1 \frac{1}{6}\) in simplest form, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert each mixed number into an improper fraction.

- For \(2 \frac{1}{3}\):
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

- For \(3 \frac{1}{4}\):
\[ 3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \]

- For \(1 \frac{1}{6}\):
\[ 1 \frac{1}{6} = \frac{1 \times 6 + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6} \]

Now we have the ratios as:
\[ \frac{7}{3} : \frac{13}{4} : \frac{7}{6} \]

### Step 2: Find the LCM of Denominators
Next, we need to find the least common multiple (LCM) of the denominators \(3\), \(4\), and \(6\).

- The prime factorization of the numbers is:
- \(3 = 3^1\)
- \(4 = 2^2\)
- \(6 = 2^1 \times 3^1\)

The LCM is found by taking the highest power of each prime:
- From \(2\): \(2^2\)
- From \(3\): \(3^1\)

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have the denominator of \(12\).

- For \(\frac{7}{3}\):
\[ \frac{7}{3} \times \frac{4}{4} = \frac{28}{12} \]

- For \(\frac{13}{4}\):
\[ \frac{13}{4} \times \frac{3}{3} = \frac{39}{12} \]

- For \(\frac{7}{6}\):
\[ \frac{7}{6} \times \frac{2}{2} = \frac{14}{12} \]

Now we have the ratios as:
\[ \frac{28}{12} : \frac{39}{12} : \frac{14}{12} \]

### Step 4: Write the Ratios in Simplest Form
Since all fractions have the same denominator, we can express the ratios as:
\[ 28 : 39 : 14 \]

### Final Answer
Thus, the simplest form of the ratios \(2 \frac{1}{3} : 3 \frac{1}{4} : 1 \frac{1}{6}\) is:
\[ 28 : 39 : 14 \]

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