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

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Convert \(1 \frac{3}{4}\)**:
\[ 1 \frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4} \]

2. **Convert \(2 \frac{1}{2}\)**:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]

3. **Convert \(1 \frac{1}{6}\)**:
\[ 1 \frac{1}{6} = \frac{6 \times 1 + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6} \]

Now we have the ratios as:
\[ \frac{7}{4} : \frac{5}{2} : \frac{7}{6} \]

### Step 2: Eliminate the Fractions
To eliminate the fractions, we can find a common denominator. The least common multiple (LCM) of the denominators \(4\), \(2\), and \(6\) is \(12\).

Now we will multiply each term by \(12\):
1. \(\frac{7}{4} \times 12 = 7 \times 3 = 21\)
2. \(\frac{5}{2} \times 12 = 5 \times 6 = 30\)
3. \(\frac{7}{6} \times 12 = 7 \times 2 = 14\)

Thus, the ratios become:
\[ 21 : 30 : 14 \]

### Step 3: Simplify the Ratios
Now, we will simplify the ratios \(21 : 30 : 14\) by finding the greatest common divisor (GCD) of the numbers.

1. The GCD of \(21\), \(30\), and \(14\) is \(1\) (since there are no common factors other than \(1\)).
2. Therefore, the simplest form remains \(21 : 30 : 14\).

### Final Answer
The simplest form of the ratio \(1 \frac{3}{4} : 2 \frac{1}{2} : 1 \frac{1}{6}\) is:
\[ 21 : 30 : 14 \]

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