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

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert each mixed number into an improper fraction.

1. **For \(1 \frac{2}{3}\)**:
\[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \]

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

3. **For \(1 \frac{3}{4}\)**:
\[ 1 \frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{7}{4} \]

So, we have:
\[ \frac{5}{3} : \frac{5}{2} : \frac{7}{4} \]

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
Identify the denominators: 3, 2, and 4. We will find the LCM.

- The prime factorization of the denominators:
- \(3 = 3^1\)
- \(2 = 2^1\)
- \(4 = 2^2\)

The LCM is obtained by taking the highest power of each prime:
\[ \text{LCM} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

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

1. **Convert \(\frac{5}{3}\)**:
\[ \frac{5}{3} \times \frac{4}{4} = \frac{20}{12} \]

2. **Convert \(\frac{5}{2}\)**:
\[ \frac{5}{2} \times \frac{6}{6} = \frac{30}{12} \]

3. **Convert \(\frac{7}{4}\)**:
\[ \frac{7}{4} \times \frac{3}{3} = \frac{21}{12} \]

### Step 4: Write the Ratios
Now, we can express the ratios:
\[ 20 : 30 : 21 \]

### Step 5: Simplify the Ratios
To simplify the ratio \(20 : 30 : 21\), we need to find the greatest common divisor (GCD) of the numbers.

- The GCD of 20, 30, and 21 is 1 (since they have no common factors other than 1).

Thus, the simplest form of the ratio is:
\[ 20 : 30 : 21 \]

### Final Answer
The simplest form of the ratio \(1 \frac{2}{3} : 2 \frac{1}{2} : 1 \frac{3}{4}\) is:
\[ 20 : 30 : 21 \]

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