Fill in the box with `gt, lt or =` signs,
`1:12 square 2:9`

Answer

To solve the problem of comparing the ratios \(1:12\) and \(2:9\), we will follow these steps:

### Step 1: Write the Ratios as Fractions
We can express the ratios as fractions:
- \(1:12\) can be written as \(\frac{1}{12}\)
- \(2:9\) can be written as \(\frac{2}{9}\)

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
The denominators are \(12\) and \(9\). We need to find the LCM of these two numbers.
- The prime factorization of \(12\) is \(2^2 \times 3^1\)
- The prime factorization of \(9\) is \(3^2\)

To find the LCM, we take the highest power of each prime:
- For \(2\), the highest power is \(2^2\)
- For \(3\), the highest power is \(3^2\)

Thus, the LCM is:
\[ LCM = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 3: Convert Both Fractions to Have the Same Denominator
Now we will convert both fractions to have a denominator of \(36\).

For \(\frac{1}{12}\):
\[ \frac{1}{12} = \frac{1 \times 3}{12 \times 3} = \frac{3}{36} \]

For \(\frac{2}{9}\):
\[ \frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36} \]

### Step 4: Compare the Two Fractions
Now we compare \(\frac{3}{36}\) and \(\frac{8}{36}\):
Since \(8 > 3\), we have:
\[ \frac{2}{9} > \frac{1}{12} \]

### Step 5: Fill in the Box
Since \(1:12\) is less than \(2:9\), we can fill in the box with:
\[ 1:12 \, \text{lt} \, 2:9 \]

### Final Answer
The answer is:
\[ 1:12 \, < \, 2:9 \]

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