Compare the ratios:
`12:35 square 14:39`

Answer

To compare the ratios \( 12:35 \) and \( 14:39 \), we can follow these steps:

### Step 1: Write the Ratios as Fractions
We can express the ratios as fractions:
\[ \frac{12}{35} \quad \text{and} \quad \frac{14}{39} \]

### Step 2: Find a Common Denominator
To compare the two fractions, we can find a common denominator. The denominators are 35 and 39. The least common multiple (LCM) of 35 and 39 can be calculated as follows:
- The prime factorization of 35 is \( 5 \times 7 \).
- The prime factorization of 39 is \( 3 \times 13 \).
- Therefore, the LCM of 35 and 39 is \( 5 \times 7 \times 3 \times 13 = 1365 \).

### Step 3: Convert Both Fractions to Have the Same Denominator
Now we convert both fractions to have the common denominator of 1365:
- For \( \frac{12}{35} \):
\[ \frac{12 \times 39}{35 \times 39} = \frac{468}{1365} \]
- For \( \frac{14}{39} \):
\[ \frac{14 \times 35}{39 \times 35} = \frac{490}{1365} \]

### Step 4: Compare the Numerators
Now that both fractions have the same denominator, we can compare the numerators:
- \( 468 \) (from \( \frac{12}{35} \)) and \( 490 \) (from \( \frac{14}{39} \)).

Since \( 468 < 490 \), we conclude that:
\[ \frac{12}{35} < \frac{14}{39} \]

### Step 5: State the Conclusion
Thus, we can say that the ratio \( 12:35 \) is less than the ratio \( 14:39 \).

### Final Answer
\[ 12:35 < 14:39 \]

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