what are all the ratios that are equivalent to 12 : 3

To find ratios equivalent to \( 12:3 \), we can simplify or multiply both terms of the ratio by the same number. 

### Simplifying the ratio \( 12:3 \):

1. Divide both numbers by their greatest common divisor (GCD), which is 3:
   \[
   \frac{12}{3} = 4:1
   \]
   So, \( 12:3 \) is equivalent to \( 4:1 \).

### Multiplying by the same factor:

If we multiply both terms of \( 12:3 \) by any positive integer, we will get equivalent ratios. For example:

1. Multiply both by 2:
   \[
   12 \times 2 : 3 \times 2 = 24:6
   \]

2. Multiply both by 3:
   \[
   12 \times 3 : 3 \times 3 = 36:9
   \]

3. Multiply both by 4:
   \[
   12 \times 4 : 3 \times 4 = 48:12
   \]

4. Multiply both by 5:
   \[
   12 \times 5 : 3 \times 5 = 60:15
   \]

Therefore, the ratios equivalent to \( 12:3 \) include \( 4:1 \), \( 24:6 \), \( 36:9 \), \( 48:12 \), \( 60:15 \), and many more by scaling with other integers.

Would you like more examples or details on this?

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Here are five related questions to deepen your understanding:

1. What is the process for finding the simplest form of any ratio?
2. Can a ratio be represented as a fraction, and if so, how would \( 12:3 \) appear?
3. How can you determine if two ratios are equivalent?
4. What is the significance of the greatest common divisor in simplifying ratios?
5. How does the concept of proportions relate to equivalent ratios?

**Tip:** Always check if both terms in a ratio can be divided by a common factor to simplify it effectively.

Discover how to find all the ratios equivalent to 12:3 by simplifying and multiplying. This guide explains the process of finding equivalent ratios and includes step-by-step examples, suitable for students in Grades 5-7.

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