Select all ratios equivalent to 14:12.

Understand the Problem

The question is asking to identify which of the provided ratios are equivalent to the ratio 14:12. This involves simplifying the ratio and comparing it with the options given.

Answer

The ratios equivalent to $14:12$ are $35:30$ and $112:96$.

Steps to Solve

Simplify the Ratio 14:12

First, we simplify the ratio $14:12$. To do this, we find the greatest common divisor (GCD) of 14 and 12, which is 2.

We divide both parts of the ratio by their GCD: $$ \frac{14 \div 2}{12 \div 2} = \frac{7}{6} $$

Thus, the simplified ratio is $7:6$.
Check Each Option Against 7:6

Now, we will compare the simplified ratio $7:6$ with each of the given options to see if they are equivalent.
- For 2:1:
  - Simplify: $2:1$ is not equal to $7:6$.
- For 35:30:
  - Simplify: Divide both by 5: $$ \frac{35 \div 5}{30 \div 5} = \frac{7}{6} $$
  - This is equal to $7:6$.
- For 112:96:
  - Simplify: Divide both by 16: $$ \frac{112 \div 16}{96 \div 16} = \frac{7}{6} $$
  - This is equal to $7:6$.
- For 45:18:
  - Simplify: Divide both by 9: $$ \frac{45 \div 9}{18 \div 9} = \frac{5}{2} $$
  - This is not equal to $7:6$.
List All Equivalent Ratios

The ratios that are equivalent to $14:12$ (or $7:6$) are:
- $35:30$
- $112:96$

The equivalent ratios to $14:12$ are $35:30$ and $112:96$.

More Information

When ratios are simplified, they can reveal equivalent relationships that are not immediately obvious. Understanding how to simplify ratios helps in identifying equivalencies in various mathematical contexts, such as proportions in recipes or scales in maps.

Tips

Not simplifying correctly: Ensure you always divide by the GCD.
Misidentifying ratios: Double-check each option's simplification to ensure accurate comparison.

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