Compare these three ratios using fractions. 3 to 2 5:6 8 to 12 Think about the sieps you could take to compares ith ratios using fractions. The first step is to write the ratios as fractions. The next step is to use a $$ \square $$ rewrite the fractions. Finatly, compare the $$ \square $$ common denominator order the ratios. common numerator equivalent satio

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Sure, let's compare the three ratios $$ 3:2 $$, $$ 5:6 $$, and $$ 8:12 $$ by following a step-by-step process using fractions.

### **Step 1: Write the Ratios as Fractions**

First, convert each ratio into a fraction by placing the first number over the second number:

- $$ 3:2 = \frac{3}{2} $$
- $$ 5:6 = \frac{5}{6} $$
- $$ 8:12 = \frac{8}{12} $$

### **Step 2: Simplify the Fractions (If Possible)**

Before comparing, simplify the fractions to their lowest terms:

- $$ \frac{3}{2} $$ is already in its simplest form.
- $$ \frac{5}{6} $$ is already in its simplest form.
- $$ \frac{8}{12} $$ can be simplified by dividing both the numerator and the denominator by 4:
  $$
  \frac{8 \div 4}{12 \div 4} = \frac{2}{3}
  $$

So, the simplified fractions are:

- $$ \frac{3}{2} $$
- $$ \frac{5}{6} $$
- $$ \frac{2}{3} $$

### **Step 3: Find a Common Denominator**

To compare these fractions easily, convert them to have the same denominator. The least common denominator (LCD) for 2, 6, and 3 is **6**.

- Convert $$ \frac{3}{2} $$ to a denominator of 6:
  $$
  \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}
  $$
  
- $$ \frac{5}{6} $$ already has the denominator 6.

- Convert $$ \frac{2}{3} $$ to a denominator of 6:
  $$
  \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
  $$

Now, all fractions have the same denominator:

- $$ \frac{9}{6} $$
- $$ \frac{5}{6} $$
- $$ \frac{4}{6} $$

### **Step 4: Compare the Fractions**

With a common denominator, you can compare the numerators directly:

- $$ \frac{9}{6} $$ (which is 1.5)
- $$ \frac{5}{6} $$ (which is approximately 0.833)
- $$ \frac{4}{6} $$ (which is approximately 0.667)

Ordering them from largest to smallest:

$$
\frac{9}{6} > \frac{5}{6} > \frac{4}{6}
$$

### **Final Order of Ratios**

Translating back to the original ratios:

$$
3:2 > 5:6 > 8:12
$$

### **Summary of Steps**

1. **Write the ratios as fractions:** $$ 3:2 = \frac{3}{2} $$, $$ 5:6 = \frac{5}{6} $$, $$ 8:12 = \frac{8}{12} $$.
2. **Simplify the fractions:** $$ \frac{8}{12} = \frac{2}{3} $$.
3. **Use a common denominator (6) to rewrite the fractions:** $$ \frac{9}{6} $$, $$ \frac{5}{6} $$, $$ \frac{4}{6} $$.
4. **Compare the fractions based on the common denominator and order the ratios:** $$ 3:2 > 5:6 > 8:12 $$.

By following these steps, you can effectively compare any set of ratios using fractions.
To compare the ratios 3:2, 5:6, and 8:12 using fractions, follow these steps: 1. **Convert Ratios to Fractions:** - 3:2 = 3/2 - 5:6 = 5/6 - 8:12 = 2/3 2. **Simplify Fractions:** - 3/2 remains the same. - 5/6 remains the same. - 2/3 remains the same. 3. **Find a Common Denominator (6):** - 3/2 = 9/6 - 5/6 remains 5/6 - 2/3 = 4/6 4. **Compare Numerators:** - 9/6 > 5/6 > 4/6 5. **Final Order:** - 3:2 > 5:6 > 8:12

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