7. $$ \frac{2}{3}, \frac{1}{4} $$ and $$ \frac{1}{6} $$ are three given rational numbers. Answer the following questions, a) List out the terminating and non-terminating recurring decimals of these rational numbers. b) What is the multiplicative identify of $$ \frac{2}{3} $$ ? c) Use the given rational numbers to verify the associative properties of addition and multiplication of rational numbers.

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### a) List out the terminating and non-terminating recurring decimals of the given rational numbers.

To determine whether a rational number has a terminating or non-terminating recurring decimal, we can convert each fraction to its decimal form.

1. **For $$ \frac{2}{3} $$**:
   - Dividing 2 by 3 gives $$ 0.6666... $$ (which is $$ 0.\overline{6} $$).
   - This is a non-terminating recurring decimal.

2. **For $$ \frac{1}{4} $$**:
   - Dividing 1 by 4 gives $$ 0.25 $$.
   - This is a terminating decimal.

3. **For $$ \frac{1}{6} $$**:
   - Dividing 1 by 6 gives $$ 0.1666... $$ (which is $$ 0.1\overline{6} $$).
   - This is a non-terminating recurring decimal.

**Summary**:
- **Terminating decimals**: $$ \frac{1}{4} = 0.25 $$
- **Non-terminating recurring decimals**: $$ \frac{2}{3} = 0.\overline{6} $$, $$ \frac{1}{6} = 0.1\overline{6} $$

### b) What is the multiplicative identity of $$ \frac{2}{3} $$?

The multiplicative identity is the number that, when multiplied by any number, leaves that number unchanged. For rational numbers, the multiplicative identity is $$ 1 $$.

Thus, the multiplicative identity of $$ \frac{2}{3} $$ is $$ 1 $$.

### c) Use the given rational numbers to verify the associative properties of addition and multiplication of rational numbers.

**Associative Property of Addition**:
The associative property states that for any three numbers $$ a, b, c $$:
$$
(a + b) + c = a + (b + c)
$$

Let $$ a = \frac{2}{3}, b = \frac{1}{4}, c = \frac{1}{6} $$.

1. Calculate $$ (a + b) + c $$:
   $$
   a + b = \frac{2}{3} + \frac{1}{4}
   $$
   To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.
   $$
   \frac{2}{3} = \frac{8}{12}, \quad \frac{1}{4} = \frac{3}{12}
   $$
   Thus,
   $$
   a + b = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}
   $$
   Now add $$ c $$:
   $$
   (a + b) + c = \frac{11}{12} + \frac{1}{6}
   $$
   The least common multiple of 12 and 6 is 12.
   $$
   \frac{1}{6} = \frac{2}{12}
   $$
   Therefore,
   $$
   (a + b) + c = \frac{11}{12} + \frac{2}{12} = \frac{13}{12}
   $$

2. Calculate $$ a + (b + c) $$:
   $$
   b + c = \frac{1}{4} + \frac{1}{6}
   $$
   The least common multiple of 4 and 6 is 12.
   $$
   \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}
   $$
   Thus,
   $$
   b + c = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}
   $$
   Now add $$ a $$:
   $$
   a + (b + c) = \frac{2}{3} + \frac{5}{12}
   $$
   The least common multiple of 3 and 12 is 12.
   $$
   \frac{2}{3} = \frac{8}{12}
   $$
   Therefore,
   $$
   a + (b + c) = \frac{8}{12} + \frac{5}{12} = \frac{13}{12}
   $$

Since $$ (a + b) + c = a + (b + c) = \frac{13}{12} $$, the associative property of addition holds.

**Associative Property of Multiplication**:
The associative property states that for any three numbers $$ a, b, c $$:
$$
(a \cdot b) \cdot c = a \cdot (b \cdot c)
$$

1. Calculate $$ (a \cdot b) \cdot c $$:
   $$
   a \cdot b = \frac{2}{3} \cdot \frac{1}{4} = \frac{2 \cdot 1}{3 \cdot 4} = \frac{2}{12} = \frac{1}{6}
   $$
   Now multiply by $$ c $$:
   $$
   (a \cdot b) \cdot c = \frac{1}{6} \cdot \frac{1}{6} = \frac{1 \cdot 1}{6 \cdot 6} = \frac{1}{36}
   $$

2. Calculate $$ a \cdot (b \cdot c) $$:
   $$
   b \cdot c = \frac{1}{4} \cdot \frac{1}{6} = \frac{1 \cdot 1}{4 \cdot 6} = \frac{1}{24}
   $$
   Now multiply by $$ a $$:
   $$
   a \cdot (b \cdot c) = \frac{2}{3} \cdot \frac{1}{24} = \frac{2 \cdot 1}{3 \cdot 24} = \frac{2}{72} = \frac{1}{36}
   $$

Since $$ (a \cdot b) \cdot c = a \cdot (b \cdot c) = \frac{1}{36} $$, the associative property of multiplication holds.

### Final Summary:
- **Terminating decimals**: $$ \frac{1}{4} = 0.25 $$
- **Non-terminating recurring decimals**: $$ \frac{2}{3} = 0.\overline{6} $$, $$ \frac{1}{6} = 0.1\overline{6} $$
- **Multiplicative identity of $$ \frac{2}{3} $$**: $$ 1 $$
- **Associative
- **Terminating decimals**: $$ \frac{1}{4} = 0.25 $$ - **Non-terminating recurring decimals**: $$ \frac{2}{3} = 0.\overline{6} $$, $$ \frac{1}{6} = 0.1\overline{6} $$ - **Multiplicative identity of $$ \frac{2}{3} $$**: $$ 1 $$ - **Associative properties verified**: Addition and multiplication of the given rational numbers satisfy the associative property.

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