Let a, b, c be three rational numbers, where `a = (3)/(5), b = (2)/(3) " and " = (-5)/(6)`, which one of the following is true?

Options

A. `a + (b + c) = b + (a + c)`
B. `a + (b + c) = c + (a + b)`
C. `a - (b - c) = c - (a - b)`
D. `a xx (b +c) = b xx (a + c)`

Answer

To determine which of the statements involving the rational numbers \( a = \frac{3}{5} \), \( b = \frac{2}{3} \), and \( c = -\frac{5}{6} \) is true, we will calculate the sum \( a + b + c \) and check the properties of addition and multiplication.

### Step 1: Calculate \( a + b + c \)

We start with the values:
- \( a = \frac{3}{5} \)
- \( b = \frac{2}{3} \)
- \( c = -\frac{5}{6} \)

To add these fractions, we first need to find a common denominator. The denominators are 5, 3, and 6. The least common multiple (LCM) of these numbers is 30.

### Step 2: Convert each fraction to have a denominator of 30

- For \( a = \frac{3}{5} \):
\[ a = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \]

- For \( b = \frac{2}{3} \):
\[ b = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \]

- For \( c = -\frac{5}{6} \):
\[ c = -\frac{5 \times 5}{6 \times 5} = -\frac{25}{30} \]

### Step 3: Add the fractions

Now we can add them:
\[ a + b + c = \frac{18}{30} + \frac{20}{30} - \frac{25}{30} \]

Combine the numerators:
\[ = \frac{18 + 20 - 25}{30} = \frac{13}{30} \]

### Step 4: Check the commutative property

Next, we check if \( b + a + c \) gives the same result:
\[ b + a + c = \frac{20}{30} + \frac{18}{30} - \frac{25}{30} = \frac{20 + 18 - 25}{30} = \frac{13}{30} \]

### Step 5: Check other properties

1. **Commutative Property**:
- \( a + b + c = c + a + b \)
- This is true since addition is commutative.

2. **Associative Property**:
- \( (a + b) + c = a + (b + c) \)
- This is also true since addition is associative.

3. **Distributive Property**:
- \( a(b + c) = ab + ac \)
- We can calculate both sides to verify.

### Conclusion

From the calculations, we see that:
- \( a + b + c = \frac{13}{30} \)
- \( b + a + c = \frac{13}{30} \)
- The properties of addition hold true.

Thus, the true statements include the commutative and associative properties of addition, and the distributive property holds under multiplication.