The additive and multiplicative inverse of `1/4 + 1/3` is

Options

A. `((11)/(30) , (30)/(11))`
B. `(( -7)/(12) , (12)/(7))`
C. `((11)/(20) , (-30)/(11))`
D. `((11)/(30), (-30)/(11))`

Answer

To find the additive and multiplicative inverse of \( \frac{1}{4} + \frac{1}{3} \), we will follow these steps:

### Step 1: Calculate \( \frac{1}{4} + \frac{1}{3} \)

To add these two fractions, we need to find a common denominator. The least common multiple (LCM) of 4 and 3 is 12.

\[ \frac{1}{4} = \frac{3}{12} \quad \text{(by multiplying numerator and denominator by 3)} \]
\[ \frac{1}{3} = \frac{4}{12} \quad \text{(by multiplying numerator and denominator by 4)} \]

Now we can add the two fractions:

\[ \frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{3 + 4}{12} = \frac{7}{12} \]

### Step 2: Find the Additive Inverse

The additive inverse of a number \( x \) is the value that, when added to \( x \), results in zero.

For \( \frac{7}{12} \), the additive inverse is:

\[ -\frac{7}{12} \]

### Step 3: Find the Multiplicative Inverse

The multiplicative inverse of a number \( x \) is the value that, when multiplied by \( x \), results in one.

For \( \frac{7}{12} \), the multiplicative inverse is:

\[ \frac{12}{7} \]

### Final Answer

Thus, the additive inverse of \( \frac{1}{4} + \frac{1}{3} \) is \( -\frac{7}{12} \) and the multiplicative inverse is \( \frac{12}{7} \).

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