`{:("Quantity A","Quantity B"),((1)/(3)+(1)/(4)+(7)/(12),(1)/((1)/(3)+(1)/(4)+(7)/(12))):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to compare Quantity A and Quantity B.

**Step 1: Calculate Quantity A**

Quantity A is given by:

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

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

Now, we convert each fraction to have a denominator of 12:

\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]
\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]
\[ \frac{7}{12} = \frac{7}{12} \quad \text{(already has the common denominator)} \]

Now, we can add them:

\[ \text{Quantity A} = \frac{4}{12} + \frac{3}{12} + \frac{7}{12} = \frac{4 + 3 + 7}{12} = \frac{14}{12} = \frac{7}{6} \]

**Step 2: Calculate Quantity B**

Quantity B is given by:

\[ \text{Quantity B} = \frac{1}{\left(\frac{1}{3} + \frac{1}{4} + \frac{7}{12}\right)} \]

Since we already calculated the value of Quantity A as \(\frac{7}{6}\), we can substitute this value into Quantity B:

\[ \text{Quantity B} = \frac{1}{\frac{7}{6}} = \frac{6}{7} \]

**Step 3: Compare Quantity A and Quantity B**

Now we have:

- Quantity A = \(\frac{7}{6}\)
- Quantity B = \(\frac{6}{7}\)

To compare these two quantities, we can convert them to decimal form or find a common denominator.

Calculating the decimal values:

\[ \frac{7}{6} \approx 1.1667 \]
\[ \frac{6}{7} \approx 0.8571 \]

Since \(1.1667 > 0.8571\), we conclude that:

\[ \text{Quantity A} > \text{Quantity B} \]

Thus, the answer is that Quantity A is greater than Quantity B.