`16x=12(x+1/6)`
Jay V left his house at 2:00 P.M. and rode his bicycle down his street at a speed of 12 mph (miles per hour). When his friend Tamika arrived at his house at 2:10 P.M. , Jay V's mother sent her off in Jay V's direction down the same street , and Tamika cycled after him at 16 mph. At what time did Tamika catch up with Jay V ?
The equation above is used to solve this problem . What is the term `12(x+1/6)` equal to ?

Options

A. The time , in hours , Tamika took to catch up with Jay V
B. The time , in hours , Jay V cycled before Tamika caught up with him
C. The distance, in miles , traveled by Tamika
D. The average speed, in miles per hour , of Tamika and Jay V

Answer

To solve the equation \( 16x = 12(x + \frac{1}{6}) \) and determine what the term \( 12(x + \frac{1}{6}) \) represents, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation**:
The equation \( 16x = 12(x + \frac{1}{6}) \) relates to the distances traveled by Jay V and Tamika. Here, \( x \) represents the time in hours that Tamika has been cycling after she started.

2. **Expand the Right Side**:
We need to expand the term \( 12(x + \frac{1}{6}) \):
\[ 12(x + \frac{1}{6}) = 12x + 12 \cdot \frac{1}{6} \]
Simplifying the multiplication:
\[ 12 \cdot \frac{1}{6} = 2 \]
Therefore, we can rewrite the equation as:
\[ 16x = 12x + 2 \]

3. **Rearranging the Equation**:
To isolate \( x \), we can subtract \( 12x \) from both sides:
\[ 16x - 12x = 2 \]
This simplifies to:
\[ 4x = 2 \]

4. **Solve for \( x \)**:
Dividing both sides by 4 gives us:
\[ x = \frac{2}{4} = \frac{1}{2} \]
This means Tamika took \( \frac{1}{2} \) hours (or 30 minutes) to catch up with Jay V.

5. **Interpret the Term \( 12(x + \frac{1}{6}) \)**:
The term \( 12(x + \frac{1}{6}) \) represents the distance that Jay V traveled during the time \( x + \frac{1}{6} \). Since \( x \) is the time Tamika took to catch up, \( \frac{1}{6} \) hours is the additional time Jay V had cycled before Tamika started (10 minutes).

### Conclusion:
Thus, the term \( 12(x + \frac{1}{6}) \) is equal to the distance traveled by Jay V before Tamika catches up with him.