Which of the following expressions is equivalent to 216 1/3/27 1/3

To simplify the expression \( 216 \frac{1}{3} / 27 \frac{1}{3} \), we first convert both mixed numbers into improper fractions.

1. **Convert \( 216 \frac{1}{3} \)**:
- \( 216 \times 3 + 1 = 648 + 1 = 649 \)
- So, \( 216 \frac{1}{3} = \frac{649}{3} \).

2. **Convert \( 27 \frac{1}{3} \)**:
- \( 27 \times 3 + 1 = 81 + 1 = 82 \)
- So, \( 27 \frac{1}{3} = \frac{82}{3} \).

Now substituting these values back into the expression:

\[
216 \frac{1}{3} / 27 \frac{1}{3} = \frac{649}{3} / \frac{82}{3}
\]

To divide by a fraction, we multiply by its reciprocal:

\[
\frac{649}{3} \times \frac{3}{82} = \frac{649 \times 3}{3 \times 82}
\]

The \( 3 \) in the numerator and denominator cancel out:

\[
= \frac{649}{82}
\]

Now, to check if this fraction can be simplified further: 649 and 82 have no common factors (since 82 is \( 2 \times 41 \), and \( 649 \) is not divisible by either \( 2 \) or \( 41 \)), so this is in its simplest form.

Thus, the expression \( 216 \frac{1}{3} / 27 \frac{1}{3} \) is equivalent to:

\[
\frac{649}{82}
\]

So the final answer is \( \frac{649}{82} \).

wrong try again please

Let's carefully simplify the expression \( 216 \frac{1}{3} / 27 \frac{1}{3} \) step by step.

1. **Convert mixed numbers to improper fractions**:

a. For \( 216 \frac{1}{3} \):
\[
216 \frac{1}{3} = 216 + \frac{1}{3} = \frac{216 \times 3 + 1}{3} = \frac{648 + 1}{3} = \frac{649}{3}
\]

b. For \( 27 \frac{1}{3} \):
\[
27 \frac{1}{3} = 27 + \frac{1}{3} = \frac{27 \times 3 + 1}{3} = \frac{81 + 1}{3} = \frac{82}{3}
\]

2. **Set up the division**:
\[
\frac{649}{3} \div \frac{82}{3}
\]

3. **Dividing by a fraction is the same as multiplying by its reciprocal**:
\[
\frac{649}{3} \times \frac{3}{82}
\]

4. **Cancel out the common factors**:
The \( 3 \) in the numerator and denominator cancels out:
\[
= \frac{649}{82}
\]

Now, let’s calculate \( 649 \div 82 \):
- Performing the division, \( 649 \div 82 \approx 7.9 \), and since \( 649 = 7 \times 82 + 53 \), it can be written as:
\[
649 = 7 \times 82 + 53 \quad (\text{Remainder: } 53)
\]
Thus, \( \frac{649}{82} = 7 \frac{53}{82} \).

5. **Final Answer**:
Thus the expression \( 216 \frac{1}{3} / 27 \frac{1}{3} = 7 \frac{53}{82} \).

Please check if you are looking for a specific form of the answer or something else. If you have specific choices you'd like me to consider, kindly provide those options.

out of these 4 choices , which is the right one ? or the closest ?

8

18

1/2

2

To solve the expression \( 216 \frac{1}{3} / 27 \frac{1}{3} \), let's break down the numbers:

1. **Convert the mixed numbers to improper fractions**:
- \( 216 \frac{1}{3} = 216 + \frac{1}{3} = \frac{649}{3} \)
- \( 27 \frac{1}{3} = 27 + \frac{1}{3} = \frac{82}{3} \)

2. **Set up the division**:
\[
\frac{649/3}{82/3} = \frac{649}{82}
\]

3. **Now we can simplify \( \frac{649}{82} \)**:
- \( 649 \) divided by \( 82 \) gives approximately \( 7.9 \).
- We can calculate the exact division: \( 649 \div 82 = 7\) remainder approximately \( 53\).

The answer \( 7.9 \) is closer to **8** in your options.

### Conclusion
Among the options given:
- **8**
- 18
- \( \frac{1}{2} \)
- 2

The closest match to \( 7.9 \) is **8**.

So the correct choice is **8**.