To solve this problem, we can use the ratio provided, which is 1:6 (teachers to students), and the number of students given (684).

Step 1: Understand the ratio

The ratio 1:6 means that for every 1 teacher, there are 6 students.

Step 2: Set up the proportion

We can represent the number of teachers as $T$ and the number of students as 684. According to the ratio: $\frac{T}{684} = \frac{1}{6}$

Step 3: Solve for $T$

To find $T$, multiply both sides of the equation by 684: $T = \frac{684}{6}$

Step 4: Calculate

$T = 114$

Final Answer:

There are 114 teachers in the school.

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Would you like more details on how ratios work, or do you have other questions? Here are some related questions you might find helpful:

1. What is the relationship between ratios and proportions in word problems?
2. How would you adjust this problem if the ratio were different, such as 2:9?
3. How can you check if your answer to a ratio problem is correct?
4. What happens to the number of teachers if the number of students changes, but the ratio stays the same?
5. How do you handle situations with multiple ratios in a single problem?

Tip: Always make sure to simplify ratios before setting up the proportion to avoid unnecessary complexity!