the ratio of buoys to gulls was 14 to 3, and four times the number of gulls was 6 less than the number of buoys. How many gulls were there?

Let the number of buoys be \( b \) and the number of gulls be \( g \).

### Step 1: Translate the given ratio into an equation.
We are told the ratio of buoys to gulls is 14:3. This gives us the equation:
\[
\frac{b}{g} = \frac{14}{3}
\]
or equivalently:
\[
b = \frac{14}{3}g
\]

### Step 2: Translate the second condition into an equation.
The second condition says that four times the number of gulls is 6 less than the number of buoys. This translates to the equation:
\[
4g = b - 6
\]

### Step 3: Substitute \( b = \frac{14}{3}g \) into the second equation.
Substitute \( b \) in the second equation:
\[
4g = \frac{14}{3}g - 6
\]

### Step 4: Solve for \( g \).
First, eliminate the fraction by multiplying through by 3:
\[
3 \times 4g = 3 \times \left(\frac{14}{3}g - 6\right)
\]
\[
12g = 14g - 18
\]

Now, subtract \( 14g \) from both sides:
\[
12g - 14g = -18
\]
\[
-2g = -18
\]

Finally, divide both sides by -2:
\[
g = 9
\]

### Conclusion:
The number of gulls is \( \boxed{9} \).

Do you want more details or have any questions?

Here are some related questions to explore:
1. What is the number of buoys based on this information?
2. How do you verify the solution using both conditions of the problem?
3. How would the problem change if the ratio was 14:4 instead?
4. How can you generalize the process for any ratio and condition like this?
5. What happens if the number of gulls is doubled?

**Tip:** In ratio problems, setting up equations from the conditions early on helps to solve them systematically.

The ratio of buoys to gulls was 14 to 3, and four times the number of gulls was 6 less than the number of buoys. How many gulls were there?

This problem explores how to solve a ratio of buoys to gulls (14:3) and apply algebra to find the number of gulls. It involves setting up and solving linear equations and understanding ratio relationships. Suitable for Grades 6-8 students learning basic algebraic concepts.

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