The ratio of seventh grade students to eighth grade students in the school band is given as 3:4. This means that for every 3 seventh grade students, there are 4 eighth grade students.

Given:

• There are 36 seventh grade students.
• The ratio of seventh grade to eighth grade students is 3:4.

Let’s set up the problem using the ratio:

$\frac{\text{Seventh Grade Students}}{\text{Eighth Grade Students}} = \frac{3}{4}$

We can represent the number of eighth grade students as $x$. Using the ratio:

$\frac{36}{x} = \frac{3}{4}$

Now, solve for $x$ by cross-multiplying:

$36 \times 4 = 3 \times x$

$144 = 3x$

Now, divide both sides by 3:

$x = \frac{144}{3} = 48$

So, there are 48 eighth grade students.

Now, the total number of students in the band is:

$36 \text{ (seventh grade)} + 48 \text{ (eighth grade)} = 84 \text{ students}$

Thus, the total number of students in the school band is 84.

Would you like more details on this solution? Here are some related questions:

1. How can I solve other ratio-based problems?
2. What are other ways to represent ratios besides fractions?
3. How does cross-multiplication work in ratio problems?
4. What if the ratio of seventh to eighth graders changes to 5:6, how would that impact the number of students?
5. Can ratios be used to solve problems in other subjects, like physics?

Tip: When dealing with ratios, always check if the total can be simplified before solving!