Sure, let's start by choosing one of the ingredients from the recipe — we will use sugar for this purpose.
1. Identify the given quantity for a serving of 8 people:
The recipe calls for [tex]\(\frac{3}{4}\)[/tex] cup of sugar to serve 8 people.
2. Set up the proportion to find the required amount for 22 people:
We need to determine how much sugar is needed for 22 people. Let's denote:
- [tex]\(s_8\)[/tex] as the amount of sugar for 8 people.
- [tex]\(s_{22}\)[/tex] as the amount of sugar for 22 people.
The proportion can be written as:
[tex]\[
\frac{s_8}{8} = \frac{s_{22}}{22}
\][/tex]
3. Substitute the values into the proportion:
We know that [tex]\(s_8 = \frac{3}{4}\)[/tex] cup.
So we have:
[tex]\[
\frac{\frac{3}{4}}{8} = \frac{s_{22}}{22}
\][/tex]
4. Solve the proportion for [tex]\(s_{22}\)[/tex]:
To isolate [tex]\(s_{22}\)[/tex], we multiply both sides of the equation by 22:
[tex]\[
s_{22} = 22 \times \frac{\frac{3}{4}}{8}
\][/tex]
5. Simplify the right-hand side:
- First, simplify the fraction inside the multiplication:
[tex]\[
\frac{\frac{3}{4}}{8} = \frac{3}{4 \times 8} = \frac{3}{32}
\][/tex]
- Then, multiply by 22:
[tex]\[
s_{22} = 22 \times \frac{3}{32}
\][/tex]
6. Calculate [tex]\(s_{22}\)[/tex]:
- Multiply the numerators and denominators:
[tex]\[
s_{22} = \frac{22 \times 3}{32} = \frac{66}{32}
\][/tex]
- Simplify the fraction:
[tex]\[
\frac{66}{32} = 2.0625
\][/tex]
Therefore, [tex]\(s_{22} = 2.0625\)[/tex] cups.
7. Conclusion:
The amount of sugar needed for 22 people is [tex]\(2.0625\)[/tex] cups.
