Certainly! Here's a detailed, step-by-step solution:
1. Understand the problem: The original soup recipe requires 3 chopped onions. You only want to make [tex]\(\frac{1}{5}\)[/tex] of the recipe. We need to determine how many onions are required for [tex]\(\frac{1}{5}\)[/tex] of the full recipe.
2. Set up the proportion: We know that 3 onions are required for the full recipe (which is 1 whole recipe). To find out how many onions, [tex]\(x\)[/tex], are required for [tex]\(\frac{1}{5}\)[/tex] of the recipe, we set up the proportion:
[tex]\[
\frac{3}{1} = \frac{x}{\frac{1}{5}}
\][/tex]
3. Solve the proportion: To find [tex]\(x\)[/tex], we solve the proportion by cross-multiplying:
[tex]\[
3 \times \frac{1}{5} = x \times 1
\][/tex]
[tex]\[
x = 3 \times \frac{1}{5}
\][/tex]
4. Perform the multiplication: Multiply 3 by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
x = 3 \times \frac{1}{5} = \frac{3}{5}
\][/tex]
So, [tex]\(x\)[/tex] is [tex]\(\frac{3}{5}\)[/tex] of an onion.
5. Interpret the numerical result: Numerically, [tex]\(\frac{3}{5}\)[/tex] is approximately 0.6. Hence, for [tex]\(\frac{1}{5}\)[/tex] of the recipe, you will need approximately 0.6 onions.
6. Verify the proportion statements: We need to identify which of the given proportions are accurate.
- The first proportion [tex]\(\frac{3(5)}{1} = \frac{2(5)}{\frac{1}{5}}\)[/tex] makes no sense for our context.
- The second proportion [tex]\(\frac{3}{1} = \frac{x}{3}\)[/tex] does not relate correctly to our situation.
- The third proportion [tex]\(\frac{3}{\frac{1}{5}} = \frac{x}{1}\)[/tex] simplifies to [tex]\(\frac{3 \times 5}{1} = x\)[/tex], which is incorrect for our goal.
The correct proportion is the one initially solved:
[tex]\[
\boxed{\frac{3}{1} = \frac{x}{\frac{1}{5}}}
\][/tex]
Therefore:
- You will need [tex]\(\frac{3}{5}\)[/tex] or 0.6 onions.
- The correct proportion that will yield the appropriate answer is [tex]\(\frac{3}{1} = \frac{x}{\frac{1}{5}}\)[/tex].
