Answer :
Sure, let's solve each part of the question step-by-step:
### Part A: Finding the Unit Rate and Writing the Equation
Step 1: Calculate the unit rate (cost per red pepper).
Sophia bought 9 red peppers for [tex]$5.40. To find the unit rate, we divide the total cost by the number of red peppers:
\[ \text{Unit rate} = \frac{\$[/tex]5.40}{9 \text{ peppers}} = \[tex]$0.60 \text{ per pepper} \]
Step 2: Write the equation relating the cost \( c \) to the number of red peppers \( p \).
Since each red pepper costs $[/tex]0.60:
[tex]\[ c = 0.60p \][/tex]
This equation tells us that the total cost [tex]\( c \)[/tex] in dollars is equal to 0.60 times the number of red peppers [tex]\( p \)[/tex].
### Part B: Calculating the Cost for 14 Red Peppers
Using the unit rate we found and the equation:
[tex]\[ c = 0.60p \][/tex]
If Sophia wants to buy 14 red peppers:
[tex]\[ c = 0.60 \times 14 = \$8.40 \][/tex]
So, Sophia would pay [tex]$8.40 for 14 red peppers.
### Question 15: Checking Proportional Relationships
The original salad dressing recipe uses 12 parts oil, 3 parts vinegar, and 2 parts honey. We need to check which of the following sets of ingredients are in the same proportional relationship.
Step 1: Check each set of ingredients one by one.
1. Set A: 9 Tbsp oil, 2 Tbsp vinegar, 1.5 Tbsp honey
\[ \frac{9}{12} = 0.75, \ \frac{2}{3} ≈ 0.67, \ \frac{1.5}{2} = 0.75 \]
The ratios are not the same, so Set A is not proportional.
2. Set B: 6 Tbsp oil, 1.5 Tbsp vinegar, 1 Tbsp honey
\[ \frac{6}{12} = 0.5, \ \frac{1.5}{3} = 0.5, \ \frac{1}{2} = 0.5 \]
The ratios are the same, so Set B is proportional.
3. Set C: 15 Tbsp oil, 4 Tbsp vinegar, 3 Tbsp honey
\[ \frac{15}{12} = 1.25, \ \frac{4}{3} ≈ 1.33, \ \frac{3}{2} = 1.5 \]
The ratios are not the same, so Set C is not proportional.
4. Set D: 18 Tbsp oil, 4.5 Tbsp vinegar, 3 Tbsp honey
\[ \frac{18}{12} = 1.5, \ \frac{4.5}{3} = 1.5, \ \frac{3}{2} = 1.5 \]
The ratios are the same, so Set D is proportional.
5. Set E: 24 Tbsp oil, 6 Tbsp vinegar, 4 Tbsp honey
\[ \frac{24}{12} = 2, \ \frac{6}{3} = 2, \ \frac{4}{2} = 2 \]
The ratios are the same, so Set E is proportional.
### Summary
- Unit rate: $[/tex]0.60 per red pepper
- Equation: [tex]\( c = 0.60p \)[/tex]
- Cost for 14 red peppers: $8.40
- Proportional sets: Set B, Set D, Set E
These detailed steps should help you understand how to solve this type of problem.
### Part A: Finding the Unit Rate and Writing the Equation
Step 1: Calculate the unit rate (cost per red pepper).
Sophia bought 9 red peppers for [tex]$5.40. To find the unit rate, we divide the total cost by the number of red peppers:
\[ \text{Unit rate} = \frac{\$[/tex]5.40}{9 \text{ peppers}} = \[tex]$0.60 \text{ per pepper} \]
Step 2: Write the equation relating the cost \( c \) to the number of red peppers \( p \).
Since each red pepper costs $[/tex]0.60:
[tex]\[ c = 0.60p \][/tex]
This equation tells us that the total cost [tex]\( c \)[/tex] in dollars is equal to 0.60 times the number of red peppers [tex]\( p \)[/tex].
### Part B: Calculating the Cost for 14 Red Peppers
Using the unit rate we found and the equation:
[tex]\[ c = 0.60p \][/tex]
If Sophia wants to buy 14 red peppers:
[tex]\[ c = 0.60 \times 14 = \$8.40 \][/tex]
So, Sophia would pay [tex]$8.40 for 14 red peppers.
### Question 15: Checking Proportional Relationships
The original salad dressing recipe uses 12 parts oil, 3 parts vinegar, and 2 parts honey. We need to check which of the following sets of ingredients are in the same proportional relationship.
Step 1: Check each set of ingredients one by one.
1. Set A: 9 Tbsp oil, 2 Tbsp vinegar, 1.5 Tbsp honey
\[ \frac{9}{12} = 0.75, \ \frac{2}{3} ≈ 0.67, \ \frac{1.5}{2} = 0.75 \]
The ratios are not the same, so Set A is not proportional.
2. Set B: 6 Tbsp oil, 1.5 Tbsp vinegar, 1 Tbsp honey
\[ \frac{6}{12} = 0.5, \ \frac{1.5}{3} = 0.5, \ \frac{1}{2} = 0.5 \]
The ratios are the same, so Set B is proportional.
3. Set C: 15 Tbsp oil, 4 Tbsp vinegar, 3 Tbsp honey
\[ \frac{15}{12} = 1.25, \ \frac{4}{3} ≈ 1.33, \ \frac{3}{2} = 1.5 \]
The ratios are not the same, so Set C is not proportional.
4. Set D: 18 Tbsp oil, 4.5 Tbsp vinegar, 3 Tbsp honey
\[ \frac{18}{12} = 1.5, \ \frac{4.5}{3} = 1.5, \ \frac{3}{2} = 1.5 \]
The ratios are the same, so Set D is proportional.
5. Set E: 24 Tbsp oil, 6 Tbsp vinegar, 4 Tbsp honey
\[ \frac{24}{12} = 2, \ \frac{6}{3} = 2, \ \frac{4}{2} = 2 \]
The ratios are the same, so Set E is proportional.
### Summary
- Unit rate: $[/tex]0.60 per red pepper
- Equation: [tex]\( c = 0.60p \)[/tex]
- Cost for 14 red peppers: $8.40
- Proportional sets: Set B, Set D, Set E
These detailed steps should help you understand how to solve this type of problem.
