Answer :
Sure! Let's solve this problem step-by-step.
### Given Information:
1. A cleaning spray contains [tex]\(\frac{1}{2}\)[/tex] cup of vinegar for every 4 teaspoons of lemon oil.
2. We need to find how many cups of vinegar are needed for 2 teaspoons of lemon oil.
3. We also need to find the rate in cups of vinegar per teaspoon of lemon oil.
### Step-by-Step Solution:
#### Step 1: Understand the ratio provided
The given ratio is:
[tex]\[
\frac{1}{2} \text{ cup of vinegar} : 4 \text{ teaspoons of lemon oil}
\][/tex]
#### Step 2: Find the amount of vinegar for 1 teaspoon of lemon oil
To find out how much vinegar corresponds to 1 teaspoon of lemon oil, you divide the given amount of vinegar by the number of teaspoons of lemon oil it correlates with.
[tex]\[
\text{Vinegar per teaspoon of lemon oil} = \frac{\frac{1}{2} \text{ cup}}{4 \text{ teaspoons}} = \frac{1}{2} \div 4 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \text{ cups per teaspoon}
\][/tex]
#### Step 3: Find the amount of vinegar for 2 teaspoons of lemon oil
Using the rate we found in Step 2:
[tex]\[
\text{Vinegar for 2 teaspoons} = 2 \text{ teaspoons} \times \frac{1}{8} \text{ cups per teaspoon} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \text{ cups}
\][/tex]
Thus, [tex]\(\frac{1}{4}\)[/tex] cups of vinegar are needed for 2 teaspoons of lemon oil.
#### Step 4: Summarize the rate in cups of vinegar per teaspoon of lemon oil
From Step 2, the rate is:
[tex]\[
\frac{1}{8} \text{ cups of vinegar per teaspoon of lemon oil}
\][/tex]
### Final Answers:
1. Cups of vinegar needed for 2 teaspoons of lemon oil: [tex]\(\frac{1}{4}\)[/tex] cups.
2. Rate in cups of vinegar per teaspoon of lemon oil: [tex]\(\frac{1}{8}\)[/tex] cups per teaspoon.
### Given Information:
1. A cleaning spray contains [tex]\(\frac{1}{2}\)[/tex] cup of vinegar for every 4 teaspoons of lemon oil.
2. We need to find how many cups of vinegar are needed for 2 teaspoons of lemon oil.
3. We also need to find the rate in cups of vinegar per teaspoon of lemon oil.
### Step-by-Step Solution:
#### Step 1: Understand the ratio provided
The given ratio is:
[tex]\[
\frac{1}{2} \text{ cup of vinegar} : 4 \text{ teaspoons of lemon oil}
\][/tex]
#### Step 2: Find the amount of vinegar for 1 teaspoon of lemon oil
To find out how much vinegar corresponds to 1 teaspoon of lemon oil, you divide the given amount of vinegar by the number of teaspoons of lemon oil it correlates with.
[tex]\[
\text{Vinegar per teaspoon of lemon oil} = \frac{\frac{1}{2} \text{ cup}}{4 \text{ teaspoons}} = \frac{1}{2} \div 4 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \text{ cups per teaspoon}
\][/tex]
#### Step 3: Find the amount of vinegar for 2 teaspoons of lemon oil
Using the rate we found in Step 2:
[tex]\[
\text{Vinegar for 2 teaspoons} = 2 \text{ teaspoons} \times \frac{1}{8} \text{ cups per teaspoon} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \text{ cups}
\][/tex]
Thus, [tex]\(\frac{1}{4}\)[/tex] cups of vinegar are needed for 2 teaspoons of lemon oil.
#### Step 4: Summarize the rate in cups of vinegar per teaspoon of lemon oil
From Step 2, the rate is:
[tex]\[
\frac{1}{8} \text{ cups of vinegar per teaspoon of lemon oil}
\][/tex]
### Final Answers:
1. Cups of vinegar needed for 2 teaspoons of lemon oil: [tex]\(\frac{1}{4}\)[/tex] cups.
2. Rate in cups of vinegar per teaspoon of lemon oil: [tex]\(\frac{1}{8}\)[/tex] cups per teaspoon.
