Answer :
We start with the information that the cleaning solution is made by mixing vinegar and water in the ratio
[tex]$$
1 \text{ cup vinegar} : 16 \text{ cups water}
$$[/tex]
and note that 1 gallon equals 16 cups.
─────────────────────────────
Step-by-Step Solution:
► Part A
The question asks: How many cups of water should be mixed with [tex]$\frac{1}{4}$[/tex] cup of vinegar?
Since the ratio is 1 cup vinegar to 16 cups water, for [tex]$\frac{1}{4}$[/tex] cup vinegar we multiply by 16:
[tex]$$
\text{Water needed} = \frac{1}{4} \times 16 = 4 \text{ cups water.}
$$[/tex]
─────────────────────────────
► Part B
Now, we need to find how many fluid ounces of vinegar are required if we have 80 fluid ounces of water.
Because the ratio of vinegar to water is 1:16, we set up the relation:
[tex]$$
\text{Vinegar needed} = \frac{\text{Water}}{16} = \frac{80}{16} = 5 \text{ fluid ounces.}
$$[/tex]
─────────────────────────────
► Part C
The bottle contains 1 quart of vinegar. First, recall that 1 quart equals 4 cups. The recipe uses 1 cup vinegar to make a total of [tex]$1 + 16 = 17$[/tex] cups of cleaning solution.
Thus, if we have 4 cups vinegar (from 1 quart), the total cleaning solution we can prepare is:
[tex]$$
\text{Total solution in cups} = 4 \times 17 = 68 \text{ cups.}
$$[/tex]
Since 4 cups equal 1 quart, convert the total cups into quarts:
[tex]$$
\text{Total solution in quarts} = \frac{68}{4} = 17 \text{ quarts.}
$$[/tex]
─────────────────────────────
► Part D
A spray bottle holds up to 1 cup of the cleaning solution. According to the recipe, 1 cup of vinegar is used to make 17 cups of solution. Therefore, when the spray bottle is full (1 cup of solution), the fraction of vinegar in it is:
[tex]$$
\frac{1 \text{ cup vinegar}}{17 \text{ cups solution}} = \frac{1}{17}.
$$[/tex]
─────────────────────────────
Summary of Answers:
- Part A: [tex]$\boxed{4}$[/tex] cups of water
- Part B: [tex]$\boxed{5}$[/tex] fluid ounces of vinegar
- Part C: [tex]$\boxed{17}$[/tex] quarts of cleaning solution
- Part D: [tex]$\boxed{\frac{1}{17}}$[/tex]
Each step follows directly from the given proportions and unit conversions.
[tex]$$
1 \text{ cup vinegar} : 16 \text{ cups water}
$$[/tex]
and note that 1 gallon equals 16 cups.
─────────────────────────────
Step-by-Step Solution:
► Part A
The question asks: How many cups of water should be mixed with [tex]$\frac{1}{4}$[/tex] cup of vinegar?
Since the ratio is 1 cup vinegar to 16 cups water, for [tex]$\frac{1}{4}$[/tex] cup vinegar we multiply by 16:
[tex]$$
\text{Water needed} = \frac{1}{4} \times 16 = 4 \text{ cups water.}
$$[/tex]
─────────────────────────────
► Part B
Now, we need to find how many fluid ounces of vinegar are required if we have 80 fluid ounces of water.
Because the ratio of vinegar to water is 1:16, we set up the relation:
[tex]$$
\text{Vinegar needed} = \frac{\text{Water}}{16} = \frac{80}{16} = 5 \text{ fluid ounces.}
$$[/tex]
─────────────────────────────
► Part C
The bottle contains 1 quart of vinegar. First, recall that 1 quart equals 4 cups. The recipe uses 1 cup vinegar to make a total of [tex]$1 + 16 = 17$[/tex] cups of cleaning solution.
Thus, if we have 4 cups vinegar (from 1 quart), the total cleaning solution we can prepare is:
[tex]$$
\text{Total solution in cups} = 4 \times 17 = 68 \text{ cups.}
$$[/tex]
Since 4 cups equal 1 quart, convert the total cups into quarts:
[tex]$$
\text{Total solution in quarts} = \frac{68}{4} = 17 \text{ quarts.}
$$[/tex]
─────────────────────────────
► Part D
A spray bottle holds up to 1 cup of the cleaning solution. According to the recipe, 1 cup of vinegar is used to make 17 cups of solution. Therefore, when the spray bottle is full (1 cup of solution), the fraction of vinegar in it is:
[tex]$$
\frac{1 \text{ cup vinegar}}{17 \text{ cups solution}} = \frac{1}{17}.
$$[/tex]
─────────────────────────────
Summary of Answers:
- Part A: [tex]$\boxed{4}$[/tex] cups of water
- Part B: [tex]$\boxed{5}$[/tex] fluid ounces of vinegar
- Part C: [tex]$\boxed{17}$[/tex] quarts of cleaning solution
- Part D: [tex]$\boxed{\frac{1}{17}}$[/tex]
Each step follows directly from the given proportions and unit conversions.
