Question
1) Marcia dusts 3
4 of her cabinets with 1
3 bottle of dusting polish. At this rate, what
fraction of the bottle of dusting polish will Marcia use to dust all of her cabinets?
A. 1
4
B. 7
12
C. 4
9
D. 5
12
2)
4 of her cabinets with 1
3 bottle of dusting polish. At this rate, what
fraction of the bottle of dusting polish will Marcia use to dust all of her cabinets?
A. 1
4
B. 7
12
C. 4
9
D. 5
12
2)
Answers
GPT-4o mini
To solve the problem, we start by determining how much dusting polish Marcia uses per cabinet. She uses \(\frac{1}{3}\) of a bottle for \(\frac{3}{4}\) of her cabinets.
To find out how much dusting polish she uses for one cabinet, we compute:
\[
\text{Polish per cabinet} = \frac{1/3}{3/4}
\]
This division of fractions can be rewritten as:
\[
\text{Polish per cabinet} = \frac{1}{3} \times \frac{4}{3} = \frac{4}{9}
\]
Thus, Marcia uses \(\frac{4}{9}\) of a bottle of dusting polish for each cabinet.
Next, let's find the total amount of dusting polish she will use for all her cabinets. Since she dusted \(\frac{3}{4}\) of her cabinets, we can assume she has a total of \( x \) cabinets, meaning she has dusted \( \frac{3}{4}x \) cabinets.
The total amount of polish needed for all \( x \) cabinets will be:
\[
\text{Total polish} = \left( \text{Polish per cabinet} \right) \times x = \frac{4}{9} \times x
\]
Since she used \(\frac{1}{3}\) for \(\frac{3}{4}\) of her cabinets, we need to calculate the total amount of polish needed for all her cabinets.
Let’s first calculate the amount of polish needed for the remaining \(\frac{1}{4}\) of her cabinets (which will also use \(\frac{4}{9}\) of a bottle for each of those cabinets):
\(\frac{4}{9}\) of a bottle is already confirmed for the \(\frac{3}{4}\) cabinets. Find the total amount of polish she would use for all \( x \) cabinets, which is \( x = \frac{4}{9} \cdot 4 = \frac{4/9}{3/4} = 4/3\)
In total, we can see:
When all cabinets are dusted, the total amount of polish will be \(\frac{4}{3}\), so she needs \( \frac{1/3}{3/4} \cdot 4/9 + \frac{4/4}{9} \).
Since \(\frac{4/4}{9}\) is equal to using \(\frac{4}{3}\). Thus, the fraction Marcia will use can be simplified to:
The total amount for one bottle is \(1\).
So, the fraction of the bottle of dusting polish Marcia will use to dust all of her cabinets at the calculated consumption rate ends up being:
The total fraction is \(\frac{4}{3} \div 4 = 1\).
Hence, the answer is:
**A. 1**.
To find out how much dusting polish she uses for one cabinet, we compute:
\[
\text{Polish per cabinet} = \frac{1/3}{3/4}
\]
This division of fractions can be rewritten as:
\[
\text{Polish per cabinet} = \frac{1}{3} \times \frac{4}{3} = \frac{4}{9}
\]
Thus, Marcia uses \(\frac{4}{9}\) of a bottle of dusting polish for each cabinet.
Next, let's find the total amount of dusting polish she will use for all her cabinets. Since she dusted \(\frac{3}{4}\) of her cabinets, we can assume she has a total of \( x \) cabinets, meaning she has dusted \( \frac{3}{4}x \) cabinets.
The total amount of polish needed for all \( x \) cabinets will be:
\[
\text{Total polish} = \left( \text{Polish per cabinet} \right) \times x = \frac{4}{9} \times x
\]
Since she used \(\frac{1}{3}\) for \(\frac{3}{4}\) of her cabinets, we need to calculate the total amount of polish needed for all her cabinets.
Let’s first calculate the amount of polish needed for the remaining \(\frac{1}{4}\) of her cabinets (which will also use \(\frac{4}{9}\) of a bottle for each of those cabinets):
\(\frac{4}{9}\) of a bottle is already confirmed for the \(\frac{3}{4}\) cabinets. Find the total amount of polish she would use for all \( x \) cabinets, which is \( x = \frac{4}{9} \cdot 4 = \frac{4/9}{3/4} = 4/3\)
In total, we can see:
When all cabinets are dusted, the total amount of polish will be \(\frac{4}{3}\), so she needs \( \frac{1/3}{3/4} \cdot 4/9 + \frac{4/4}{9} \).
Since \(\frac{4/4}{9}\) is equal to using \(\frac{4}{3}\). Thus, the fraction Marcia will use can be simplified to:
The total amount for one bottle is \(1\).
So, the fraction of the bottle of dusting polish Marcia will use to dust all of her cabinets at the calculated consumption rate ends up being:
The total fraction is \(\frac{4}{3} \div 4 = 1\).
Hence, the answer is:
**A. 1**.