Alright, let's work through this problem step-by-step.
First, we need to understand the rates at which Kalon and Marna clean a chimney.
### Step 1: Determine Kalon's Cleaning Rate
Kalon can clean a 20-foot chimney by himself in 4 hours. This means Kalon's cleaning rate is:
[tex]\[ \text{Kalon's rate} = \frac{1 \text{ chimney}}{4 \text{ hours}} = \frac{1}{4} \text{ chimneys per hour} \][/tex]
### Step 2: Determine Their Combined Cleaning Rate
Together, Kalon and Marna clean a 20-foot chimney in [tex]\( 1 \frac{5}{7} \)[/tex] hours. We need to convert this time into an improper fraction.
[tex]\[ 1 \frac{5}{7} = 1 + \frac{5}{7} = \frac{7}{7} + \frac{5}{7} = \frac{12}{7} \text{ hours} \][/tex]
Their combined cleaning rate is:
[tex]\[ \text{Combined rate} = \frac{1 \text{ chimney}}{\frac{12}{7} \text{ hours}} = \frac{7}{12} \text{ chimneys per hour} \][/tex]
### Step 3: Determine Marna's Cleaning Rate
The combined rate is the sum of Kalon's rate and Marna's rate:
[tex]\[ \text{Combined rate} = \text{Kalon's rate} + \text{Marna's rate} \][/tex]
[tex]\[ \frac{7}{12} = \frac{1}{4} + \text{Marna's rate} \][/tex]
Subtract Kalon's rate from the combined rate to find Marna's rate:
[tex]\[ \text{Marna's rate} = \frac{7}{12} - \frac{1}{4} = \frac{7}{12} - \frac{3}{12} = \frac{4}{12} = \frac{1}{3} \text{ chimneys per hour} \][/tex]
### Step 4: Determine the Time it Takes Marna to Clean a Chimney by Herself
Finally, we need to calculate the time it takes Marna to clean 1 chimney. Since rate and time are inversely related, we can use the formula:
[tex]\[ \text{Time} = \frac{1}{\text{Rate}} \][/tex]
[tex]\[ \text{Marna's Time} = \frac{1 \text{ chimney}}{\frac{1}{3} \text{ chimneys per hour}} = 3 \text{ hours} \][/tex]
### Conclusion
Therefore, it takes Marna 3 hours to clean a 20-foot chimney by herself.
