Certainly! Let's go through the problem step-by-step to find out how long it will take to paint one entire wall.
1. Understand the given rates:
You are able to paint [tex]\(\frac{1}{3}\)[/tex] of a wall in [tex]\(\frac{1}{4}\)[/tex] hour.
2. Calculate the rate of painting per hour:
To determine the rate at which you paint, divide the fraction of the wall you paint by the time it takes:
[tex]\[
\text{Rate of painting per hour} = \frac{\frac{1}{3} \text{ wall}}{\frac{1}{4} \text{ hour}}
\][/tex]
Simplifying this, we invert the divisor and multiply:
[tex]\[
\text{Rate of painting per hour} = \frac{1}{3} \times \frac{4}{1} = \frac{4}{3} \text{ walls per hour}
\][/tex]
This tells us that you can paint [tex]\(\frac{4}{3}\)[/tex] of a wall every hour.
3. Calculate the total time to paint one wall:
Now, we need to find out how long it takes to paint one whole wall. If you can paint [tex]\(\frac{4}{3}\)[/tex] of a wall in one hour, you need to find the reciprocal of this rate to determine the time it takes to paint one wall:
[tex]\[
\text{Time to paint one wall} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \text{ hours}
\][/tex]
So, after all the calculations, it will take you [tex]\(\frac{3}{4}\)[/tex] of an hour (or 0.75 hours) to paint one entire wall.
