What is the greatest amount of time it takes the painters to paint the room together?
The inequality to model this situation is: 1/3+1/6 ? 1/x
The Correct Answer and Explanation is:
To solve this problem, we need to find the greatest amount of time it takes the painters to paint the room together. The inequality provided is:
[
\frac{1}{3} + \frac{1}{6} \leq \frac{1}{x}
]
Here, (\frac{1}{3}) and (\frac{1}{6}) represent the rates at which the two painters work individually, and (x) represents the time it would take for them to paint the room together.
Step 1: Understanding the Rates
- The first painter can paint the room in 3 hours, meaning that in 1 hour, they can paint (\frac{1}{3}) of the room.
- The second painter can paint the room in 6 hours, so in 1 hour, they can paint (\frac{1}{6}) of the room.
Together, in 1 hour, they will paint the sum of these fractions:
[
\frac{1}{3} + \frac{1}{6}
]
Step 2: Adding the Fractions
To add (\frac{1}{3}) and (\frac{1}{6}), we need a common denominator. The least common denominator of 3 and 6 is 6. We can rewrite (\frac{1}{3}) as (\frac{2}{6}) to have the same denominator:
[
\frac{1}{3} = \frac{2}{6}
]
Now, add the two fractions:
[
\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
]
So, together, the painters can paint (\frac{1}{2}) of the room in one hour.
Step 3: Solving for (x)
The equation tells us that the combined rate of the two painters is (\frac{1}{x}), where (x) is the time it takes for them to paint the entire room together. Since they paint (\frac{1}{2}) of the room in one hour, we set up the following equation:
[
\frac{1}{x} = \frac{1}{2}
]
To solve for (x), we can take the reciprocal of both sides:
[
x = 2
]
Conclusion:
The greatest amount of time it takes the painters to paint the room together is 2 hours. This means that if both painters work together, they can finish painting the room in 2 hours.