James can paint the office by himself in 7 hours. Manny paints the office in 10 hours. How long will it take them to paint the office working together?
1/x = 1/7 + 1/10
Well, as they say, "Two heads are better than one, and two painters are better than…well, one painter." So, let's see how James and Manny team up to transform that office.
If James can paint the office by himself in 7 hours, we can say that his painting speed is 1/7 of the office per hour. Similarly, since Manny can complete the job in 10 hours, his painting speed is 1/10 of the office per hour.
Now, when they work together, we can simply add up their speeds. So, James and Manny working together will paint at a speed of 1/7 + 1/10 of the office per hour.
To find out how long it will take them to paint the entire office together, we can set up the equation:
(1/7 + 1/10) * t = 1, where 't' is the time it takes them to paint the office together.
Solving this equation will give us the answer. But hey, wouldn't it be more fun if James and Manny just painted clown faces all over the walls? That way, they can turn the office into an actual circus, and maybe even attract customers while they're at it! Just a thought.
To determine how long it will take James and Manny to paint the office together, we can use the formula:
1/T = 1/T1 + 1/T2 + ...
where T is the time taken when working together, T1 is the time taken by James, and T2 is the time taken by Manny.
Let's substitute the given information into the formula:
1/T = 1/7 + 1/10
To simplify the equation, we need to find a common denominator. The common denominator in this case is 70:
1/T = (10/70) + (7/70)
= 17/70
Now, invert both sides of the equation to solve for T:
T/1 = 70/17
Finally, divide 70 by 17 to find the time it will take them to paint the office together:
T ≈ 4.12 hours
Therefore, it will take James and Manny approximately 4.12 hours to paint the office together.
To find out how long it will take James and Manny to paint the office working together, we can use the concept of rates. A rate represents the amount of work done per unit of time.
First, let's calculate James's rate of painting. We know that he can paint the office by himself in 7 hours. So, his rate of painting is 1 office / 7 hours, which can be written as 1/7 offices per hour.
Similarly, Manny's rate of painting is 1 office / 10 hours, which is 1/10 offices per hour.
To find out how long it will take them to paint the office working together, we can add their rates of painting. So, their combined rate of painting is (1/7 + 1/10) offices per hour.
To simplify this fraction, we need to find a common denominator. In this case, the least common multiple (LCM) of 7 and 10 is 70. Thus, we can rewrite the rates as (10/70 + 7/70) offices per hour, which equals 17/70 offices per hour.
Now, we can find the time it will take them to paint the office by dividing the total amount of work (1 office) by their combined rate of painting. The equation would be:
Time = Amount of work / Combined rate
Time = 1 office / (17/70 offices per hour)
To divide by a fraction, we can multiply by its reciprocal, so the equation becomes:
Time = 1 office * (70/17 offices per hour)
Simplifying this multiplication gives us:
Time = 70/17 hours
So, it will take James and Manny approximately 4.12 hours to paint the office together.
Jame's rate = 1office/7 hrs
Manny' rate = 1office/10 hrs
combined rate = 1office/7 + 1office/10
= 17 offices/70 hours
time to do 1 office = 1office/(17offices/70 hours) = 70/17 hours
= appr 4 hours and 7 minutes