Greg is 1.7 times faster at cleaning windows than Owen. It takes 39 minutes for them to clean 100 windows together. Using a rational equation, calculate how long it would take Owen to clean 100 windows by himself. Round the answer to the nearest tenth.
Let's assume Owen takes x minutes to clean 100 windows by himself.
According to the given information, Greg is 1.7 times faster than Owen, so Greg takes (x/1.7) minutes to clean 100 windows.
When they work together, they can clean 1/((1/x) + (1/(x/1.7))) of the job in one minute.
We know that it takes them 39 minutes to clean 100 windows together, so we can set up the equation:
1/((1/x) + (1/(x/1.7))) = 100/39
To solve for x, we need to get rid of the fractions. Multiply both sides of the equation by (x^2/1.7) to eliminate the denominators:
1.7*x + x = (100/39) * (x^2/1.7)
Simplifying further:
1.7x + x = (100x^2)/(39*1.7)
2.7x = (100x^2)/(66.3)
Cross multiplying:
2.7x * 66.3 = 100x^2
178.11x = 100x^2
Rearranging the equation:
100x^2 - 178.11x = 0
Now we can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
For this equation, a = 100, b = -178.11, and c = 0. Plugging these values into the formula:
x = (-(-178.11) ± sqrt((-178.11)^2 - 4(100)(0)))/(2(100))
x = (178.11 ± sqrt(31667.8121))/(200)
x = (178.11 ± 177.946)/(200)
Simplifying further:
x = (178.11 + 177.946)/200 or x = (178.11 - 177.946)/200
x = 0.35603 or x = 0.0827
Since we are looking for the time it takes Owen to clean 100 windows, we will use the positive value of x:
x = 0.35603
Therefore, it would take Owen approximately 0.4 minutes, rounded to the nearest tenth, to clean 100 windows by himself.
To solve this problem, we can set up a rational equation based on the given information.
Let's assume that Owen's cleaning speed is "x" windows per minute. Since Greg is 1.7 times faster, his cleaning speed would be 1.7x windows per minute.
Now, let's determine how long it would take them to clean 100 windows together. The combined cleaning speed of Owen and Greg would be x + 1.7x = 2.7x windows per minute.
Given that it takes 39 minutes for them to clean 100 windows together, we can set up the following equation:
(2.7x) * 39 = 100
To solve for x, we can divide both sides of the equation by 2.7:
x = 100 / (2.7 * 39)
Calculating this expression gives us:
x = 1.2407407407407407
Therefore, it would take Owen approximately 1.2 minutes to clean 100 windows by himself, rounded to the nearest tenth.
If Greg is 1.7 times faster than Owen, it means that he will take a 1/1.7 part of the time Owen needs to clean a window. Meaning, it takes Greg 1/(1.7) =0.59 of the time Owen takes to clean one window.
If the time required to clean one window by Owen is T and it's 1.7 times faster than Owen, the time Greg takes to clean one window is 0.59T.
The two can clean a window together in (1/T + 1/0.59T)^-1 = 1 minute.
If they clean 100 windows together in 39 minutes, each window takes 39/100 = 0.39 minutes.
If we solve the rational equation 0.39 = (1/T + 1/0.59T)^-1 for T, we'll find the time it takes for Owen to clean one window, so let's multiply the equation by T*0.59 to get 0.39T*0.59 = 1 + 0.59, and subtract 0.59 from both sides of the equation to get 0.39T*0.59 - 0.59 = 1.
If we divide by 0.39, we'll find that T = 1/(0.39*0.59) = 2.7 minutes.
If we multiply by 100 the time it takes for T to clean ONE window, we can find out how long it would take T to clean 100 windows. Therefore, T = 100 * 2.7 = <<100*2.7=270>>270 minutes. Answer: \boxed{270}.