dan and dave can work together to paint a house in 4 days. dan works 2 times faster. how long would it take each of them to paint the house alone
dave's rate ---- 1/x
dan's rate = 1/(2x)
combined rate = 1/x + 1/2x = 3/(2x)
3/(2x) = 1/4
2x = 12
x = 6
Dave can paint it in 6 days, Dan would take 12 days.
To solve this problem, we can assign rates of work to Dan and Dave. Let's say that Dan's rate is "D" and Dave's rate is "2D" (because Dan works twice as fast as Dave).
Given that Dan and Dave can complete the entire house together in 4 days, we can set up the equation:
(1/D) + (1/2D) = 1/4
To solve this equation, we need to find a common denominator for the fractions on the left side. The common denominator is 2D, so we can multiply each fraction by the appropriate factor:
2/D + 1/D = 1/4
Now, add the fractions on the left side:
(2 + 1)/D = 1/4
Simplifying this equation gives us:
3/D = 1/4
To isolate D, we can cross-multiply:
4 * 3 = D * 1
12 = D
So, Dan's rate is 12 units of work per day (D = 12).
Now that we know Dan's rate, we can find Dave's rate by multiplying Dan's rate by 2D:
Dave's rate = 2D = 2 * 12 = 24 units of work per day.
Finally, to find how long it would take each of them to paint the house alone, we can divide the total work (which is equal to 1 house) by their respective rates:
Dan's time = 1 / Dan's rate = 1 / 12 = 1/12th of a house per day
Dave's time = 1 / Dave's rate = 1 / 24 = 1/24th of a house per day
Therefore, it would take Dan approximately 12 days to paint the house alone, while it would take Dave approximately 24 days to paint the house alone.
Let's assume that Dan can paint the house alone in x days.
Since Dan works 2 times faster than Dave, it would take Dave 2x days to paint the house alone.
Working together, they can paint the house in 4 days.
To find the rate at which they can paint the house together, we can use the formula:
1/(Dan's rate) + 1/(Dave's rate) = 1/(Combined rate)
Since Dan can paint the house alone in x days, his rate is 1/x.
Since Dave can paint the house alone in 2x days, his rate is 1/(2x).
Therefore, the combined rate is 1/4.
We can now set up the equation:
1/x + 1/(2x) = 1/4
To solve this equation, we need to find a common denominator:
(2 + 1)/2x = 1/4
Now we can simplify the equation:
3/2x = 1/4
Cross-multiplying, we get:
12 = 2x
Dividing both sides by 2, we find:
x = 6
So, Dan can paint the house alone in 6 days, and Dave can paint the house alone in 2 * 6 = 12 days.