Mio can build a large shed in 7 days less then yuuko can. If they build it together it would take them 12 days. How long would it take each of them working alone?
m = y-7
1/m + 1/y = 1/12
1/(y-7) + 1/y = 1/12
y = 28
so, m = 21
check:
working together, in one day they can assemble 1/28 + 1/21 = 1/12 of the whole.
To find the time it would take for each of them working alone, we can use the concept of work rates. Let's assume that Yuuko takes x days to build the shed alone.
Yuuko's work rate is represented as 1/x (shed per day), which means she can build 1 shed in x days.
Now, since Mio takes 7 days less than Yuuko to build the shed, her work rate would be represented as 1/(x-7), where x-7 represents the number of days it takes Mio to build the shed alone.
When they work together, their combined work rate is 1/12 (shed per day), as it takes them 12 days to finish the shed.
Using the concept of work rates, we can set up the equation:
1/x + 1/(x-7) = 1/12
To solve this equation, we can multiply both sides by 12x(x-7) to eliminate the denominators:
12(x-7) + 12x = x(x-7)
Expanding and simplifying:
12x - 84 + 12x = x^2 - 7x
Rearranging and simplifying further:
x^2 - 31x + 84 = 0
Now we can factor or use the quadratic formula to solve for x. Factoring the equation, we get:
(x - 3)(x - 28) = 0
From this, we have two possible solutions: x = 3 or x = 28.
Since Mio takes 7 days less than Yuuko, the only valid solution is x = 28.
Therefore, Yuuko alone would take 28 days to build the shed.
To find Mio's time, we subtract 7 from Yuuko's time, so Mio alone would take 21 days to build the shed.