1) Mio can build a large shed in 15 days less than Yuuko can. If they build it together, it would take them 18 days. How long would it take each of them working alone?

2) If Kamina can do a job in 29 hours and Simon and Kamina working together can do the same job in 10 hours, find how long it takes Simon to do the job alone.

#1

1/(y-15) + 1/y = 1/18
y = 45
so, m = 30

#2
1/29 + 1/s = 1/10
s = 15.26
odd answer. Typo?

1) Let's assume that Yuuko can build the shed in x days. According to the problem, Mio can build the shed in 15 days less than Yuuko. So, Mio can build the shed in (x - 15) days.

When they work together, they can complete the shed in 18 days. We know that their combined rate of work is 1 shed per 18 days.

Now, let's set up an equation based on their individual rates of work:

1/(x) + 1/(x - 15) = 1/18

To solve this equation, we need to find a common denominator and then combine the fractions:

[(x - 15) + x] / [x(x - 15)] = 1/18

Simplifying further:

2x - 15 / [x(x - 15)] = 1/18

Cross-multiplying:

18(2x - 15) = x(x - 15)

Expanding:

36x - 270 = x^2 - 15x

Rearranging to form a quadratic equation:

x^2 - 15x - 36x + 270 = 0

x^2 - 51x + 270 = 0

Now, we can solve this quadratic equation using factoring or the quadratic formula. Factoring this equation:

(x - 6)(x - 45) = 0

Setting each factor equal to zero:

x - 6 = 0 or x - 45 = 0

Solving for x:

x = 6 or x = 45

Now, we have the possible values for x, which represents the number of days it would take Yuuko alone to build the shed. So, it would take Yuuko either 6 or 45 days to complete the shed alone.

To find out how long it would take Mio alone, we can substitute these values into the equation (x - 15):

For x = 6: Mio = 6 - 15 = -9 (not possible, since time cannot be negative)
For x = 45: Mio = 45 - 15 = 30

Therefore, it would take Yuuko 45 days to build the shed alone, and it would take Mio 30 days to build the shed alone.

2) Let's assume that Simon can do the job in y hours.

According to the problem, Kamina can do the job in 29 hours and Simon and Kamina working together can do the same job in 10 hours.

Let's set up an equation based on their individual rates of work:

1/(29) + 1/(y) = 1/(10)

To solve this equation, we need to find a common denominator and then combine the fractions:

[(10)(y) + 29(y)] / (29)(y) = 1/10

Simplifying further:

(10y + 29y) / (29y) = 1/10

Combining like terms:

39y / (29y) = 1/10

Cross-multiplying:

(29y)(1) = (39y)(10)

Expanding:

29y = 390y

Subtracting 390y from both sides:

29y - 390y = 0

-361y = 0

Dividing by -361:

y = 0

However, this does not make sense, as time cannot be zero.

Hence, there is no possible answer for Simon's individual time.

To solve these problems, we can use the concept of work rate. The work rate is a measure of how much work can be completed in a unit of time. If we denote the work rate of Mio as "M" and the work rate of Yuuko as "Y," we can set up the following equations:

1) Mio takes 15 days less than Yuuko to build the shed alone. So, we can say that M = Y - 15.

2) When Mio and Yuuko work together, they can finish the job in 18 days. This means that their combined work rate is equal to 1/18 of the job per day. Mathematically, we can express this as 1/M + 1/Y = 1/18.

Now, let's solve these equations to find the time it takes each person to complete the job alone.

1) Substituting M = Y - 15 into the equation 1/M + 1/Y = 1/18, we get 1/(Y - 15) + 1/Y = 1/18.

To simplify this equation, we can multiply both sides by (Y - 15) * Y * 18 to eliminate the denominators.

(Y * 18) + (Y - 15) * 18 = Y * (Y - 15).

Simplifying further, we have 18Y + 18(Y - 15) = Y^2 - 15Y.

Expanding and rearranging, we get Y^2 - 15Y - 36Y + 270 = 0.

Combining like terms, we have Y^2 - 51Y + 270 = 0.

Now, we can solve this quadratic equation. Factoring or using the quadratic formula will give us the two possible values for Y.

2) To find how long it takes Simon to do the job alone, let's denote Simon's work rate as "S." Since Kamina can do the job in 29 hours, we can say that S = 1/29.

When Simon and Kamina work together, their combined work rate is equal to 1/10 of the job per hour. Hence, 1/S + 1/29 = 1/10.

Substituting S = 1/29 into the equation, we get 1/(1/29) + 1/29 = 1/10.

Simplifying, we have 29 + 1/29 = 1/10.

To solve this equation, we can multiply both sides by 290 to eliminate the denominators.

290 * (29 + 1/29) = 29 * (1/10).

Simplifying further, we get 290 * 29 + 290 * (1/29) = 29/10.

Now, we can calculate this expression to find the time it takes Simon to do the job alone.

By following the steps above, you can find the solution to both problems.