Two women Renu and Ushi are working on an embroidery design. If Ushi worked alone, she would need eight hours more to complete the design than if they both worked together. Now if Renu worked alone, it would need 4.5 hours more to complete the design than they both working together. What time would it take Renu alone to complete the design?

Let's denote:

• the time it would take for Renu to complete the design alone as R,
• the time it would take for Ushi to complete the design alone as U,
• the time it would take for both of them to complete the design together as T.

From the problem, we have two equations:

1. U = T + 8 (since Ushi would need eight hours more to complete the design than if they both worked together)
2. R = T + 4.5 (since Renu would need 4.5 hours more to complete the design than if they both worked together)

We also know that the rate of work done is the reciprocal of the time taken to complete the work. So, if they are working together, their rates of work add up. Therefore, we have:

1/R + 1/U = 1/T

Substituting the first two equations into this one, we get:

1/(T + 4.5) + 1/(T + 8) = 1/T

This is a rational equation, and we can solve it by multiplying every term by T*(T + 4.5)*(T + 8) to get rid of the denominators:

T*(T + 8) + T*(T + 4.5) = (T + 4.5)*(T + 8)

Expanding and simplifying, we get:

T^2 + 8T + T^2 + 4.5T = T^2 + 12.5T + 36

Combining like terms, we get:

T^2 - 0.5T - 36 = 0

This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve it using the quadratic formula:

T = [-(-0.5) ± sqrt((-0.5)^2 - 41(-36))]/(2*1)

Solving this, we get two solutions: T = 6 and T = -6. Since time cannot be negative, we discard T = -6.

So, T = 6 hours.

Substituting T = 6 into the second equation, we get:

R = 6 + 4.5 = 10.5 hours.

So, it would take Renu 10.5 hours to complete the design alone.