When a piece of wire of resistance of resistance r is cut into five equal parts and those parts are connected in parallel, the equivalent resistance (r′) of the combination can be calculated as follows:

For n resistors in parallel with resistance R, the formula for equivalent resistance (R_eq​) is given by:

1/R_eq=1/R₁+1/R₂+…+1/R_n​

In this case, n=5 (since there are five equal parts) and R=r/5​ (since each part has resistance r/5​).

So, for the five equal parts in parallel:

1/R_eq=1/(r/5)+1/(r/5)+1/(r/5)+1/(r/5)+1/(r/5)

Simplify:

1/R_eq=5⋅(1/(r/5))

1/R_eq=5/(r/5)

1/R_eq=25/r

Now, take the reciprocal of both sides to find Req​:

R_eq=r/25

The ratio of r to r′ is:

r/r^' = r/(r/25)

r/r' = 25.