What is the smallest length of a string from which you can cut equal pieces of 25cm,40cm or 50cm each

We are essentially looking for the LCM (least common multiple) of the lengths 25, 40 and 50 cm.

Since 50 is already a common multiple of 25, since 2*25=50, we only need to look for the LCM of 40 and 50.

One simple way to find the LCM is to factor both numbers,
40=2^3*5
50=2*5^2
and look for the product of the higher power of each factor, which in this case is 2^3*5^2=8*25=200.

Another way, looks round about in this case, is to look for the HCF, which is the product of the lowest power of each factor, which in this case is 2*5=10.
The LCM is then the product of the two numbers divided by the HCF, or
LCM=50*40/10=200, again.