The lengths of 3 pieces of wire are in the ratio of wire are in the ratio 10: 15: 8 if the length of the shortest piece of wire is 2.4m find the difference between the length of the other two

You will be using proportions to solve this.
The first proportion will be 8/2.4 = 15/x
Here the ratio given will be in the numerator and the meters of the actual wires will be in the denominator. This proportion will help you find the longest wire's length.
To continue, do cross multiplication: 8x=36
Then divide by 8: x= 4.5, OR the longest wire has a length of 4.5m.
To find the middle-length wire, the proportion will almost be the same but now the 15 will be replaced by 10. 
8/2.4 = 10/x     Then the same steps
8x= 24
x= 3, OR the middle-length wire has a length of 3m.

In conclusion, the corresponding ratio to 10:15:8 would be 3:4.5:2.4