ratio

Dan and Paul share some money in the ratio 13:5. 
Dan decides this is unfair so he gives Paul £32 of his share to make the ratio 1:1. 
How much did Paul originally have?

Call Paul's original money, P

Call Dan's original money (13/5)P

So, when Dan gives  Paul £32 their money is equal...so we have

(13/5)P  -  32  = P + 32        add 32 to both sides

(13/5)P  =  P +  64        subtract  P from both sides

(8/5)P  =   64         multiply both sides by (5/8)

P =  64 (5/8)   =  £40

Proof

Paul starts with   £40

Dan starts with (13/5) * 40  =  £104

Paul  ends up with 40 + 32  = £72

Dan ends up with  104 - 32  = £72

So....each has equal money, i.e., a 1: 1 ratio