62.5% is 5/8, so we can let S=Stanley members and B=Boyd members. The total in the team is T. S+B=T and S=5T/8, so B=3T/8.
If the team takes N new members then the percentage of Stanley members is S/(T+N)=55/100=5T/(8(T+N))=55/100.
Divide both sides by 5: T/8T+8N=11/100. Cross-multiply: 100T=88T+88N. Divide through by 4: 25T=22T+22N.
N=3T/22. N must be a whole number, so T=22 is the lowest value and N=3. However, if T=22, it exceeds 10 and 5/8 of 22=55/4 which is not a whole number, so the lowest value for T would need to be 88, made up of 55 Stanley members, and N=12, making the new team membership 100, with 55 Stanley members.