this is a nice problem on simple interest
Let the rate of interest be R% per annum
Total Amount (if paid at the end of 11 months) = P + {P*R*T/100}
= 10 + {10*R*(11/12)}/100 = 10 +11*R/120..............................(1)
Total effective payment
= (Rs.1 + interest on Rs.1 for 10 months) + (Rs.1 + interest on Rs.1 for 9 months) + ... + (Rs.1 + interest on Rs.1 for 1 month) + Rs.1
= [1 + {1*R*(10/12)}/100] + [1 + {1*R*(9/12)}/100] + .... ...+ [1 + {1*R*(1/12)}/100] + 1
=(1 + 10*R/1200) + (1 + 9*R/1200) +............+ (1 + R/1200) + 1
=11 + R(1+2+⋯+10) / 1200
= 11 + R * (10*11/2) / 1200
11 + 11*R/240................................................................(2)
so Now we have
from (1) and (2)
10+{11*R/120}=11+{11*R/240}
here we get R = 240 /11 = 21.81 %