Two equal sums were lent at 5% and 6% per annum.

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Two equal sums were lent at 5% and 6% per annum compound interest for 2 years. If the difference in the compound interest was ₹ 422, find:
(i) the equal sums
(ii) compound interest for each sum.

Solution:

Consider ₹ 100 as each equal sum
Case I –
Rate (r) = 5%
Period (n) = 2 years
We know that
A = P (1 + r/100)n
Substituting the values
= 100 (1 + 5/100)2
It can be written as
= 100 × 21/20 × 21/20
= ₹ 441/4
Here
CI = A – P
Substituting the values
= 441/4 – 100
= ₹ 41/4
Case II –
Rate of interest (R) = 6n
Period (n) = 2 years
We know that
A = P (1 + r/100)n
Substituting the values
= 100 (1 + 6/100)2
It can be written as
= 100 × 53/50 × 53/50
= ₹ 2809/25
Here
CI = A – P
Substituting the values
= 2809/25 – 100
= ₹ 309/25
So the difference between the two interests = 309/25 – 41/4
Taking LCM
= (1236 – 1025)/ 100
= ₹ 211/100
If the difference is ₹ 211/100, then equal sum = ₹ 100
If the difference is ₹ 422, then equal sum = (100 × 422 × 100)/ 211 = ₹ 20000
Here, Amount in first case = 20000 (1 + 5/100)2
So we get
= 20000 × (21/20)2
It can be written as
= 20000 × 21/20 × 21/20
So we get
= 44100/2
= ₹ 22050
CI = 22050 – 20000 = ₹ 2050
Amount in second case = 20000 (1 + 6/100)2
It can be written as
= 20000 × 53/50 × 53/50
= ₹ 22472
CI = 22472 – 20000 = ₹ 2472

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