#### 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) = 6^{n}

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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