Rs. 15,000 will amount to Rs. 19,965 in 15 months at ______ per annum and the compund interest is calculated on every 5 months.

Given:

Principal = Rs. 15,000

Amount = Rs. 19,965

Time = 15 months

Condition = on every 5 months

Concept used:

Condition = on every 5 months

New rate = Rate × 5/12

New time = Time × 12/5

Calculations:

Let the new rate be R%

According to the question,

New time = Time × 12/5

⇒ 15 × 12/5 = 36 months = 3 years

Simplifying the values by dividing it by 15 to its lowest possible values, we get Principal = 1000 and Amount = 1331

Now, new time period is 3 years, hence taking the cube roots of Principal and Amount,

⇒ R = 10%

New rate = Rate × 5/12

⇒ 10 = Rate × 5/12

⇒ Rate = (10 × 12)/5

⇒ Rate = 24%

∴ Rate is 24% per annum.

Alternate MethodGiven:

Principal = Rs. 15,000

Amount = Rs. 19,965

Time = 15 months

Condition = on every 5 months

Concept used:

Condition = on every 5 months

New rate = Rate × 5/12

New time = Time × 12/5

Formulae used:

(1) Effective rate for 3 years = 3R + 3R²/100 + R³/10000

(2) A = P(1 + R/100)^T

Where, A → Amount

P → Principal

R → Rate of interest

T → Time

Calculations:

According to the question,

Let the new rate be R%

New time = Time × 12/5

⇒ 15 × 12/5 = 36 months = 3 years

Amount = P(1 + R/100)^T

⇒ 19,965 = 15,000(1 + R/100)³

⇒ 19,965/15,000 = (1 + R/100)³

⇒ 1331/1000 = (1 + R/100)³

⇒ (11/10)³ = (1 + R/100)³

⇒ 11/10 = 1 + R/100

⇒ (11/10) – 1 = R/100

⇒ 1/10 = R/100

⇒ R = 10%

New rate = Rate × 5/12

⇒ 10 = Rate × 5/12

⇒ Rate = (10 × 12)/5

⇒ Rate = 24%

∴ Rate is 24% per annum

Additional InformationCompound Interest means interest earned on interest. Simple interest always occurs on only principal but compound interest also occurs on simple interest. So, if time period is 2 years, compound interest will also apply on simple interest of first year.