Is the difference between compound interest on Rs 1000 for 1 year at 20% per annum when compound half yearly and quarterly?

We will learn how to use the formula for calculating the compound interest when interest is compounded half-yearly.

Computation of compound interest by using growing principal becomes lengthy and complicated when the period is long. If the rate of interest is annual and the interest is compounded half-yearly (i.e., 6 months or, 2 times in a year) then the number of years (n) is doubled (i.e., made 2n) and the rate of annual interest (r) is halved (i.e., made \(\frac{r}{2}\)).  In such cases we use the following formula for compound interest when the interest is calculated half-yearly.

If the principal = P, rate of interest per unit time = \(\frac{r}{2}\)%, number of units of time = 2n, the amount = A and the compound interest = CI

Then

A = P(1 + \(\frac{\frac{r}{2}}{100}\))\(^{2n}\)

Here, the rate percent is divided by 2 and the number of years is multiplied by 2

Therefore,  CI = A - P = P{(1 + \(\frac{\frac{r}{2}}{100}\))\(^{2n}\) - 1}

Note:

A = P(1 + \(\frac{\frac{r}{2}}{100}\))\(^{2n}\) is the relation among the four quantities P, r, n and A.

Given any three of these, the fourth can be found from this formula.

CI = A - P = P{(1 + \(\frac{\frac{r}{2}}{100}\))\(^{2n}\) - 1} is the relation among the four quantities P, r, n and CI.

Given any three of these, the fourth can be found from this formula.

Word problems on compound interest when interest is compounded half-yearly:

1. Find the amount and the compound interest on $ 8,000 at 10 % per annum for 1\(\frac{1}{2}\) years if the interest is compounded half-yearly.

Solution:

Here, the interest is compounded half-yearly. So,

Principal (P) = $ 8,000

Number of years (n) = 1\(\frac{1}{2}\) × 2 = \(\frac{3}{2}\) × 2 = 3

Rate of interest compounded half-yearly (r) = \(\frac{10}{2}\)% = 5%

Now, A = P (1 + \(\frac{r}{100}\))\(^{n}\)

⟹ A = $ 8,000(1 + \(\frac{5}{100}\))\(^{3}\)

⟹ A = $ 8,000(1 + \(\frac{1}{20}\))\(^{3}\)

⟹ A = $ 8,000 × (\(\frac{21}{20}\))\(^{3}\)

⟹ A = $ 8,000 × \(\frac{9261}{8000}\)

⟹ A = $ 9,261 and

Compound interest = Amount - Principal

                          = $ 9,261 - $ 8,000

                          = $ 1,261

Therefore, the amount is $ 9,261 and the compound interest is $ 1,261

2. Find the amount and the compound interest on $ 4,000 is 1\(\frac{1}{2}\) years at 10 % per annum compounded half-yearly.

Solution:

Here, the interest is compounded half-yearly. So,

Principal (P) = $ 4,000

Number of years (n) = 1\(\frac{1}{2}\) × 2 = \(\frac{3}{2}\) × 2 = 3

Rate of interest compounded half-yearly (r) = \(\frac{10}{2}\)% = 5%

Now, A = P (1 + \(\frac{r}{100}\))\(^{n}\)

⟹ A = $ 4,000(1 + \(\frac{5}{100}\))\(^{3}\)

⟹ A = $ 4,000(1 + \(\frac{1}{20}\))\(^{3}\)

⟹ A = $ 4,000 × (\(\frac{21}{20}\))\(^{3}\)

⟹ A = $ 4,000 × \(\frac{9261}{8000}\)

⟹ A = $ 4,630.50 and

Compound interest = Amount - Principal

                          = $ 4,630.50 - $ 4,000

                          = $ 630.50

Therefore, the amount is $ 4,630.50 and the compound interest is $ 630.50

● Compound Interest

Compound Interest

Compound Interest with Growing Principal

Compound Interest with Periodic Deductions

Compound Interest by Using Formula

Compound Interest when Interest is Compounded Yearly

Problems on Compound Interest

Variable Rate of Compound Interest

Practice Test on Compound Interest

● Compound Interest - Worksheet

Worksheet on Compound Interest

Worksheet on Compound Interest with Growing Principal

Worksheet on Compound Interest with Periodic Deductions

8th Grade Math Practice 

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Given:

Principal amount = Rs. 12000

Time = 18 months

Rate of interest = 20%

Concept used:

Simple interest = (Principal × Time × Rate) ÷ 100

In the case of compound interest,

A = P × (1 + r/100)n

C.I = A - P

Here, A = Amount, P = Principal, r = Rate of interest. n = Time, C.I = Compound interest

Calculation:

When interest is compounded yearly,

Principal = Rs. 12,000

At the end of the first year, the principal becomes = 12000 × 1.21 = Rs. 14400

Then, after 6 months the interest accumulated on the principal amount of Rs. 14400

⇒ (14400 × 20 × 6/12) / 100

⇒ 1440

So, after 8 months the initial principal amount becomes = 14400 + 1440 = Rs. 15840

Interest accumuted in 18 months = Rs. (15840 - 12000) = Rs. 3840

When interest compounded half-yearly,

Rate of interest (R) = 20/2 % = 10%

Time (n) = 18 ÷ 6 = 3

Principal = 12,000

So, after 18 months the principal becomes

⇒ 12000 (1 + 10/100)3

⇒ Rs. 15972

Interest accumuted in 18 months = Rs. (15972 - 12000) = Rs. 3972

So, the difference between the interest accumulated = Rs. (3972 - 3840) = Rs. 132

∴ The difference between the compound interest compounded yearly and compounded half yearly for 18 months at 20% per annum on a sum of Rs. 12,000 is Rs. 132.

Alternate Method Formulae used:

CIhalf yearly – CIyearly = P[(1 + (r/200))2n – (1 + (r/100))n]

CIhalf yearly – CIyearly = 12000[(1 + (20/200))2(1.5) – (1 + (20/100))1.5]

CIhalf yearly – CIyearly = 12000 [(1 + (1/10))3– (1 + (1/5))1.5]

CIhalf yearly – CIyearly = 132