Compute the compound interest on Rs 4000 for 1½ years at 10% per annum compounded half yearly

For 1st half-year : P = Rs.4000, R = 14% and T = `(1)/(2)`year

Interest = Rs.`(4000 xx 14 xx 1)/(100 xx 12)`
 = Rs.280
And, amount
= Rs.4000 + Rs.280
= Rs.4280
For 2nd half-year : P = Rs.4280, R = 14% and T = `(1)/(2)`year

Interest = Rs.`(4280 xx 14 xx 1)/(100 xx 2)`
= Rs.299.60
And, amount
= Rs.4280 + Rs.299.60
= Rs.4579.60
For 3rd half-year : P = Rs.4579.60, R = 14% and T = `(1)/(2)`year

Interest = Rs.`(4579.60 xx 14 xx 1)/(100 xx 2)`
= Rs.320.572
And, amount
= Rs.4579.60 + Rs.320.572
= Rs.4900.172
Thus, the amount to be paid at the end of 1`(1)/(2)`years is Rs.4900.172.

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

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