# Compound Interest when Interest is Compounded Half-Yearly

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

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