At what rate a sum of money will become four times of itself in 2 years if the interest compounded half yearly?

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Let's discuss the concepts related to Interest and Compound Interest. Explore more from Quantitative Aptitude here. Learn now!

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

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Practice Test on Compound Interest

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Worksheet on Compound Interest with Periodic Deductions

8th Grade Math Practice 

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10 Qs. 10 Marks 10 Mins

Given:

Sum of money becomes 4 times after 2 years at CI.

Formula used:

For CI,

A = P{1 + (R/100)}n

where,

P = Principal

A = Amount

R = Rate

n = time

Calculations:

Let the sum be P.

⇒ 4P = P{1 + (R/100)}2

⇒ 4 = {1 + (R/100)}2

⇒ √4 = {1 + (R/100)}

⇒ 2 = {1 + (R/100)}

⇒ 2 - 1 = (R/100)

⇒ 1 = R/100

⇒ R = 100%

∴ The rate of interest is 100%.

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Find the rate at which a sum of money will become four times the original amount in 2 years, if the interest is compounded half-yearly.

Let the rate percent per annum be R.Then, \[A = P \left( 1 + R \right)^{2n} \]\[4P = P \left( 1 + \frac{R}{200} \right)^4 \]\[ \left( 1 + \frac{R}{200} \right)^4 = 4\]\[\left( 1 + \frac{R}{200} \right) = 1 . 4142\]\[\frac{R}{200} = 0 . 4142\]R = 82 . 84

Thus, the required rate is 82 . 84 %.

Concept: Rate Compounded Annually Or Half Yearly (Semi Annually)

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Find the rate at which a sum of money will become four times the original amount in 2 years, if the interest is compounded half yearly.

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