Question

find the effective rate of interest is paid at 8%p.a. and its compounded 1) Half yearly 2)quarterly​

Answer 1

Find Compound Interest when interest is compounded Half yearly

To find compound interest when interest is compounded half yearly, we use the following formula. 

A = P ( 1 + r/2)2n and C.I = A - P 

where, P = principal 
R = rate in percent p.a.( per annum i.e. annually)
n = number of years.

Examples : 

1) Compute the compound interest on $12,000 for 2 years ate 20% p.a. when compounded half-yearly.

Solution : 
Here, P = $12,000, R = 20% and n = 2 years.

Amount after 2 years = P ( 1 + r/2)2n 

= 12,000 ( 1 + 0.20/2)2 x 2

= 12,000 ( 1 + 0.1)4

= 12,000 (1.1)4

= 12,000 x 1.4641 

Amount = $ 17569.20 

∴ C.I = A - P 

C.I = 17569.20 - 12000

C.I = $ 5569.20 
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2) Find the compound interest on $1,000 at the rate of 10% p.a. for 18 months when interest is compounded half-yearly.

Solution : 
Here, P = $1,000, R = 10% and n = 18 months = 18/12 = 3/2 years.

Amount after 2 years = P ( 1 + r/2)2n 

= 1,000 ( 1 + 0.10 /2)2 x 3/2

= 1,000 ( 1 + 0.05)6/2

= 1,000 (1.05)3

= 1,000 x 1.157625 

Amount = $ 1157.625 

∴ C.I = A - P 

C.I = 1157.625 - 1000

C.I = $ 157.625 
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3) What sum will become $9,724.05 in 2 years, if the rate of interest is 10% compounded half yearly? 

Solution :
Here, A = $9,724.05, R = 10% and n = 2years , P =?.

Amount after 2 years = P ( 1 + r/2)2n 

9,724.05= P ( 1 + 0.10 /2)2 x 2

9,724.05= P ( 1 + 0.05)4

9,724.05= P (1.05)4

9,724.05= P x 1.21550625 

P = 9,724.05/1.21550625 P = $ 8000 


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Answer 2

Concept

A loan or deposit's interest is computed using both the initial principle and the accrued interest from prior periods. This is known as compound interest, or compounding interest. Compound interest, sometimes known as "interest on interest," is said to have its origins in Italy in the $17$th century. It will cause money to increase more quickly than basic interest, which just considers the principal.

Given

Rate of interest is paid at $8$%p.a.

To Find

We have to find the effective rate of interest compounded half yearly and annually.

Solution

According to the problem,

For half yearly:

Rate of interest is = $4+4*(1+4/100)^{1} =4+4*1.04=4+4.16=8.16$%

For quarterly rate pf interest is = $2+2*(1+2/100)+2.04*(1+2/100)+2.08*(1+2/100)$

$=2+2.04+2.08+2.12=8.24$%

Hence, rate of interest half yearly is $8.16$% and quarterly is $8.24$%.

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