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Simple Interest #13

13. An automobile financier claims to be lending money at simple interest, but he includes the interest every six months for calculating the principal. If he is charging an interest of $10\%,$ the effective rate of interest after one year becomes:
A. $10.5\%$B. $10.25\%$
C. $10\%$D. None of these

Answer: Option B

Explanation:

Solution 1

Suppose the automobile financier lends $100$

Simple interest for first $6$ months $=5$
hint

For $1$ year, simple interest is $10\%$ of $100,$ which is $10$ and for $6$ months, simple interest is half of $10,$ which is $5.$

Using formula, we get the same as $\dfrac{100×10×\dfrac{1}{2}}{100}=5$

After $6$ months, he adds the simple interest to principal. Therefore, principal after $6$ months $=100+5=105$

Simple interest for next $6$ months $=5.25$
hint

For $1$ year, simple interest is $10\%$ of $105,$ which is $10.5$ and for $6$ months, simple interest is half of $10.5$ which is $5.25$

Using formula, we get the same as $\dfrac{105×10×\dfrac{1}{2}}{100}=5.25$

Total simple interest for $1$ year $=5+5.25=10.25$

Hence, effective rate of interest $=10.25\%$
hint

$100$ gives $10.25$ as simple interest for $1$ year and therefore interest rate is $10.25\%.$

Using formula, we get the same as $\dfrac{100×10.25}{100×1}=10.25$

Solution 2

Suppose the automobile financier lends $100$

Simple interest for first $6$ months $=5$

So, additional amount he gets (when compared to the normal simple interest on $₹100$ for $1$ year at $10\%$) is the simple interest on this $5$ for next $6$ months at $10\%$

simple interest on $5$ for $6$ months at $10\%$
$\equiv$ simple interest on $5$ for $1$ year at $5\%$
$\equiv$ simple interest on $100$ for $1$ year at $\dfrac{5}{20}\%$

So, effective rate of interest
$=10\%+\dfrac{5}{20}\%=10\%+0.25\%=10.25\%$

Solution 3

Refer formula

Suppose the automobile financier lends $100$

Amount after $1$ year $=100\left(1+\dfrac{10/2}{100}\right)^{2×1}=100\left(\dfrac{21}{20}\right)^2=110.25$

Total simple interest for $1$ year $=110.25-100=10.25$

Therefore, effective rate of interest $=10.25\%$

Comments(2)

profileGautam
2015-03-22 14:55:49 
The explanation is incorrect.

Actually it is a CI problem.
Let actual rate be r. 
Effective rate be R. 
and time period be n years.Then

SI = CI -P
P*R*n/100 = P(1+r/200)^2n - P

P*R*n/100 = P[(1+10/200)^2n - 1]

R = [(1.05) ^ 2n  - 1] * 100 /n

Now if n=1 then only R =10.25. Clearly result depends on the time period.

So, correct answer is (d) None of these
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profileSupport, careerbless.com
2015-04-05 17:58:03 
Dear Gautam, you are right. The question was supposed to be "the effective rate of interest after one year" which is corrected now. Thanks.
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