Solution: (c)

Principal(P)=Rs. 6000

Time(t) = 1\(\displaystyle \frac{1}{2}\) = \(\displaystyle \frac{3}{2}\)  years

Rate(r) =10%

Amount= \(\displaystyle =P\times {{\left( {1+\frac{r}{{100}}} \right)}^{n}}\)

Hence, for 1 year R = 10% and n = 1

Amount= \(\displaystyle =P\times {{\left( {1+\frac{r}{{100}}} \right)}^{n}}\)

\(\displaystyle \begin{array}{l}=6000\times {{\left( {1+\frac{{10}}{{100}}} \right)}^{1}}\\=6000\times \left( {\frac{{11}}{{10}}} \right)\\=Rs.6600\end{array}\)

Now, for the remaining 1/2 year P = 6600, R = 5%

Amount= \(\displaystyle =P\times {{\left( {1+\frac{r}{{100}}} \right)}^{n}}\)

\(\displaystyle \begin{array}{l}=6600\times {{\left( {1+\frac{5}{{100}}} \right)}^{1}}\\=6600\times \left( {\frac{{105}}{{100}}} \right)\\=Rs.6930\end{array}\)

C.I.= Amount-Principal

= Rs. 6930 – 6000= 930

Alternate method

Principal(P)=Rs. 6000

Time(t) = 1\(\displaystyle \frac{1}{2}\) = \(\displaystyle \frac{3}{2}\)  years

Rate(r) =10%

2nd year CI = 660

6 months 2nd year CI = 660/2 = 330

Total CI = (600+330)=Rs. 930

Solution: (b)
CI = \(\displaystyle P[{{(1+\frac{R}{{100}})}^{T}}-1]\)
\(\displaystyle \Rightarrow 1411.4=P[{{(1+\frac{8}{{100}})}^{2}}-1]\)
\(\displaystyle \Rightarrow 1411.4=P\times 0.1664\)
\(\displaystyle \Rightarrow \)P = \(\displaystyle \frac{{141434}}{{0.1664}}=8500\)
Therefore, Amount = ₹ (8500 + 1414.4)
= ₹ 9914.4

Solution: (b)
\(\displaystyle \begin{array}{l}Amount=1600\times {{\left( {1+\frac{5}{{2\times 100}}} \right)}^{2}}+1600\times \left( {1+\frac{5}{{2\times 100}}} \right)\\=1600\times \frac{{41}}{{40}}\times \frac{{41}}{{40}}+1600\times \frac{{41}}{{40}}\\=1600\times \frac{{41}}{{40}}\left( {\frac{{41}}{{40}}+1} \right)\\=\frac{{1600\times 41\times 81}}{{40\times 40}}\\=Rs.3321\\CI=Rs(321-3200)=Rs.121\end{array}\)

Solution: (c)

Rate % = 16%,

Time = 1 year

Case (I) : When interest is calculated yearly, Rate = 16%

Case (II) : When interest is calculated half yearly

⇒New rate %=16/2=8%

Time = 1×2=2 years

Effective rate% = \(\displaystyle 8+8+\frac{{8\times 8}}{{100}}\)=16.64%

Difference in rates = (16.64−16)%=0.64%

As per the question,

0.64% of sum = Rs 56

Sum = \(\displaystyle \frac{{56}}{{0.64}}\times 100\)=Rs. 8750

Alternate method

N=1year

R=10 %

We have SI\(\displaystyle =\frac{{PRT}}{{100}}\)=\(\displaystyle =\frac{{P\times 1\times 16}}{{100}}\)=0.16P

When interest being compounded for half yearly, for 1 year

We have, N=2

And R=16/2 =8 %

And Amount= \(\displaystyle =P\times {{\left( {1+\frac{r}{{100}}} \right)}^{n}}\)

\(\displaystyle \begin{array}{l}=P\times {{\left( {1+\frac{8}{{100}}} \right)}^{2}}\\=P\times {{\left( {1+0.08} \right)}^{2}}\\=P\times {{\left( {1.08} \right)}^{2}}\end{array}\)

Amount=1.1664P

And C.I.=A−P=1.1664P−P=0.1664P

Given, C.I.−S.I.=Rs. 56

\(\displaystyle \Rightarrow \)0.1664P −0.16P=56

\(\displaystyle \Rightarrow \)0.0064P=56

\(\displaystyle \Rightarrow \)P=56/0.0064=Rs. 8750

One more Alternate method

When the money is compounded half yearly the effective rate of interest for 6 months =\(\displaystyle \frac{16}{2}\) =8%

=\(\displaystyle \frac{8}{100}\)= \(\displaystyle \frac{2}{25}\)

Let principal = \(\displaystyle {{\left( {25} \right)}^{2}}\)= 625

\(\displaystyle \begin{array}{l}\Rightarrow 4units\to 56\\\Rightarrow 1units\to 14\\\Rightarrow \Pr incipal=14\times 625\\=Rs.8750\end{array}\)

CI for 1st year

\(\displaystyle \begin{array}{l}=Rs.\left[ {5000\left( {1+\frac{5}{{100}}} \right)-5000} \right]\\=Rs.\left[ {5000\left( {\frac{{21}}{{20}}} \right)-5000} \right]\end{array}\)

=Rs. (5250 – 5000)

=Rs. 250

Amount after 1st year

=Rs. (5250−20% of 250)

=Rs. (5250−50)

= Rs. 5200

CI after 2nd year

\(\displaystyle \begin{array}{l}=Rs.\left[ {5200\left( {1+\frac{5}{{100}}} \right)-5200} \right]\\=Rs.\left[ {5200\left( {\frac{{21}}{{20}}} \right)-5200} \right]\end{array}\)

=Rs. (5460 – 5200)

=Rs. 260

Amount after 2nd year

= Rs. (5460−20% of 260)

= Rs. (5460−52)

=Rs. 5408

CI after 3rd year

\(\displaystyle \begin{array}{l}=Rs.\left[ {5408\left( {1+\frac{5}{{100}}} \right)- 5408} \right]\\=Rs.\left[ {5408\left( {\frac{{21}}{{20}}} \right)- 5408} \right]\end{array}\)

=Rs.(5678.40 – 5408)

=Rs. 270.40

Amount after 3rd year

=Rs. (5678.40−20%of 270.40)

=Rs. (5678.40−54.08)

=Rs. 5624.32

Alternate shortcut method

5% is the interest rate.

20% of the interest amount is paid as tax

i:e 80% of the interest amount stays back

If we compute the rate of interest as 80% of 5%=4% p.a., we will get the dame value

The interest occurred for 3 years in compound interest= 3 x simple interest on principal + 3 x interest on simple interest + 1 x interest on the interest 

= \(\displaystyle =3\times (200)+3\times (8)+1\times (0.32)\)

=600+24+0.32 = 624.32

Amount at the end of 3 years=5000+624.32=5624.32

Solution: (a)
CI = \(\displaystyle p[{{(1+\frac{R}{{100}})}^{T}}-1]\)
=\(\displaystyle 53000[{{(1+\frac{4}{{100}})}^{2}}-1]\)
=\(\displaystyle 53000[{{(\frac{{26}}{{25}})}^{2}}-1]\)
=\(\displaystyle 53000[\frac{{676}}{{625}}-1]\)
=\(\displaystyle \frac{{53000\times 51}}{{625}}=4324.8\)