C.I –S.I = P\({ \left( \frac { r }{ 100 } \right) }^{ 2 }\)= 5000 x \(\frac { 4 }{ 100 } \times \frac { 4 }{ 100 } \) = Rs 8
(ii) C.I –S.I = P\({ \left( \frac { r }{ 100 } \right) }^{ 2 }\left( 2+\frac { r }{ 100 } \right) \)
= 8000 x \(\frac { 5 }{ 100 } \times \frac { 5 }{ 100 } \) x \(\left( 3+\frac { 5 }{ 100 } \right) \)
= 20 x \(\frac { 61 }{ 20 } \) = Rs.61

Let the population in 2016 be ‘P’.
Then, A = P\({ \left( 1+\frac { r }{ 100 } \right) }^{ n }\)
⇒ 238765 = p\({ \left( 1+\frac { 6 }{ 100 } \right) }^{ 2 }=P{ \left( \frac { 53 }{ 50 } \right) }^{ 2 }\)
⇒ P = 238765 x \(\frac { 50 }{ 53 } \times \frac { 50 }{ 53 } \)
∴ P = 212500
Let the population in 2020 be ‘A’
Then, A= P\({ \left( 1+\frac { r }{ 100 } \right) }^{ n }\)
∴ A = 238765\({ \left( 1+\frac { 6 }{ 100 } \right) }^{ 2 }\)
= 238765 x \(\frac { 53 }{ 50 } \times \frac { 53 }{ 50 } \)
= 95.506 x 53 x 53
A = 268276
∴ The population in the year 2016 is 212500 and that in the year 2020 is 268276.

Increase in population = 25000 − 20000
= 5000
∴ Percentage increase in population =\(\frac { 5000 }{ 20000 } \)x 100
= 25%

Let his income be rs x.
Income after 10% increase is
\(100+100 \times \frac{10}{100}=Rs. 110\)
Now, income after 10% decrease is
\(110-110 \times \frac{10}{100}=110-11=Rs. 99\)
Net change in his income = 100 – 99 = 1
Percentage change \(=\frac{1}{100} \times 100 \%=1 \%\)
That is, income of the person is reduced by 1%.
Aliter
Let his income be Rs. 100
Income after 10% increase is
100 + 100 × \(\frac{10}{100}\) = Rs.110
Now, income after 10% decrease is,
110 – 110 × \(\frac{10}{100}\) = 110 – 11 = Rs. 99
∴ Net change in his income = 100 – 99 = 1
Percentage change = × = \(\frac{1 }{100}\) × 100% = 1% 
That is, income of the person is reduced by 1%.

Number of students who did not pass = 30% of boys + 15% of girls
=\(\frac { 30 }{ 100 } \) x 900 + \(\frac { 15 }{ 100 } \) x 600
= 270 + 90 = 360
∴ Percentage of students who did not pass =\(\frac { 360 }{ 1500 } \) x 100 = 24%

Let x be the cost price of the article.
810 - x = x - 530
2x = 810 + 530
2x = 1340
x = 670
The cost price of the article is Rs. 670.

Let the numerator be x and the denominator be y.
The original fraction = x/y
Given the numerator of a fraction is increased by 50%
\( \therefore \mathrm{Nr} =x+50 \% \text { of } x \)
\(=x+\frac{50}{100} x=x+\frac{1}{2} x \\ \mathrm{Nr} =\frac{3}{2} x \)
Also given the denominator is decreased by 20 %
\( \text { Dr } =y-20 \% \text { of } y \)
\(=y-\frac{20}{100} y=y-\frac{1}{5} y=\frac{4}{5} y \)
By given data
\( \frac{\frac{3}{2} x}{\frac{4}{5} y}=\frac{3}{5} \Rightarrow \frac{3}{2} x \times \frac{5}{4 y}=\frac{3}{5} \)
\(\frac{x}{y}=\frac{8}{25} \)
The original fraction is 8/25

Let x and y be the increased and decreased percentage
The change in that number \(=\left(x+y+\frac{x y}{100}\right) \%\)
\( =\left(25+(-20)+\frac{25(-20)}{100}\right) \% \)
\(=\left(25-20-\frac{500}{100}\right) \)
\(=(25-20-5) \% \)
= 0%
There is no percentage changes in that number.

\(\frac{Pnr}{100}\)
\(240 = \frac{3000\times2 \times r}{100}\\ = \frac{240}{60}\)
R = 4%

Let the number be x.
\( \frac{32}{100} \times x =48 \)
\(x =\frac{48 \times 100}{32}=150\)
The number is 150

Let the marked price be Rs x.
Now, x + \(\frac { 18x }{ 100 } \) = 10502
\(\frac { 118x }{ 100 } \) = 10502
∴ Marked price, x = Rs. 8900.
GST at 18% = 8900 x \(\frac { 18 }{ 100 } \)
= Rs. 10502 – Rs. 8900
= Rs.1602

Given,
Discount = Marked Price –Selling Price
= 1060 – 901
= Rs. 159
∴ Discount = \(\frac { 159 }{ 1060 } \)× 100%
= 15%

Total C.P of the washing machine
= C.P + Overhead Expenses
= 16150 + 1350 = Rs. 17500
S.P = Rs. 19250
Therefore, we find S.P > C.P.
Gain % = \(\left( \frac { Gain }{ C.P } \times 100 \right) %\)% = \(\left( \frac { 19250-17500 }{ 17500 } \times 100 \right) %\)
=\(\frac { 1750 }{ 17500 } \times 100\) = 10%

S.P of the bicycle = Rs. 4275
Loss = 5%
∴ C.P =\(\frac { 100 }{ 100-loss% } \times S.P\)
= \(\frac { 100 }{ 95 } \times 4275\)
= Rs. 4500
Now,
C.P = Rs. 4500 and the desired profi t = 5%
∴ Desired S.P = \(\frac { 100+gain% }{ 100 } \times C.P\)
=\(\frac { 100+5 }{ 100 } \times 4500\)
= 105 x 45
= Rs. 4725
Hence, the desired selling price is Rs. 4725.