Percentage - 01

Explanation:-

$$\eqalign{ & {\text{Increase}}\,{\text{in}}\,{\text{10}}\,{\text{years}} \cr & = \left( {262500 - 175000} \right) \cr & = 87500. \cr & {\text{Increase}}\% \cr & = \left( {\frac{{87500}}{{175000}} \times 100} \right)\% = 50\% . \cr & \therefore {\text{Required}}\,{\text{average}} \cr & = \left( {\frac{{50}}{{10}}} \right)\% = 5\% . \cr}$$

Explanation:-

$$\eqalign{ & {\text{Rebate}} = 6\% \,{\text{of}}\,Rs.\,6650 \cr & = Rs.\,\left( {\frac{6}{{100}} \times 6650} \right) = Rs.\,399. \cr & {\text{Sales}}\,{\text{tax}} = \,10\% \,{\text{of}}\,Rs.\,\left( {6650 - 399} \right) \cr & = Rs.\,\left( {\frac{{10}}{{100}} \times 6251} \right) = Rs.\,625.10 \cr & \therefore {\text{Final}}\,{\text{amount}} \cr & = Rs.\,\left( {6251 + 625.10} \right) \cr & = Rs.\,6876.10 \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{sum}}\,{\text{paid}}\,{\text{to}}\,{\text{y}}\,{\text{per}}\,{\text{week}}\,{\text{be}}\,Rs.\,z. \cr & {\text{Then}},\,z + 120\% \,of\,z = 550. \cr & \Rightarrow z + \frac{{120}}{{100}}z = 550 \cr & \Rightarrow \frac{{11}}{5}z = 550 \cr & \Rightarrow z = \left( {\frac{{550 \times 5}}{{11}}} \right) = 250. \cr}$$

Explanation:-

Total number of votes polled = (1136 + 7636 + 11628) = 20400.
∴ Required percentage = $$\left( {\frac{{11628}}{{20400}} \times 100} \right)\% = 57\% .$$

Explanation:-

Number of valid votes = 80% of 7500 = 6000.
∴ Valid votes polled by other candidate = 45% of 6000 $$= \left( {\frac{{45}}{{100}} \times 6000} \right) = 2700.$$

Explanation:-

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{number}}\,{\text{be}}\,x. \cr & {\text{Then,}}\,{\text{error}} = \frac{5}{3}x - \frac{3}{5}x = \frac{{16}}{{15}}x. \cr & {\text{Error}}\% = \left( {\frac{{16x}}{{15}} \times \frac{3}{{5x}} \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 64\% . \cr}$$

Explanation:-

Let the number of students be x. Then,
Number of students above 8 years of age $$\eqalign{ & = \left( {100 - 20} \right)\% \,{\text{of}}\,x = 80\% \,{\text{of}}\,x. \cr & \therefore 80\% \,{\text{of}}\,x = 48 + \frac{2}{3}\,{\text{of}}\,48 \cr & \Rightarrow \frac{{80}}{{100}}x = 80 \cr & \Rightarrow x = 100. \cr}$$

Explanation:-

$$\eqalign{ & 20\% \,{\text{of}}\,a = b \Rightarrow \frac{{20}}{{100}}a = b. \cr & \therefore b\% \,{\text{of}}\,20 = \left( {\frac{b}{{100}} \times 20} \right) \cr & = \left( {\frac{{20}}{{100}}a \times \frac{1}{{100}} \times 20} \right) \cr & = \frac{4}{{100}}a = 4\% \,{\text{of}}\,a. \cr}$$

Explanation:-

$$\eqalign{ & x\% \,{\text{of}}\,y = \left( {\frac{x}{{100}} \times y} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{y}{{100}} \times x} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = y\% \,{\text{of}}\,x \cr & \therefore A = B \cr}$$

Explanation:-

Clearly, the numbers which have 1 or 9 in the unit's digit, have squares that end in the digit 1. Such numbers from 1 to 70 are 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69.
Number of such number =14 $$= \left( {\frac{{14}}{{70}} \times 100} \right)\% = 20\% .$$