Percentage and its Applications

To convert a percentage into a fraction, replace the % sign with $\frac{1}{100}$ and reduce the fraction to simplest form.

$$\begin{aligned}\Rightarrow 356 \times \frac{1}{100}\\[1em] \Rightarrow \frac{356}{100}\\[1em] \Rightarrow \frac{89}{25}\\[1em] \Rightarrow 3\frac{14}{25}\end{aligned}$$

Hence, 356% = $\mathbf{3\frac{14}{25}}$.

To convert a percentage into a fraction, replace the % sign with $\frac{1}{100}$ and reduce the fraction to simplest form.

$2\frac{1}{2}$% = $\frac{5}{2}$%

$$\begin{aligned}\Rightarrow \frac{5}{2} \times \frac{1}{100}\\[1em] \Rightarrow \frac{5}{200}\\[1em] \Rightarrow \frac{1}{40}\end{aligned}$$

Hence, $\mathbf{2\frac{1}{2}}$% = $\mathbf{\frac{1}{40}}$.

To convert a percentage into a fraction, replace the % sign with $\frac{1}{100}$ and reduce the fraction to simplest form.

$16\frac{2}{3}$% = $\frac{50}{3}$%

$$\begin{aligned}\Rightarrow \frac{50}{3} \times \frac{1}{100}\\[1em] \Rightarrow \frac{50}{300}\\[1em] \Rightarrow \frac{1}{6}\end{aligned}$$

Hence, $\mathbf{16\frac{2}{3}}$% = $\mathbf{\frac{1}{6}}$.

To convert a fraction into a percentage, multiply the fraction by 100 and put the % sign.

$\Rightarrow \frac{3}{2} \times 100$%

$\Rightarrow \frac{300}{2}$%

⇒ 150%

Hence, $\mathbf{\frac{3}{2}}$ = 150%.

To convert a fraction into a percentage, multiply the fraction by 100 and put the % sign.

$\Rightarrow \frac{9}{20} \times 100$%

$\Rightarrow \frac{900}{20}$%

⇒ 45%

Hence, $\mathbf{\frac{9}{20}}$ = 45%.

To convert a fraction into a percentage, multiply the fraction by 100 and put the % sign.

$1\frac{1}{4}$ = $\frac{5}{4}$

$\Rightarrow \frac{5}{4} \times 100$%

$\Rightarrow \frac{500}{4}$%

⇒ 125%

Hence, $\mathbf{1\frac{1}{4}}$ = 125%.

$\frac{3}{4}$ = 0.75

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.75 = (0.75 x 100)% = 75%

Hence, $\mathbf{\frac{3}{4}}$ = 0.75 = 75%.

$\frac{5}{8}$ = 0.625

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.625 = (0.625 x 100)% = 62.5%

Hence, $\mathbf{\frac{5}{8}}$ = 0.625 = 62.5%.

$\frac{3}{16}$ = 0.1875

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.1875 = (0.1875 x 100)% = 18.75%

Hence, $\mathbf{\frac{3}{16}}$ = 0.1875 = 18.75%.

$\frac{2}{3}$ = 0.66666.....

Correct to four decimal places,

$\frac{2}{3}$ = 0.6667

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.6667 = (0.6667 x 100)% = 66.67%

Hence, $\mathbf{\frac{2}{3}}$ = 0.6667 = 66.67%.

$\frac{5}{6}$ = 0.83333.....

Correct to four decimal places,

$\frac{5}{6}$ = 0.8333

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.8333 = (0.8333 x 100)% = 83.33%

Hence, $\mathbf{\frac{5}{6}}$ = 0.8333 = 83.33%.

$\frac{4}{7}$ = 0.571428.....

Correct to four decimal places,

$\frac{4}{7}$ = 0.5714

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

⇒ 0.5714 = (0.5714 x 100)% = 57.14%

Hence, $\mathbf{\frac{4}{7}}$ = 0.5714 = 57.14%.

To convert a ratio into a percentage, first convert the given ratio into a fraction and then convert the fraction into a percentage.

⇒ 17 : 20

$\Rightarrow \frac{17}{20}$

$\Rightarrow \frac{17}{20} \times 100$%

$\Rightarrow \frac{1700}{20}$%

⇒ 85%

Hence, 17 : 20 = 85%.

To convert a ratio into a percentage, first convert the given ratio into a fraction and then convert the fraction into a percentage.

⇒ 13 : 18

$\Rightarrow \frac{13}{18}$

$\Rightarrow \frac{13}{18} \times 100$%

$\Rightarrow \frac{1300}{18}$%

$\Rightarrow \frac{650}{9}$%

$\Rightarrow 72\frac{2}{9}$%

Hence, 13 : 18 = $\mathbf{72\frac{2}{9}}$%.

To convert a ratio into a percentage, first convert the given ratio into a fraction and then convert the fraction into a percentage.

⇒ 93 : 80

$\Rightarrow \frac{93}{80}$

$\Rightarrow \frac{93}{80} \times 100$%

$\Rightarrow \frac{9300}{80}$%

⇒ 116.25%

Hence, 93 : 80 = 116.25%.

To convert a percentage into a decimal, first convert the percentage into a fraction by replacing the % sign with $\frac{1}{100}$, then convert the fraction into a decimal.

$\Rightarrow \frac{20}{100}$

⇒ 0.2

Hence, 20% = 0.2.

To convert a percentage into a decimal, first convert the percentage into a fraction by replacing the % sign with $\frac{1}{100}$, then convert the fraction into a decimal.

$\Rightarrow \frac{2}{100}$

⇒ 0.02

Hence, 2% = 0.02.

To convert a percentage into a decimal, first convert the percentage into a fraction by replacing the % sign with $\frac{1}{100}$, then convert the fraction into a decimal.

$3\frac{1}{4}$% = $\frac{13}{4}$%

$$\begin{aligned}\Rightarrow \frac{13}{4} \times \frac{1}{100}\\[1em] \Rightarrow \frac{13}{400}\\[1em] \Rightarrow 0.0325\end{aligned}$$

Hence, $\mathbf{3\frac{1}{4}}$% = 0.0325.

To find a percentage of a given quantity, change the percentage into a fraction and multiply it by the given quantity.

$$\begin{aligned}\Rightarrow \frac{27}{100} \times 50\\[1em] \Rightarrow \frac{1350}{100}\\[1em] \Rightarrow ₹13.50\end{aligned}$$

Hence, 27% of ₹50 = ₹13.50.

To find a percentage of a given quantity, change the percentage into a fraction and multiply it by the given quantity.

$6\frac{1}{4}$% of 25 kg = $\frac{25}{4}$% of 25 kg

$$\begin{aligned}\Rightarrow \frac{25}{4} \times \frac{1}{100} \times 25\\[1em] \Rightarrow \frac{625}{400}\\[1em] \Rightarrow \frac{25}{16}\\[1em] \Rightarrow 1\frac{9}{16} \text{ kg}\end{aligned}$$

Hence, $\mathbf{6\frac{1}{4}}$% of 25 kg = $\mathbf{1\frac{9}{16}}$ kg.

To express $x$ as a percentage of $y$, percentage = $\Big(\frac{x}{y} \times 100\Big)$%. Both the quantities must be of the same kind (in the same units).

2 kg = (2 x 1000) g = 2000 g

Required percentage = $\frac{300}{2000} \times 100$%

= $\frac{30000}{2000}$%

= 15%

Hence, 300 g is 15% of 2 kg.

To express $x$ as a percentage of $y$, percentage = $\Big(\frac{x}{y} \times 100\Big)$%. Both the quantities must be of the same kind (in the same units).

Required percentage = $\frac{7.50}{6} \times 100$%

= $\frac{750}{6}$%

= 125%

Hence, ₹7.50 is 125% of ₹6.

To express $x$ as a percentage of $y$, percentage = $\Big(\frac{x}{y} \times 100\Big)$%. Both the quantities must be of the same kind (in the same units).

Required percentage = $\frac{65}{50} \times 100$%

= $\frac{6500}{50}$%

= 130%

Hence, 65 kg is 130% of 50 kg.

To express $x$ as a percentage of $y$, percentage = $\Big(\frac{x}{y} \times 100\Big)$%. Both the quantities must be of the same kind (in the same units).

Required percentage = $\frac{4}{9} \times 100$%

= $\frac{400}{9}$%

= $44\frac{4}{9}$%

Hence, ₹4 is $\mathbf{44\frac{4}{9}}$% of ₹9.

Let the required number be $x$.

$16\frac{2}{3}$% = $\frac{50}{3}$%

According to the given condition,

$\frac{50}{3}$% of $x$ = 25

$$\begin{aligned}\Rightarrow \frac{50}{3} \times \frac{1}{100} \times x = 25 \\[1em] \Rightarrow \frac{50x}{300} = 25 \\[1em] \Rightarrow \frac{x}{6} = 25 \\[1em] \Rightarrow x = 25 \times 6 \\[1em] \Rightarrow x = 150\end{aligned}$$

Hence, the required number is 150.

Let the required number be $x$.

According to the given condition,

13.25% of $x$ = 159

$$\begin{aligned}\Rightarrow \frac{13.25}{100} \times x = 159 \\[1em] \Rightarrow \frac{13.25x}{100} = 159 \\[1em] \Rightarrow x = \frac{159 \times 100}{13.25} \\[1em] \Rightarrow x = \frac{15900}{13.25} \\[1em] \Rightarrow x = 1200\end{aligned}$$

Hence, the required number is 1200.

If we increase the number 60 by 30%, we get

$$\begin{aligned}\text{New number} = 60 + 60 \times \frac{30}{100} \\[1em] = \Big(1 + \frac{30}{100}\Big) \times 60 \\[1em] = \frac{130}{100} \times 60 \\[1em] = \frac{13}{10} \times 60 \\[1em] = \frac{780}{10}\\[1em] = 78\end{aligned}$$

Hence, on increasing 60 by 30% we get 78.

If we decrease the number 750 by 10%, we get

$$\begin{aligned}\text{New number} = 750 - 750 \times \frac{10}{100} \\[1em] = \Big(1 - \frac{10}{100}\Big) \times 750 \\[1em] = \frac{90}{100} \times 750 \\[1em] = \frac{9}{10} \times 750 \\[1em] = \frac{6750}{10}\\[1em] = 675\end{aligned}$$

Hence, on decreasing 750 by 10% we get 675.

Let the required number be $x$.

Since the number is increased by 15%, we get

$$\begin{aligned}\text{New number} = \Big(1 + \frac{15}{100}\Big) \times x \\[1em] \Rightarrow 299 = \frac{115}{100} \times x \\[1em] \Rightarrow x = \frac{299 \times 100}{115} \\[1em] \Rightarrow x = \frac{29900}{115} \\[1em] \Rightarrow x = 260\end{aligned}$$

Hence, the required number is 260.

Let the required number be $x$.

Since the number is decreased by 18%, we get

$$\begin{aligned}\text{New number} = \Big(1 - \frac{18}{100}\Big) \times x \\[1em] \Rightarrow 697 = \frac{82}{100} \times x \\[1em] \Rightarrow x = \frac{697 \times 100}{82} \\[1em] \Rightarrow x = \frac{69700}{82} \\[1em] \Rightarrow x = 850\end{aligned}$$

Hence, the required number is 850.

Given:

Percentage of salary spent = 83%

Savings = ₹9,350

Percentage of salary saved = (100 - 83)% = 17%

Let the monthly salary of Mr. Khanna be ₹$x$.

According to the given condition,

17% of $x$ = 9350

$$\begin{aligned}\Rightarrow \frac{17}{100} \times x = 9350 \\[1em] \Rightarrow \frac{17x}{100} = 9350 \\[1em] \Rightarrow x = \frac{9350 \times 100}{17} \\[1em] \Rightarrow x = \frac{935000}{17} \\[1em] \Rightarrow x = 55000\end{aligned}$$

Hence, the monthly salary of Mr. Khanna is ₹55,000.

Given:

Percentage of girls in the school = 38%

Number of boys = 1023

Percentage of boys in the school = (100 - 38)% = 62%

Let the total strength of the school be $x$.

According to the given condition,

62% of $x$ = 1023

$$\begin{aligned}\Rightarrow \frac{62}{100} \times x = 1023 \\[1em] \Rightarrow \frac{62x}{100} = 1023 \\[1em] \Rightarrow x = \frac{1023 \times 100}{62} \\[1em] \Rightarrow x = \frac{102300}{62} \\[1em] \Rightarrow x = 1650\end{aligned}$$

Hence, the total strength of the school is 1650.

Given:

Original price of the article = ₹960

New price of the article = ₹1,080

Increase in price = New price - Original price

= ₹1,080 - ₹960

= ₹120

Percentage increase = $\frac{\text{Increase in price}}{\text{Original price}} \times 100$%

= $\frac{120}{960} \times 100$%

= $\frac{12000}{960}$%

= 12.5%

Hence, the percentage increase in the price is 12.5%.

Given:

Percentage of votes polled by the loser = 42%

The loser lost by 14400 votes.

There are only two candidates in a straight contest.

Since the loser got 42% votes, therefore, the winner got (100 - 42)% votes = 58% votes

Difference in percentage of votes = (58 - 42)% = 16%

As the loser lost by 14400 votes, so difference in votes = 14400

∴ 16% of total votes polled = 14400

Let the total votes polled be $x$.

$$\begin{aligned}\Rightarrow \frac{16}{100} \times x = 14400 \\[1em] \Rightarrow \frac{16x}{100} = 14400 \\[1em] \Rightarrow x = \frac{14400 \times 100}{16} \\[1em] \Rightarrow x = \frac{1440000}{16} \\[1em] \Rightarrow x = 90000\end{aligned}$$

∴ Total number of votes polled = 90000

Total number of eligible voters = 1 lakh = 100000

Number of voters who did not vote = 100000 - 90000 = 10000

Percentage of voters who did not vote = $\frac{10000}{100000} \times 100$%

= $\frac{1000000}{100000}$%

= 10%

Hence, the total number of votes polled is 90000 and 10% of the voters did not vote.

Given:

Total number of candidates = 8000

Percentage of boys = 60%

Number of boys = 60% of 8000

$$\begin{aligned}= \frac{60}{100} \times 8000 \\[1em] = \frac{480000}{100} \\[1em] = 4800\end{aligned}$$

Number of girls = 8000 - 4800 = 3200

Number of boys who passed = 80% of 4800

$$\begin{aligned}= \frac{80}{100} \times 4800 \\[1em] = \frac{384000}{100} \\[1em] = 3840\end{aligned}$$

Number of girls who passed = 90% of 3200

$$\begin{aligned}= \frac{90}{100} \times 3200 \\[1em] = \frac{288000}{100} \\[1em] = 2880\end{aligned}$$

Total number of candidates who passed = 3840 + 2880 = 6720

Number of candidates who failed = Total candidates - Candidates who passed

= 8000 - 6720

= 1280

Hence, the number of candidates who failed is 1280.

Given:

New price of the article = ₹1,479

Increase in price = 16%

Let the original price of the article be ₹$x$.

Since the price has been increased by 16%, we get

$$\begin{aligned}\text{New price} = \Big(1 + \frac{16}{100}\Big) \text{ of original price} \\[1em] \Rightarrow 1479 = \frac{116}{100} \times x \\[1em] \Rightarrow x = \frac{1479 \times 100}{116} \\[1em] \Rightarrow x = \frac{147900}{116} \\[1em] \Rightarrow x = 1275\end{aligned}$$

Hence, the original price of the article is ₹1,275.

Given:

Present weight of Prachi = 59.5 kg

Reduction in weight = 15%

Let the earlier weight of Prachi be $x$ kg.

Since the weight has been reduced by 15%, we get

$$\begin{aligned}\text{Present weight} = \Big(1 - \frac{15}{100}\Big) \text{ of earlier weight} \\[1em] \Rightarrow 59.5 = \frac{85}{100} \times x \\[1em] \Rightarrow x = \frac{59.5 \times 100}{85} \\[1em] \Rightarrow x = \frac{5950}{85} \\[1em] \Rightarrow x = 70\end{aligned}$$

Hence, the earlier weight of Prachi was 70 kg.

Given:

Reduction in all the prices = 15%

(i) Original price of the article = ₹40

$$\begin{aligned}\text{Cost of the article in the sale} = \Big(1 - \frac{15}{100}\Big) \times 40 \\[1em] = \frac{85}{100} \times 40 \\[1em] = \frac{3400}{100} \\[1em] = ₹34\end{aligned}$$

Hence, the cost of the article in the sale is ₹34.

(ii) Sale price of the article = ₹20.40

Let the original price of the article be ₹$x$.

Since the price has been reduced by 15%, we get

$$\begin{aligned}\text{Sale price} = \Big(1 - \frac{15}{100}\Big) \text{ of original price} \\[1em] \Rightarrow 20.40 = \frac{85}{100} \times x \\[1em] \Rightarrow x = \frac{20.40 \times 100}{85} \\[1em] \Rightarrow x = \frac{2040}{85} \\[1em] \Rightarrow x = 24\end{aligned}$$

Hence, the original price of the article is ₹24.

Given:

Original price = ₹200

Increasing the price of ₹200 by 10%,

$$\begin{aligned}\text{New price} = \Big(1 + \frac{10}{100}\Big) \times 200 \\[1em] = \frac{110}{100} \times 200 \\[1em] = \frac{22000}{100} \\[1em] = ₹220\end{aligned}$$

Now decreasing the new price of ₹220 by 10%,

$$\begin{aligned}\text{Final price} = \Big(1 - \frac{10}{100}\Big) \times 220 \\[1em] = \frac{90}{100} \times 220 \\[1em] = \frac{19800}{100} \\[1em] = ₹198\end{aligned}$$

Original price = ₹200 and final price = ₹198.

As ₹198 < ₹200, the final price is not the same as the original price.

Hence, the final price ₹198 is not the same as the original price ₹200.

Given:

First candidate gets 36% marks and fails by 24 marks.

Second candidate gets 43% marks and gets 18 marks more than the pass marks.

Let the maximum marks be $x$.

In the first case,

Passing marks = 36% of $x$ + 24

$= \frac{36}{100} \times x + 24$

In the second case,

Passing marks = 43% of $x$ - 18

$= \frac{43}{100} \times x - 18$

Since pass marks are same in both the cases,

$$\begin{aligned}\Rightarrow \frac{43}{100} \times x - 18 = \frac{36}{100} \times x + 24 \\[1em] \Rightarrow \frac{43x}{100} - \frac{36x}{100} = 24 + 18 \\[1em] \Rightarrow \frac{7x}{100} = 42 \\[1em] \Rightarrow x = \frac{42 \times 100}{7} \\[1em] \Rightarrow x = \frac{4200}{7} \\[1em] \Rightarrow x = 600\end{aligned}$$

∴ Maximum marks = 600

Passing marks = 36% of 600 + 24

$$\begin{aligned}= \frac{36}{100} \times 600 + 24 \\[1em] = \frac{21600}{100} + 24 \\[1em] = 216 + 24 \\[1em] = 240\end{aligned}$$

Percentage of passing marks = $\frac{240}{600} \times 100$%

= $\frac{24000}{600}$%

= 40%

Hence, the maximum marks are 600 and the percentage of passing marks is 40%.

Given:

Length of the rectangle = 20 cm

Breadth of the rectangle = 15 cm

Original area of the rectangle = Length x Breadth

= 20 x 15

= 300 cm²

Each side is increased by 20%.

$$\begin{aligned}\text{New length} = \Big(1 + \frac{20}{100}\Big) \times 20 \\[1em] = \frac{120}{100} \times 20 \\[1em] = \frac{2400}{100} \\[1em] = 24 \text{ cm}\\[1em] \text{New breadth} = \Big(1 + \frac{20}{100}\Big) \times 15 \\[1em] = \frac{120}{100} \times 15 \\[1em] = \frac{1800}{100} \\[1em] = 18 \text{ cm}\end{aligned}$$

New area of the rectangle = 24 x 18 = 432 cm²

Increase in area = New area - Original area

= 432 - 300

= 132 cm²

Percentage increase in area = $\frac{\text{Increase in area}}{\text{Original area}} \times 100$%

= $\frac{132}{300} \times 100$%

= $\frac{13200}{300}$%

= 44%

Hence, the percentage increase in the area is 44%.

Given:

Pocket money given to the two sons = 1% of monthly income

Share of the elder son = 80% of the pocket money

Amount spent by the elder son = 80% of his share

Savings of the elder son = ₹60

Let Imran's monthly income be ₹$x$.

Total pocket money given = 1% of $x$

$$\begin{aligned}= \frac{1}{100} \times x \\[1em] = \frac{x}{100}\end{aligned}$$

Share of the elder son = 80% of $\frac{x}{100}$

$$\begin{aligned}= \frac{80}{100} \times \frac{x}{100} \\[1em] = \frac{80x}{10000} \\[1em] = \frac{x}{125}\end{aligned}$$

The elder son spends 80% of his share, so he saves (100 - 80)% = 20% of his share.

Savings of the elder son = 20% of $\frac{x}{125}$

$$\begin{aligned}= \frac{20}{100} \times \frac{x}{125} \\[1em] = \frac{20x}{12500} \\[1em] = \frac{x}{625}\end{aligned}$$

According to the given condition,

$$\begin{aligned}\Rightarrow \frac{x}{625} = 60 \\[1em] \Rightarrow x = 60 \times 625 \\[1em] \Rightarrow x = 37500\end{aligned}$$

Hence, Imran's monthly income is ₹37,500.

(i) C.P. of the article = ₹400

S.P. of the article = ₹468

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

= ₹468 - ₹400

= ₹68

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{68}{400} \times 100$%

= $\frac{6800}{400}$%

= 17%

Hence, there is a profit of 17%.

(ii) C.P. of the article = ₹13,600

S.P. of the article = ₹12,104

As S.P. < C.P., there is a loss.

Loss = C.P. - S.P.

= ₹13,600 - ₹12,104

= ₹1,496

Loss percentage = $\frac{\text{Loss}}{\text{C.P.}} \times 100$%

= $\frac{1496}{13600} \times 100$%

= $\frac{149600}{13600}$%

= 11%

Hence, there is a loss of 11%.

Given:

S.P. of the article = ₹1,636.25

Gain = ₹96.25

C.P. of the article = S.P. - Gain

= ₹1,636.25 - ₹96.25

= ₹1,540

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{96.25}{1540} \times 100$%

= $\frac{9625}{1540}$%

= 6.25%

Hence, the gain percentage is 6.25%.

Given:

S.P. of the article = ₹770

Loss = ₹110

C.P. of the article = S.P. + Loss

= ₹770 + ₹110

= ₹880

Loss percentage = $\frac{\text{Loss}}{\text{C.P.}} \times 100$%

= $\frac{110}{880} \times 100$%

= $\frac{11000}{880}$%

= 12.5%

Hence, the loss percentage is 12.5%.

Given:

Number of dozens of eggs bought = 25

Rate of buying = ₹48 per dozen

Number of eggs broken = 30

Selling rate of the remaining eggs = ₹5 each

Total number of eggs bought = 25 x 12 = 300

C.P. of 300 eggs = 25 x ₹48 = ₹1,200

Number of eggs left after breakage = 300 - 30 = 270

S.P. of 270 eggs = 270 x ₹5 = ₹1,350

As S.P. > C.P., there is a gain.

Gain = S.P. - C.P.

= ₹1,350 - ₹1,200

= ₹150

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{150}{1200} \times 100$%

= $\frac{15000}{1200}$%

= 12.5%

Hence, Rashida made a gain of 12.5%.

Given:

Cost of the article = ₹20,000

Overhead expenses on repairs = ₹1,400

Profit = 20%

Total C.P. of the article = Cost + Overheads

= ₹20,000 + ₹1,400

= ₹21,400

As the article is sold at a profit of 20%,

Profit = 20% of C.P.

S.P. of the article = C.P. + Profit

= C.P. + 20% of C.P.

= $\Big(1 + \frac{20}{100}\Big)$ of C.P.

In general,

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 + \frac{20}{100}\Big)\text{ of C.P.} \\[1em] = \frac{120}{100} \times 21400 \\[1em] = \frac{2568000}{100} \\[1em] = ₹25{,}680\end{aligned}$$

Hence, the selling price of the article is ₹25,680.

Given:

Number of fans bought = 200

Cost of one fan = ₹1,200

Transportation cost per fan = ₹30

Advertising cost = ₹4,000

Selling price of one fan = ₹1,350

Cost of 200 fans = 200 x ₹1,200 = ₹2,40,000

Transportation cost of 200 fans = 200 x ₹30 = ₹6,000

Total C.P. of 200 fans = Cost + Transportation + Advertising

= ₹2,40,000 + ₹6,000 + ₹4,000

= ₹2,50,000

S.P. of 200 fans = 200 x ₹1,350 = ₹2,70,000

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

= ₹2,70,000 - ₹2,50,000

= ₹20,000

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{20000}{250000} \times 100$%

= $\frac{2000000}{250000}$%

= 8%

Hence, the shopkeeper made a profit of ₹20,000 and the profit percentage is 8%.

Given:

C.P. of the article = 90% of its S.P.

Let the S.P. of the article be ₹$x$.

C.P. of the article = 90% of ₹$x$

= $\frac{90}{100} \times x$

= ₹$\frac{9x}{10}$

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

$$\begin{aligned}= ₹\Big(x - \frac{9x}{10}\Big) \\[1em] = ₹\frac{10x - 9x}{10} \\[1em] = ₹\frac{x}{10}\end{aligned}$$

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{\frac{x}{10}}{\frac{9x}{10}} \times 100$%

= $\frac{x}{10} \times \frac{10}{9x} \times 100$%

= $\frac{100}{9}$%

= $11\frac{1}{9}$%

Hence, the profit percentage is $\mathbf{11\frac{1}{9}}$%.

Given:

C.P. of 4 ball pens = ₹35

S.P. of 5 ball pens = ₹58

(i) C.P. of 1 ball pen = ₹$\frac{35}{4}$

S.P. of 1 ball pen = ₹$\frac{58}{5}$

Gain on the sale of one ball pen = S.P. - C.P.

$$\begin{aligned}= ₹\Big(\frac{58}{5} - \frac{35}{4}\Big) \\[1em] = ₹\frac{232 - 175}{20} \\[1em] = ₹\frac{57}{20}\end{aligned}$$

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{\frac{57}{20}}{\frac{35}{4}} \times 100$%

= $\frac{57}{20} \times \frac{4}{35} \times 100$%

= $\frac{22800}{700}$%

= $\frac{228}{7}$%

= $32\frac{4}{7}$%

Hence, the gain percentage is $\mathbf{32\frac{4}{7}}$%.

(ii) Number of ball pens to be sold = $\frac{\text{Total profit}}{\text{Profit on one ball pen}}$

$$\begin{aligned}= \frac{171}{\frac{57}{20}} \\[1em] = 171 \times \frac{20}{57} \\[1em] = \frac{3420}{57} \\[1em] = 60\end{aligned}$$

Hence, he should sell 60 ball pens to earn a profit of ₹171.

Given:

C.P. of 3 bananas = ₹1

S.P. of 4 bananas = ₹1

C.P. of 1 banana = ₹$\frac{1}{3}$

S.P. of 1 banana = ₹$\frac{1}{4}$

As S.P. < C.P., there is a loss.

Loss on the sale of one banana = C.P. - S.P.

$$\begin{aligned}= ₹\Big(\frac{1}{3} - \frac{1}{4}\Big) \\[1em] = ₹\frac{4 - 3}{12} \\[1em] = ₹\frac{1}{12}\end{aligned}$$

Loss percentage = $\frac{\text{Loss}}{\text{C.P.}} \times 100$%

= $\frac{\frac{1}{12}}{\frac{1}{3}} \times 100$%

= $\frac{1}{12} \times \frac{3}{1} \times 100$%

= $\frac{300}{12}$%

= 25%

Hence, the vendor made a loss of 25%.

Given:

S.P. of 5 geometry boxes = C.P. of 7 geometry boxes

Let the cost price of one geometry box be ₹$x$.

C.P. of 7 geometry boxes = ₹$7x$

It is given that S.P. of 5 geometry boxes = C.P. of 7 geometry boxes

∴ S.P. of 5 geometry boxes = ₹$7x$

∴ S.P. of one geometry box = ₹$\frac{7x}{5}$

As the cost price of one geometry box is ₹$x$ and the selling price of one geometry box is ₹$\frac{7x}{5}$, the shopkeeper makes a profit.

Profit on selling one geometry box = S.P. - C.P.

$$\begin{aligned}= ₹\Big(\frac{7x}{5} - x\Big) \\[1em] = ₹\frac{7x - 5x}{5} \\[1em] = ₹\frac{2x}{5}\end{aligned}$$

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{\frac{2x}{5}}{x} \times 100$%

= $\frac{2x}{5} \times \frac{1}{x} \times 100$%

= $\frac{200}{5}$%

= 40%

Hence, the shopkeeper made a profit of 40%.

(i) C.P. of the article = ₹2,360

Profit = 8%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 + \frac{8}{100}\Big)\text{ of C.P.} \\[1em] = \frac{108}{100} \times 2360 \\[1em] = \frac{254880}{100} \\[1em] = ₹2{,}548.80\end{aligned}$$

Hence, the selling price of the article is ₹2,548.80.

(ii) C.P. of the article = ₹380

Loss = 7.5%

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 - \frac{7.5}{100}\Big)\text{ of C.P.} \\[1em] = \frac{92.5}{100} \times 380 \\[1em] = \frac{35150}{100} \\[1em] = ₹351.50\end{aligned}$$

Hence, the selling price of the article is ₹351.50.

Given:

C.P. of one dozen eggs = ₹36

Profit = 50%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of one dozen eggs} = \Big(1 + \frac{50}{100}\Big)\text{ of C.P.} \\[1em] = \frac{150}{100} \times 36 \\[1em] = \frac{5400}{100} \\[1em] = ₹54\end{aligned}$$

S.P. of one egg = $\frac{54}{12}$

= ₹4.50

Hence, the selling price per egg is ₹4.50.

Given:

Total C.P. of the wrist watches = ₹60,000

One-third sold at a profit of 30%, one-third at a profit of 20% and the remaining one-third at a loss of 5%.

C.P. of one-third of the wrist watches = $\frac{1}{3} \times 60000$ = ₹20,000

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

S.P. of the first one-third at a profit of 30% = $\Big(1 + \frac{30}{100}\Big) \times 20000$

= $\frac{130}{100} \times 20000$

= ₹26,000

S.P. of the second one-third at a profit of 20% = $\Big(1 + \frac{20}{100}\Big) \times 20000$

= $\frac{120}{100} \times 20000$

= ₹24,000

S.P. of the remaining one-third at a loss of 5% = $\Big(1 - \frac{5}{100}\Big) \times 20000$

= $\frac{95}{100} \times 20000$

= ₹19,000

Total S.P. of all the wrist watches = ₹26,000 + ₹24,000 + ₹19,000 = ₹69,000

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

= ₹69,000 - ₹60,000

= ₹9,000

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{9000}{60000} \times 100$%

= $\frac{900000}{60000}$%

= 15%

Hence, Mr. Ghosh made an overall profit of 15%.

Given:

C.P. of the laptop = ₹40,000, profit on laptop = 8%

C.P. of the mobile phone = ₹24,000, loss on mobile phone = 12%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the laptop} = \Big(1 + \frac{8}{100}\Big) \times 40000 \\[1em] = \frac{108}{100} \times 40000 \\[1em] = ₹43{,}200\end{aligned}$$

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the mobile phone} = \Big(1 - \frac{12}{100}\Big) \times 24000 \\[1em] = \frac{88}{100} \times 24000 \\[1em] = ₹21{,}120\end{aligned}$$

Total C.P. = ₹40,000 + ₹24,000 = ₹64,000

Total S.P. = ₹43,200 + ₹21,120 = ₹64,320

As S.P. > C.P., there is a gain.

Gain = S.P. - C.P.

= ₹64,320 - ₹64,000

= ₹320

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{320}{64000} \times 100$%

= $\frac{32000}{64000}$%

= 0.5%

Hence, the shopkeeper made a gain of 0.5% on the whole transaction.

Given:

Number of chairs bought = 40

C.P. of each chair = ₹900

One-fourth of the chairs sold at a loss of 8%

Desired gain on the whole deal = 10%

C.P. of all the 40 chairs = 40 x ₹900 = ₹36,000

S.P. = $\Big(1 + \frac{\text{Gain Percentage}}{100}\Big) \times$ C.P.

S.P. of all the chairs to gain 10% = $\Big(1 + \frac{10}{100}\Big) \times 36000$

= $\frac{110}{100} \times 36000$

= ₹39,600

One-fourth of all the chairs = $\frac{1}{4} \times 40$ = 10

C.P. of 10 chairs = 10 x ₹900 = ₹9,000

As these chairs are sold at a loss of 8%,

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of these 10 chairs} = \Big(1 - \frac{8}{100}\Big) \times 9000 \\[1em] = \frac{92}{100} \times 9000 \\[1em] = ₹8{,}280\end{aligned}$$

Number of remaining chairs = 40 - 10 = 30

S.P. of the remaining 30 chairs = ₹39,600 - ₹8,280 = ₹31,320

S.P. of each of the remaining chairs = $\frac{31320}{30}$

= ₹1,044

Hence, he must sell each of the remaining chairs at ₹1,044.

Given:

S.P. of each gadget = ₹44,000

Loss on the first gadget = 12%

Profit on the second gadget = 10%

For the first gadget :

S.P. = ₹44,000, loss = 12%

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 44000 = \Big(1 - \frac{12}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 44000 = \frac{88}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{44000 \times 100}{88} \\[1em] \Rightarrow \text{C.P.} = ₹50{,}000\end{aligned}$$

For the second gadget :

S.P. = ₹44,000, profit = 10%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 44000 = \Big(1 + \frac{10}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 44000 = \frac{110}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{44000 \times 100}{110} \\[1em] \Rightarrow \text{C.P.} = ₹40{,}000\end{aligned}$$

Total C.P. of both the gadgets = ₹50,000 + ₹40,000 = ₹90,000

Total S.P. of both the gadgets = ₹44,000 + ₹44,000 = ₹88,000

As S.P. < C.P., there is a loss.

Loss = C.P. - S.P.

= ₹90,000 - ₹88,000

= ₹2,000

Hence, the shopkeeper suffered an overall loss of ₹2,000.

Given:

Manufacturing price of the T.V. set = ₹24,000

Profit of the company = 20%

Profit of the dealer = 12.5%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the T.V. set for the company} = \Big(1 + \frac{20}{100}\Big) \times 24000 \\[1em] = \frac{120}{100} \times 24000 \\[1em] = \frac{2880000}{100} \\[1em] = ₹28{,}800\end{aligned}$$

This becomes the C.P. of the T.V. set for the dealer.

$$\begin{aligned}\text{S.P. of the T.V. set for the dealer} = \Big(1 + \frac{12.5}{100}\Big) \times 28800 \\[1em] = \frac{112.5}{100} \times 28800 \\[1em] = \frac{3240000}{100} \\[1em] = ₹32{,}400\end{aligned}$$

Hence, the customer has to pay ₹32,400 for the T.V. set.

(i) S.P. of the article = ₹450, loss = 10%

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 450 = \Big(1 - \frac{10}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 450 = \frac{90}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{450 \times 100}{90} \\[1em] \Rightarrow \text{C.P.} = \frac{45000}{90} \\[1em] \Rightarrow \text{C.P.} = ₹500\end{aligned}$$

Hence, the cost price of the article is ₹500.

(ii) S.P. of the article = ₹690, profit = 15%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 690 = \Big(1 + \frac{15}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 690 = \frac{115}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{690 \times 100}{115} \\[1em] \Rightarrow \text{C.P.} = \frac{69000}{115} \\[1em] \Rightarrow \text{C.P.} = ₹600\end{aligned}$$

Hence, the cost price of the article is ₹600.

Given:

S.P. of the shoe rack = ₹3,920, gain = 12%

New S.P. of the shoe rack = ₹4,375

S.P. = $\Big(1 + \frac{\text{Gain Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 3920 = \Big(1 + \frac{12}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 3920 = \frac{112}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{3920 \times 100}{112} \\[1em] \Rightarrow \text{C.P.} = \frac{392000}{112} \\[1em] \Rightarrow \text{C.P.} = ₹3{,}500\end{aligned}$$

Now the shoe rack is sold for ₹4,375.

As S.P. > C.P., there is a gain.

Gain = S.P. - C.P.

= ₹4,375 - ₹3,500

= ₹875

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{875}{3500} \times 100$%

= $\frac{87500}{3500}$%

= 25%

Hence, the shopkeeper made a gain of 25%.

Given:

S.P. of the plastic coffee table = ₹1,334, loss = 8%

Desired profit = $12\frac{1}{2}$% = 12.5%

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 1334 = \Big(1 - \frac{8}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 1334 = \frac{92}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{1334 \times 100}{92} \\[1em] \Rightarrow \text{C.P.} = \frac{133400}{92} \\[1em] \Rightarrow \text{C.P.} = ₹1{,}450\end{aligned}$$

Now C.P. = ₹1,450, profit = 12.5%,

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{New S.P.} = \Big(1 + \frac{12.5}{100}\Big) \times 1450 \\[1em] = \frac{112.5}{100} \times 1450 \\[1em] = \frac{163125}{100} \\[1em] = ₹1{,}631.25\end{aligned}$$

Hence, he should sell the plastic coffee table for ₹1,631.25.

Given:

S.P. of the tie = ₹252, gain = 5%

Desired gain = 35%

S.P. = $\Big(1 + \frac{\text{Gain Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 252 = \Big(1 + \frac{5}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 252 = \frac{105}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{252 \times 100}{105} \\[1em] \Rightarrow \text{C.P.} = \frac{25200}{105} \\[1em] \Rightarrow \text{C.P.} = ₹240\end{aligned}$$

Now C.P. = ₹240, gain = 35%,

$$\begin{aligned}\text{New S.P.} = \Big(1 + \frac{35}{100}\Big) \times 240 \\[1em] = \frac{135}{100} \times 240 \\[1em] = \frac{32400}{100} \\[1em] = ₹324\end{aligned}$$

Hence, he should sell the tie for ₹324.

Given:

Profit on selling the bag = 12%

On selling it for ₹39 more, profit = 18%

Let the cost price of the bag be ₹$x$.

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

S.P. of the bag at 12% profit = $\Big(1 + \frac{12}{100}\Big)$ of ₹$x$

$$\begin{aligned}= \frac{112}{100} \times x \\[1em] = ₹\frac{28x}{25}\end{aligned}$$

To obtain 18% profit,

$$\begin{aligned}\text{S.P. of the bag} = \Big(1 + \frac{18}{100}\Big)\text{ of ₹}x \\[1em] = \frac{118}{100} \times x \\[1em] = ₹\frac{59x}{50}\end{aligned}$$

According to the given information,

$$\begin{aligned}\Rightarrow \frac{59x}{50} = \frac{28x}{25} + 39 \\[1em] \Rightarrow \frac{59x}{50} - \frac{28x}{25} = 39 \\[1em] \Rightarrow \frac{59x - 56x}{50} = 39 \\[1em] \Rightarrow \frac{3x}{50} = 39 \\[1em] \Rightarrow x = \frac{39 \times 50}{3} \\[1em] \Rightarrow x = \frac{1950}{3} \\[1em] \Rightarrow x = 650\end{aligned}$$

Hence, the cost price of the bag is ₹650.

Given:

Loss on selling the sweater = 5%

On selling it for ₹260 more, profit = 15%

Let the purchase price of the sweater be ₹$x$.

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

S.P. of the sweater at 5% loss = $\Big(1 - \frac{5}{100}\Big)$ of ₹$x$

$$\begin{aligned}= \frac{95}{100} \times x \\[1em] = ₹\frac{19x}{20}\end{aligned}$$

To obtain 15% profit,

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the sweater} = \Big(1 + \frac{15}{100}\Big)\text{ of ₹}x \\[1em] = \frac{115}{100} \times x \\[1em] = ₹\frac{23x}{20}\end{aligned}$$

According to the given information,

$$\begin{aligned}\Rightarrow \frac{23x}{20} = \frac{19x}{20} + 260 \\[1em] \Rightarrow \frac{23x}{20} - \frac{19x}{20} = 260 \\[1em] \Rightarrow \frac{4x}{20} = 260 \\[1em] \Rightarrow \frac{x}{5} = 260 \\[1em] \Rightarrow x = 260 \times 5 \\[1em] \Rightarrow x = 1300\end{aligned}$$

Hence, the purchase price of the sweater is ₹1,300.

Given:

Loss on selling the purse = 8%

On selling it for ₹150 more, profit = 12%

Let the cost price of the purse be ₹$x$.

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

S.P. of the purse at 8% loss = $\Big(1 - \frac{8}{100}\Big)$ of ₹$x$

$$\begin{aligned}= \frac{92}{100} \times x \\[1em] = ₹\frac{23x}{25}\end{aligned}$$

To obtain 12% profit,

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the purse} = \Big(1 + \frac{12}{100}\Big)\text{ of ₹}x \\[1em] = \frac{112}{100} \times x \\[1em] = ₹\frac{28x}{25}\end{aligned}$$

According to the given information,

$$\begin{aligned}\Rightarrow \frac{28x}{25} = \frac{23x}{25} + 150 \\[1em] \Rightarrow \frac{28x}{25} - \frac{23x}{25} = 150 \\[1em] \Rightarrow \frac{5x}{25} = 150 \\[1em] \Rightarrow \frac{x}{5} = 150 \\[1em] \Rightarrow x = 150 \times 5 \\[1em] \Rightarrow x = 750\end{aligned}$$

∴ C.P. of the purse = ₹750

$$\begin{aligned}\text{S.P. of the purse} = \frac{23x}{25} \\[1em] = \frac{23 \times 750}{25} \\[1em] = \frac{17250}{25} \\[1em] = ₹690\end{aligned}$$

Hence, the selling price of the purse is ₹690.

(i) M.P. of the article = ₹575

Rate of discount = 12%

Discount = 12% of ₹575

$$\begin{aligned}= \frac{12}{100} \times 575 \\[1em] = \frac{6900}{100} \\[1em] = ₹69\end{aligned}$$

S.P. of the article = M.P. - Discount

= ₹575 - ₹69

= ₹506

Hence, the discount is ₹69 and the selling price is ₹506.

(ii) Printed price of the article = ₹12,750

Rate of discount = $8\frac{1}{3}$% = $\frac{25}{3}$%

Discount = $\frac{25}{3}$% of ₹12,750

$$\begin{aligned}= \frac{25}{3} \times \frac{1}{100} \times 12750 \\[1em] = \frac{318750}{300} \\[1em] = ₹1{,}062.50\end{aligned}$$

S.P. of the article = Printed price - Discount

= ₹12,750 - ₹1,062.50

= ₹11,687.50

Hence, the discount is ₹1,062.50 and the selling price is ₹11,687.50.

(i) M.P. of the article = ₹780

S.P. of the article = ₹721.50

Discount = M.P. - S.P.

= ₹780 - ₹721.50

= ₹58.50

Discount percentage = $\frac{\text{Discount}}{\text{M.P.}} \times 100$%

= $\frac{58.50}{780} \times 100$%

= $\frac{5850}{780}$%

= 7.5%

Hence, the discount is ₹58.50 and the discount percentage is 7.5%.

(ii) Advertised price of the article = ₹28,500

S.P. of the article = ₹24,510

Discount = Advertised price - S.P.

= ₹28,500 - ₹24,510

= ₹3,990

Discount percentage = $\frac{\text{Discount}}{\text{Advertised price}} \times 100$%

= $\frac{3990}{28500} \times 100$%

= $\frac{399000}{28500}$%

= 14%

Hence, the discount is ₹3,990 and the discount percentage is 14%.

Given:

M.P. of one notebook = ₹30

Rate of discount = 15%

M.P. of a dozen notebooks = 12 x ₹30 = ₹360

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{S.P. of a dozen notebooks} = \Big(1 - \frac{15}{100}\Big)\text{ of M.P.} \\[1em] = \frac{85}{100} \times 360 \\[1em] = \frac{30600}{100} \\[1em] = ₹306\end{aligned}$$

Hence, the student pays ₹306 for a dozen notebooks.

Given:

S.P. of the sunglasses = ₹728

Rate of discount = 9%

Let the marked price of the sunglasses be ₹$x$.

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{S.P.} = \Big(1 - \frac{9}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 728 = \frac{91}{100} \times x \\[1em] \Rightarrow x = \frac{728 \times 100}{91} \\[1em] \Rightarrow x = \frac{72800}{91} \\[1em] \Rightarrow x = 800\end{aligned}$$

Hence, the marked price of the sunglasses is ₹800.

Given:

List price of the article = ₹800

Rate of discount = 20%

Profit = 25%

(i) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ List price

S.P. of the article = $\Big(1 - \frac{20}{100}\Big)$ of list price

$$\begin{aligned}= \frac{80}{100} \times 800 \\[1em] = \frac{64000}{100} \\[1em] = ₹640\end{aligned}$$

Hence, the selling price of the article is ₹640.

(ii) S.P. of the article = ₹640, profit = 25%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 640 = \Big(1 + \frac{25}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 640 = \frac{125}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{640 \times 100}{125} \\[1em] \Rightarrow \text{C.P.} = \frac{64000}{125} \\[1em] \Rightarrow \text{C.P.} = ₹512\end{aligned}$$

Hence, the cost price of the article is ₹512.

Given:

M.P. of the article = ₹2,250

Rate of discount = 12%

Profit = 10%

(i) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the article = $\Big(1 - \frac{12}{100}\Big)$ of M.P.

$$\begin{aligned}= \frac{88}{100} \times 2250 \\[1em] = \frac{198000}{100} \\[1em] = ₹1{,}980\end{aligned}$$

Hence, the selling price of the article is ₹1,980.

(ii) S.P. of the article = ₹1,980, profit = 10%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 1980 = \Big(1 + \frac{10}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 1980 = \frac{110}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{1980 \times 100}{110} \\[1em] \Rightarrow \text{C.P.} = \frac{198000}{110} \\[1em] \Rightarrow \text{C.P.} = ₹1{,}800\end{aligned}$$

Hence, the cost price of the article is ₹1,800.

Given:

C.P. of the badminton racket = ₹650

Rate of discount = 20%

Profit = 20%

(i) S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

S.P. of the badminton racket = $\Big(1 + \frac{20}{100}\Big)$ of C.P.

$$\begin{aligned}= \frac{120}{100} \times 650 \\[1em] = \frac{78000}{100} \\[1em] = ₹780\end{aligned}$$

Hence, the selling price of the badminton racket is ₹780.

(ii) S.P. of the badminton racket = ₹780, discount = 20%

Let the marked price of the badminton racket be ₹$x$.

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\Rightarrow 780 = \Big(1 - \frac{20}{100}\Big) \times x \\[1em] \Rightarrow 780 = \frac{80}{100} \times x \\[1em] \Rightarrow x = \frac{780 \times 100}{80} \\[1em] \Rightarrow x = \frac{78000}{80} \\[1em] \Rightarrow x = 975\end{aligned}$$

Hence, the marked price of the badminton racket is ₹975.

Given:

C.P. of the brass vase = ₹1,200

The vase is marked 80% above the cost price.

Rate of discount = 15%

(i) M.P. of the brass vase = C.P. + 80% of C.P.

$$\begin{aligned}= 1200 + \frac{80}{100} \times 1200 \\[1em] = 1200 + \frac{96000}{100} \\[1em] = 1200 + 960 \\[1em] = ₹2{,}160\end{aligned}$$

Hence, the marked price of the brass vase is ₹2,160.

(ii) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the brass vase = $\Big(1 - \frac{15}{100}\Big)$ of M.P.

$$\begin{aligned}= \frac{85}{100} \times 2160 \\[1em] = \frac{183600}{100} \\[1em] = ₹1{,}836\end{aligned}$$

Hence, the selling price of the brass vase is ₹1,836.

(iii) Profit = S.P. - C.P.

= ₹1,836 - ₹1,200

= ₹636

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{636}{1200} \times 100$%

= $\frac{63600}{1200}$%

= 53%

Hence, his profit percentage is 53%.

Given:

C.P. of the article = ₹1,600, which is 20% below the marked price.

Rate of discount = 16%

(i) Since the cost price is 20% below the marked price,

C.P. = M.P. - 20% of M.P.

$$\begin{aligned}\Rightarrow 1600 = \Big(1 - \frac{20}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 1600 = \frac{80}{100}\text{ of M.P.} \\[1em] \Rightarrow \text{M.P.} = \frac{1600 \times 100}{80} \\[1em] \Rightarrow \text{M.P.} = \frac{160000}{80} \\[1em] \Rightarrow \text{M.P.} = ₹2{,}000\end{aligned}$$

Hence, the marked price of the article is ₹2,000.

(ii) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the article = $\Big(1 - \frac{16}{100}\Big)$ of M.P.

$$\begin{aligned}= \frac{84}{100} \times 2000 \\[1em] = \frac{168000}{100} \\[1em] = ₹1{,}680\end{aligned}$$

Hence, the selling price of the article is ₹1,680.

(iii) Profit = S.P. - C.P.

= ₹1,680 - ₹1,600

= ₹80

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{80}{1600} \times 100$%

= $\frac{8000}{1600}$%

= 5%

Hence, the profit percentage is 5%.

Given:

S.P. of the article = ₹360

Rate of discount = 20%

Profit = 20%

(i) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

Let the marked price of the article be ₹$x$.

$$\begin{aligned}\text{S.P.} = \Big(1 - \frac{20}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 360 = \frac{80}{100} \times x \\[1em] \Rightarrow x = \frac{360 \times 100}{80} \\[1em] \Rightarrow x = \frac{36000}{80} \\[1em] \Rightarrow x = 450\end{aligned}$$

Hence, the marked price of the article is ₹450.

(ii) S.P. of the article = ₹360, profit = 20%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 360 = \Big(1 + \frac{20}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 360 = \frac{120}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{360 \times 100}{120} \\[1em] \Rightarrow \text{C.P.} = \frac{36000}{120} \\[1em] \Rightarrow \text{C.P.} = ₹300\end{aligned}$$

Hence, the cost price of the article is ₹300.

Given:

Printed price of the refrigerator = ₹28,600

Two successive discounts = 10% and 5%

S.P. = $\Big(1 - \frac{d_1}{100}\Big)\Big(1 - \frac{d_2}{100}\Big) \times$ M.P., where $d_1$% and $d_2$% are the two successive discounts.

$$\begin{aligned}\text{S.P. of the refrigerator} = \Big(1 - \frac{10}{100}\Big)\Big(1 - \frac{5}{100}\Big)\text{ of M.P.} \\[1em] = \frac{90}{100} \times \frac{95}{100} \times 28600 \\[1em] = \frac{9}{10} \times \frac{19}{20} \times 28600 \\[1em] = \frac{4890600}{200} \\[1em] = ₹24{,}453\end{aligned}$$

Hence, the customer has to pay ₹24,453 for the refrigerator.

Given:

Both the dealers have marked the article at the same price.

First dealer allows two successive discounts of 15% and 5%.

Second dealer allows a single discount of 20%.

Let the marked price of the article be ₹100.

For the first dealer :

S.P. = $\Big(1 - \frac{d_1}{100}\Big)\Big(1 - \frac{d_2}{100}\Big) \times$ M.P., where $d_1$% and $d_2$% are the two successive discounts.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 - \frac{15}{100}\Big)\Big(1 - \frac{5}{100}\Big) \times 100 \\[1em] = \frac{85}{100} \times \frac{95}{100} \times 100 \\[1em] = \frac{8075}{100} \\[1em] = ₹80.75\end{aligned}$$

For the second dealer :

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 - \frac{20}{100}\Big) \times 100 \\[1em] = \frac{80}{100} \times 100 \\[1em] = ₹80\end{aligned}$$

As ₹80 < ₹80.75, the article is cheaper with the second dealer.

Hence, the offer of the second dealer is the better offer.

Given:

Two successive discounts = 30% and 10%

Let the marked price of the article be ₹$x$ and a single discount of $d$% be equivalent to the two given successive discounts of 30% and 10%.

Both the selling prices must be equal.

$$\begin{aligned}\Rightarrow \Big(1 - \frac{d}{100}\Big)\text{ of }x = \Big(1 - \frac{30}{100}\Big)\Big(1 - \frac{10}{100}\Big)\text{ of }x \\[1em] \Rightarrow 1 - \frac{d}{100} = \frac{70}{100} \times \frac{90}{100} \\[1em] \Rightarrow 1 - \frac{d}{100} = \frac{7}{10} \times \frac{9}{10} \\[1em] \Rightarrow 1 - \frac{d}{100} = \frac{63}{100} \\[1em] \Rightarrow \frac{100 - d}{100} = \frac{63}{100} \\[1em] \Rightarrow 100 - d = 63 \\[1em] \Rightarrow d = 100 - 63 \\[1em] \Rightarrow d = 37\end{aligned}$$

Hence, a single discount of 37% is equivalent to the two given successive discounts.

Given:

Rate of GST = 5%

(i) Cost of the spice packet = ₹50

GST = 5% of ₹50

$$\begin{aligned}= \frac{5}{100} \times 50 \\[1em] = \frac{250}{100} \\[1em] = ₹2.50\end{aligned}$$

Buying price = Cost of item + GST

= ₹50 + ₹2.50

= ₹52.50

Hence, the buying price of the spice packet is ₹52.50.

(ii) Cost of 5 kg of refined oil = 5 x ₹150 = ₹750

GST = 5% of ₹750

$$\begin{aligned}= \frac{5}{100} \times 750 \\[1em] = \frac{3750}{100} \\[1em] = ₹37.50\end{aligned}$$

Buying price = Cost of item + GST

= ₹750 + ₹37.50

= ₹787.50

Hence, the buying price of 5 kg of refined oil is ₹787.50.

Given:

Rate of GST = 18%

(i) Buying price of the small TV = ₹12,980

Let the original price of the small TV be ₹$x$.

GST = 18% of ₹$x$ = ₹$\frac{18x}{100}$

Buying price = Original price + GST

$$\begin{aligned}\Rightarrow 12980 = x + \frac{18x}{100} \\[1em] \Rightarrow 12980 = \frac{100x + 18x}{100} \\[1em] \Rightarrow 12980 = \frac{118x}{100} \\[1em] \Rightarrow x = \frac{12980 \times 100}{118} \\[1em] \Rightarrow x = \frac{1298000}{118} \\[1em] \Rightarrow x = 11000\end{aligned}$$

Hence, the original price of the small TV is ₹11,000.

(ii) Buying price of the shampoo bottle = ₹177

Let the original price of the shampoo bottle be ₹$x$.

GST = 18% of ₹$x$ = ₹$\frac{18x}{100}$

Buying price = Original price + GST

$$\begin{aligned}\Rightarrow 177 = x + \frac{18x}{100} \\[1em] \Rightarrow 177 = \frac{100x + 18x}{100} \\[1em] \Rightarrow 177 = \frac{118x}{100} \\[1em] \Rightarrow x = \frac{177 \times 100}{118} \\[1em] \Rightarrow x = \frac{17700}{118} \\[1em] \Rightarrow x = 150\end{aligned}$$

Hence, the original price of the shampoo bottle is ₹150.

Given:

Buying price of the AC = ₹41,472

Rate of GST = 28%

Let the price of the AC before GST be ₹$x$.

GST = 28% of ₹$x$ = ₹$\frac{28x}{100}$

Buying price = Price before GST + GST

$$\begin{aligned}\Rightarrow 41472 = x + \frac{28x}{100} \\[1em] \Rightarrow 41472 = \frac{100x + 28x}{100} \\[1em] \Rightarrow 41472 = \frac{128x}{100} \\[1em] \Rightarrow x = \frac{41472 \times 100}{128} \\[1em] \Rightarrow x = \frac{4147200}{128} \\[1em] \Rightarrow x = 32400\end{aligned}$$

Hence, the price of the AC before GST was added is ₹32,400.

Given:

Buying price of the shirt = ₹1,344

Original price of the shirt = ₹1,200

GST = Buying price - Original price

= ₹1,344 - ₹1,200

= ₹144

Rate of GST = $\frac{\text{GST}}{\text{Original price}} \times 100$%

= $\frac{144}{1200} \times 100$%

= $\frac{14400}{1200}$%

= 12%

Hence, the rate of GST is 12%.

Given:

Buying price of the purse = ₹560 including 12% GST

New rate of GST = 18%

Let the original price of the purse be ₹$x$.

GST = 12% of ₹$x$ = ₹$\frac{12x}{100}$

Buying price = Original price + GST

$$\begin{aligned}\Rightarrow 560 = x + \frac{12x}{100} \\[1em] \Rightarrow 560 = \frac{100x + 12x}{100} \\[1em] \Rightarrow 560 = \frac{112x}{100} \\[1em] \Rightarrow x = \frac{560 \times 100}{112} \\[1em] \Rightarrow x = \frac{56000}{112} \\[1em] \Rightarrow x = 500\end{aligned}$$

∴ Original price of the purse = ₹500

Now GST = 18% of ₹500

$$\begin{aligned}= \frac{18}{100} \times 500 \\[1em] = \frac{9000}{100} \\[1em] = ₹90\end{aligned}$$

New selling price of the purse = Original price + GST

= ₹500 + ₹90

= ₹590

Hence, the new selling price of the purse is ₹590.

Given:

M.P. of the painting = ₹4,800 excluding GST

Rate of discount = 10%

Rate of GST = 5%

Discount = 10% of ₹4,800

$$\begin{aligned}= \frac{10}{100} \times 4800 \\[1em] = \frac{48000}{100} \\[1em] = ₹480\end{aligned}$$

S.P. of the painting = M.P. - Discount

= ₹4,800 - ₹480

= ₹4,320

Now GST = 5% of ₹4,320

$$\begin{aligned}= \frac{5}{100} \times 4320 \\[1em] = \frac{21600}{100} \\[1em] = ₹216\end{aligned}$$

Amount paid by the customer = S.P. + GST

= ₹4,320 + ₹216

= ₹4,536

Hence, the amount paid by the customer to purchase the painting is ₹4,536.

Given:

M.P. of the solar cooker = ₹7,840 excluding 12% GST

Final cost paid by Amit = ₹7,840 including GST

Let the reduced price of the solar cooker be ₹$x$.

GST = 12% of ₹$x$ = ₹$\frac{12x}{100}$

Amount paid by Amit = Reduced price + GST

$$\begin{aligned}\Rightarrow 7840 = x + \frac{12x}{100} \\[1em] \Rightarrow 7840 = \frac{100x + 12x}{100} \\[1em] \Rightarrow 7840 = \frac{112x}{100} \\[1em] \Rightarrow x = \frac{7840 \times 100}{112} \\[1em] \Rightarrow x = \frac{784000}{112} \\[1em] \Rightarrow x = 7000\end{aligned}$$

∴ Reduced price of the solar cooker = ₹7,000

Amount reduced by the shopkeeper = ₹7,840 - ₹7,000 = ₹840

Hence, the amount reduced by the shopkeeper is ₹840.

(i) Percentage is the numerator of a fraction with denominator 100.

(ii) Let the required number be $x$.

15% of $x$ = 135

$$\begin{aligned}\Rightarrow \frac{15}{100} \times x = 135 \\[1em] \Rightarrow \frac{15x}{100} = 135 \\[1em] \Rightarrow x = \frac{135 \times 100}{15} \\[1em] \Rightarrow x = \frac{13500}{15} \\[1em] \Rightarrow x = 900\end{aligned}$$

Hence, 15% of 900 is 135.

(iii) $\frac{3}{4} = \frac{3}{4} \times 100$%

= $\frac{300}{4}$%

= 75%

Hence, $\frac{3}{4}$ = 75%.

(iv) Both the quantities must be in the same units.

8 rupees = (8 x 100) paise = 800 paise

Required percentage = $\frac{50}{800} \times 100$%

= $\frac{5000}{800}$%

= 6.25%

Hence, 50 paise = 6.25% of 8 rupees.

(v) A shopkeeper earns a profit if S.P. is more than C.P.

(vi) Loss or profit% is always calculated on cost price.

(vii) The additional expenses on transportation, labour, repairing etc. are called overhead expenses.

(viii) Amount deducted from the marked price of an article is called discount.

(ix) Discount = Marked price - selling price

(i) Discount is the amount deducted from the marked price of an article, so it is always calculated on the marked price.

Hence, the given statement is true.

(ii) Let the marked price of the article be ₹100.

S.P. after two successive discounts of 20% and 20% = $\Big(1 - \frac{20}{100}\Big)\Big(1 - \frac{20}{100}\Big) \times 100$

$$\begin{aligned}= \frac{80}{100} \times \frac{80}{100} \times 100 \\[1em] = \frac{6400}{100} \\[1em] = ₹64\end{aligned}$$

Single discount = ₹100 - ₹64 = ₹36

Single discount percentage = $\frac{36}{100} \times 100$% = 36%

As 36% ≠ 40%, the two successive discounts of 20% and 20% are equivalent to a single discount of 36% and not 40%.

Hence, the given statement is false.

(iii) Overhead expenses such as transportation, labour and repairing charges are always added to the original cost to get the total cost.

Hence, the given statement is true.

(iv) When the selling price of an article is less than its cost price, the shopkeeper suffers a loss.

Hence, the given statement is true.

(v) GST is calculated on the selling price of an item and is added to the bill.

Hence, the given statement is true.

$3\frac{1}{8}$% of 75 kg = $\frac{25}{8}$% of 75 kg

$$\begin{aligned}\Rightarrow \frac{25}{8} \times \frac{1}{100} \times 75 \\[1em] \Rightarrow \frac{1875}{800} \\[1em] \Rightarrow \frac{75}{32} \\[1em] \Rightarrow 2\frac{11}{32}\text{ kg}\end{aligned}$$

Hence, option 2 is the correct option.

$2\frac{1}{12}$% = $\frac{25}{12}$%

$$\begin{aligned}\Rightarrow \frac{25}{12} \times \frac{1}{100} \\[1em] \Rightarrow \frac{25}{1200} \\[1em] \Rightarrow \frac{1}{48}\end{aligned}$$

Hence, option 3 is the correct option.

$1\frac{11}{16} = \frac{27}{16}$

$\Rightarrow \frac{27}{16} \times 100$%

$\Rightarrow \frac{2700}{16}$%

⇒ 168.75%

Hence, option 1 is the correct option.

21 : 80 = $\frac{21}{80}$

$\Rightarrow \frac{21}{80} \times 100$%

$\Rightarrow \frac{2100}{80}$%

⇒ 26.25%

Hence, option 4 is the correct option.

Let the required number be $x$.

According to the given condition,

5% of $x$ = 42

$$\begin{aligned}\Rightarrow \frac{5}{100} \times x = 42 \\[1em] \Rightarrow \frac{5x}{100} = 42 \\[1em] \Rightarrow x = \frac{42 \times 100}{5} \\[1em] \Rightarrow x = \frac{4200}{5} \\[1em] \Rightarrow x = 840\end{aligned}$$

Hence, option 3 is the correct option.

Required percentage = $\frac{30}{150} \times 100$%

= $\frac{3000}{150}$%

= 20%

Hence, option 4 is the correct option.

Let the required number be $x$.

New number = $\Big(1 + \frac{\text{Increase Percentage}}{100}\Big) \times$ Original number

Since the number is increased by 25%, we get

$$\begin{aligned}\text{New number} = \Big(1 + \frac{25}{100}\Big)\text{ of }x \\[1em] \Rightarrow 155 = \frac{125}{100} \times x \\[1em] \Rightarrow x = \frac{155 \times 100}{125} \\[1em] \Rightarrow x = \frac{15500}{125} \\[1em] \Rightarrow x = 124\end{aligned}$$

Hence, option 3 is the correct option.

C.P. of the game = ₹600

S.P. of the game = ₹750

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

= ₹750 - ₹600

= ₹150

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\frac{150}{600} \times 100$%

= $\frac{15000}{600}$%

= 25%

Hence, option 3 is the correct option.

C.P. of the watch = ₹680, profit = 35%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the watch} = \Big(1 + \frac{35}{100}\Big)\text{ of C.P.} \\[1em] = \frac{135}{100} \times 680 \\[1em] = \frac{91800}{100} \\[1em] = ₹918\end{aligned}$$

Hence, option 1 is the correct option.

S.P. of the article = ₹150

Gain = ₹30

C.P. of the article = S.P. - Gain

= ₹150 - ₹30

= ₹120

Gain percentage = $\frac{\text{Gain}}{\text{C.P.}} \times 100$%

= $\frac{30}{120} \times 100$%

= $\frac{3000}{120}$%

= 25%

Hence, option 3 is the correct option.

Let the cost price of one article be ₹$x$.

C.P. of 5 articles = ₹$5x$

It is given that S.P. of 6 articles = C.P. of 5 articles

∴ S.P. of 6 articles = ₹$5x$

∴ S.P. of one article = ₹$\frac{5x}{6}$

As $\frac{5x}{6} \lt x$, the selling price of one article is less than its cost price.

Therefore, there is a loss.

Hence, option 2 is the correct option.

M.P. of the article = ₹675, discount = 20%

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{S.P. of the article} = \Big(1 - \frac{20}{100}\Big)\text{ of M.P.} \\[1em] = \frac{80}{100} \times 675 \\[1em] = \frac{54000}{100} \\[1em] = ₹540\end{aligned}$$

Hence, option 1 is the correct option.

M.P. of the jacket = ₹2,590

S.P. of the jacket = ₹2,331

Discount = M.P. - S.P.

= ₹2,590 - ₹2,331

= ₹259

Discount percentage = $\frac{\text{Discount}}{\text{M.P.}} \times 100$%

= $\frac{259}{2590} \times 100$%

= $\frac{25900}{2590}$%

= 10%

Hence, option 2 is the correct option.

M.P. of the showpiece = ₹950

Two successive discounts = 10% and 10%

S.P. = $\Big(1 - \frac{d_1}{100}\Big)\Big(1 - \frac{d_2}{100}\Big) \times$ M.P., where $d_1$% and $d_2$% are the two successive discounts.

$$\begin{aligned}\text{S.P. of the showpiece} = \Big(1 - \frac{10}{100}\Big)\Big(1 - \frac{10}{100}\Big)\text{ of M.P.} \\[1em] = \frac{90}{100} \times \frac{90}{100} \times 950 \\[1em] = \frac{9}{10} \times \frac{9}{10} \times 950 \\[1em] = \frac{76950}{100} \\[1em] = ₹769.50\end{aligned}$$

Hence, option 1 is the correct option.

S.P. of the board game = ₹784 including 12% GST

Let the original price of the board game be ₹$x$.

GST = 12% of ₹$x$ = ₹$\frac{12x}{100}$

S.P. = Original price + GST

$$\begin{aligned}\Rightarrow 784 = x + \frac{12x}{100} \\[1em] \Rightarrow 784 = \frac{100x + 12x}{100} \\[1em] \Rightarrow 784 = \frac{112x}{100} \\[1em] \Rightarrow x = \frac{784 \times 100}{112} \\[1em] \Rightarrow x = \frac{78400}{112} \\[1em] \Rightarrow x = 700\end{aligned}$$

Hence, option 3 is the correct option.

List price of the article = ₹1,050

Rate of GST = 18%

GST = 18% of ₹1,050

$$\begin{aligned}= \frac{18}{100} \times 1050 \\[1em] = \frac{18900}{100} \\[1em] = ₹189\end{aligned}$$

Bill amount = List price + GST

= ₹1,050 + ₹189

= ₹1,239

Hence, option 2 is the correct option.

Profit percentage = $\frac{\text{Profit}}{\text{C.P.}} \times 100$% and loss percentage = $\frac{\text{Loss}}{\text{C.P.}} \times 100$%

In both the formulae, the cost price is taken as the base.

So, Statement I is true.

When the selling price is greater than the cost price, the seller makes a profit and not a loss.

So, Statement II is false.

Hence, option 1 is the correct option.

Let the C.P. of an item be ₹100 and its S.P. be ₹150.

Profit = ₹150 - ₹100 = ₹50

Profit percentage = $\frac{50}{100} \times 100$% = 50%

On doubling the S.P., new S.P. = 2 x ₹150 = ₹300

New profit = ₹300 - ₹100 = ₹200

New profit percentage = $\frac{200}{100} \times 100$% = 200%

As 200% ≠ 2 x 50%, the profit percentage does not double.

So, Statement I is false.

On halving the C.P., new C.P. = $\frac{100}{2}$ = ₹50

New profit = ₹150 - ₹50 = ₹100

New profit percentage = $\frac{100}{50} \times 100$% = 200%

As 200% ≠ 2 x 50%, the profit percentage does not double.

So, Statement II is false.

Hence, option 4 is the correct option.

List price of the item = ₹950 and rate of GST = 6%

GST = 6% of ₹950

$$\begin{aligned}= \frac{6}{100} \times 950 \\[1em] = \frac{5700}{100} \\[1em] = ₹57\end{aligned}$$

S.P. of the item = List price + GST

= ₹950 + ₹57

= ₹1,007

So, Statement I is true.

GST is calculated on the selling price of an item and is added to the bill, so S.P. = list price + taxes.

So, Statement II is true.

Hence, option 3 is the correct option.

Converting,

$\Rightarrow \frac{7}{24} \times 100$%

$\Rightarrow \frac{700}{24}$%

$\Rightarrow \frac{175}{6}$%

⇒ 29.1666.....%

Correct to two decimal places,

$\frac{7}{24}$ = 29.17%

Hence, $\mathbf{\frac{7}{24}}$ = 29.17%.

Given:

Quantity = 750 mL

Total quantity = 4 litres

Both the quantities must be in the same units.

4 litres = (4 x 1000) mL = 4000 mL

Required percentage = $\frac{750}{4000} \times 100$%

= $\frac{75000}{4000}$%

= 18.75%

Hence, 750 mL is 18.75% of 4 litres.

Given:

Number of matches won = 9

Win percentage = 45%

Let the total number of matches played be $x$.

According to the given condition,

45% of $x$ = 9

$$\begin{aligned}\Rightarrow \frac{45}{100} \times x = 9 \\[1em] \Rightarrow \frac{45x}{100} = 9 \\[1em] \Rightarrow x = \frac{9 \times 100}{45} \\[1em] \Rightarrow x = \frac{900}{45} \\[1em] \Rightarrow x = 20\end{aligned}$$

Hence, the football team played 20 matches in all.

Given:

Percentage of votes polled by Rahul = 66%

Votes got by the defeated student = 187

There are only two candidates in a straight election.

Percentage of votes polled by the defeated student = (100 - 66)% = 34%

Let the total number of votes polled be $x$.

According to the given condition,

34% of $x$ = 187

$$\begin{aligned}\Rightarrow \frac{34}{100} \times x = 187 \\[1em] \Rightarrow \frac{34x}{100} = 187 \\[1em] \Rightarrow x = \frac{187 \times 100}{34} \\[1em] \Rightarrow x = \frac{18700}{34} \\[1em] \Rightarrow x = 550\end{aligned}$$

Hence, the total number of votes polled is 550.

Given:

Total quantity of gunpowder = 36 kg

Quantity of nitre = 27 kg

Quantity of charcoal = 5.4 kg

Quantity of sulphur = Total gunpowder - (Nitre + Charcoal)

= 36 - (27 + 5.4)

= 36 - 32.4

= 3.6 kg

Percentage content of sulphur = $\frac{\text{Quantity of sulphur}}{\text{Total quantity of gunpowder}} \times 100$%

= $\frac{3.6}{36} \times 100$%

= $\frac{360}{36}$%

= 10%

Hence, the percentage content of sulphur in the gunpowder is 10%.

Given:

Percentage of students who passed = 75%

If 2 more students had passed, the pass percentage would be 80%.

Let the total number of students in the class be $x$.

Number of students who passed = 75% of $x$ = $\frac{75x}{100}$

If 2 more students had passed, then

$$\begin{aligned}\Rightarrow \frac{75}{100} \times x + 2 = \frac{80}{100} \times x \\[1em] \Rightarrow \frac{80x}{100} - \frac{75x}{100} = 2 \\[1em] \Rightarrow \frac{5x}{100} = 2 \\[1em] \Rightarrow \frac{x}{20} = 2 \\[1em] \Rightarrow x = 2 \times 20 \\[1em] \Rightarrow x = 40\end{aligned}$$

Hence, there are 40 students in the class.

Given:

Monthly salary in 2022 = ₹24,000

Increase in 2023 = 10%

Increase in 2024 = 10%

New salary = $\Big(1 + \frac{\text{Increase Percentage}}{100}\Big) \times$ Original salary

$$\begin{aligned}\text{Salary in 2023} = \Big(1 + \frac{10}{100}\Big) \times 24000 \\[1em] = \frac{110}{100} \times 24000 \\[1em] = \frac{2640000}{100} \\[1em] = ₹26{,}400\\[1em] \text{Salary in 2024} = \Big(1 + \frac{10}{100}\Big) \times 26400 \\[1em] = \frac{110}{100} \times 26400 \\[1em] = \frac{2904000}{100} \\[1em] = ₹29{,}040\end{aligned}$$

Hence, his salary in 2024 is ₹29,040.

Given:

Decrease in price last year = 10%

Increase in price this year = 20%

Let the original price of the commodity be ₹100.

New price = $\Big(1 - \frac{\text{Decrease Percentage}}{100}\Big) \times$ Original price

Price after a decrease of 10% = $\Big(1 - \frac{10}{100}\Big) \times 100$

= $\frac{90}{100} \times 100$

= ₹90

New price = $\Big(1 + \frac{\text{Increase Percentage}}{100}\Big) \times$ Original price

Price after an increase of 20% = $\Big(1 + \frac{20}{100}\Big) \times 90$

$$\begin{aligned}= \frac{120}{100} \times 90 \\[1em] = \frac{10800}{100} \\[1em] = ₹108\end{aligned}$$

Change in price = Final price - Original price

= ₹108 - ₹100

= ₹8

As the change is positive, the price has increased.

Percentage change = $\frac{\text{Change in price}}{\text{Original price}} \times 100$%

= $\frac{8}{100} \times 100$%

= 8%

Hence, the price of the commodity increased by 8% in two years.

Given:

Rate of discount = 15%

(i) M.P. of the cap = ₹150

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{Sale price of the cap} = \Big(1 - \frac{15}{100}\Big)\text{ of M.P.} \\[1em] = \frac{85}{100} \times 150 \\[1em] = \frac{12750}{100} \\[1em] = ₹127.50\end{aligned}$$

Hence, the sale price of the cap is ₹127.50.

(ii) M.P. of the pair of shoes = ₹800

S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$$\begin{aligned}\text{Sale price of the pair of shoes} = \Big(1 - \frac{15}{100}\Big)\text{ of M.P.} \\[1em] = \frac{85}{100} \times 800 \\[1em] = \frac{68000}{100} \\[1em] = ₹680\end{aligned}$$

Hence, the sale price of the pair of shoes is ₹680.

Given:

M.P. of the washing machine = ₹18,000

S.P. of the washing machine = ₹17,100

Discount = M.P. - S.P.

= ₹18,000 - ₹17,100

= ₹900

Discount percentage = $\frac{\text{Discount}}{\text{M.P.}} \times 100$%

= $\frac{900}{18000} \times 100$%

= $\frac{90000}{18000}$%

= 5%

Hence, the discount given is ₹900 and the discount percent is 5%.

Given:

C.P. of each table fan = ₹1,800

Loss on the first table fan = 8%

Profit on the second table fan = 12%

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the first table fan} = \Big(1 - \frac{8}{100}\Big) \times 1800 \\[1em] = \frac{92}{100} \times 1800 \\[1em] = \frac{165600}{100} \\[1em] = ₹1{,}656\end{aligned}$$

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the second table fan} = \Big(1 + \frac{12}{100}\Big) \times 1800 \\[1em] = \frac{112}{100} \times 1800 \\[1em] = \frac{201600}{100} \\[1em] = ₹2{,}016\end{aligned}$$

Total C.P. of both the table fans = ₹1,800 + ₹1,800 = ₹3,600

Total S.P. of both the table fans = ₹1,656 + ₹2,016 = ₹3,672

As S.P. > C.P., there is a profit.

Profit = S.P. - C.P.

= ₹3,672 - ₹3,600

= ₹72

Hence, the selling prices are ₹1,656 and ₹2,016, and the total profit is ₹72.

Given:

Cost of the old car = ₹78,000

Overhead expenses on repairs and repainting = ₹2,000

Gain of Ajit = 15%

Loss of Anuj = 5%

Total C.P. of the car for Ajit = Cost + Overheads

= ₹78,000 + ₹2,000

= ₹80,000

S.P. = $\Big(1 + \frac{\text{Gain Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the car for Ajit} = \Big(1 + \frac{15}{100}\Big) \times 80000 \\[1em] = \frac{115}{100} \times 80000 \\[1em] = \frac{9200000}{100} \\[1em] = ₹92{,}000\end{aligned}$$

This becomes the C.P. of the car for Anuj.

S.P. = $\Big(1 - \frac{\text{Loss Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\text{S.P. of the car for Anuj} = \Big(1 - \frac{5}{100}\Big) \times 92000 \\[1em] = \frac{95}{100} \times 92000 \\[1em] = \frac{8740000}{100} \\[1em] = ₹87{,}400\end{aligned}$$

Hence, the car cost ₹87,400 to Anthony.

Given:

Profit on selling the article = 12%

On selling it for ₹16 more, profit = 20%

Let the cost price of the article be ₹$x$.

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

S.P. of the article at 12% profit = $\Big(1 + \frac{12}{100}\Big)$ of ₹$x$

$$\begin{aligned}= \frac{112}{100} \times x \\[1em] = ₹\frac{28x}{25}\end{aligned}$$

To obtain 20% profit,

$$\begin{aligned}\text{S.P. of the article} = \Big(1 + \frac{20}{100}\Big)\text{ of ₹}x \\[1em] = \frac{120}{100} \times x \\[1em] = ₹\frac{6x}{5}\end{aligned}$$

According to the given information,

$$\begin{aligned}\Rightarrow \frac{6x}{5} = \frac{28x}{25} + 16 \\[1em] \Rightarrow \frac{6x}{5} - \frac{28x}{25} = 16 \\[1em] \Rightarrow \frac{30x - 28x}{25} = 16 \\[1em] \Rightarrow \frac{2x}{25} = 16 \\[1em] \Rightarrow x = \frac{16 \times 25}{2} \\[1em] \Rightarrow x = \frac{400}{2} \\[1em] \Rightarrow x = 200\end{aligned}$$

Hence, the C.P. of the article is ₹200.

Given:

S.P. of the article = ₹460

Rate of discount = 8%

Profit = 15%

(i) S.P. = $\Big(1 - \frac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

Let the marked price of the article be ₹$x$.

$$\begin{aligned}\text{S.P.} = \Big(1 - \frac{8}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 460 = \frac{92}{100} \times x \\[1em] \Rightarrow x = \frac{460 \times 100}{92} \\[1em] \Rightarrow x = \frac{46000}{92} \\[1em] \Rightarrow x = 500\end{aligned}$$

Hence, the marked price of the article is ₹500.

(ii) S.P. of the article = ₹460, profit = 15%

S.P. = $\Big(1 + \frac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$$\begin{aligned}\Rightarrow 460 = \Big(1 + \frac{15}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 460 = \frac{115}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \frac{460 \times 100}{115} \\[1em] \Rightarrow \text{C.P.} = \frac{46000}{115} \\[1em] \Rightarrow \text{C.P.} = ₹400\end{aligned}$$

Hence, the cost price of the article is ₹400.

Given:

Buying price of the laptop = ₹41,300 including 18% GST

Let the price of the laptop before GST be ₹$x$.

GST = 18% of ₹$x$ = ₹$\frac{18x}{100}$

Buying price = Price before GST + GST

$$\begin{aligned}\Rightarrow 41300 = x + \frac{18x}{100} \\[1em] \Rightarrow 41300 = \frac{100x + 18x}{100} \\[1em] \Rightarrow 41300 = \frac{118x}{100} \\[1em] \Rightarrow x = \frac{41300 \times 100}{118} \\[1em] \Rightarrow x = \frac{4130000}{118} \\[1em] \Rightarrow x = 35000\end{aligned}$$

Hence, the price of the laptop before GST was added is ₹35,000.

Given:

M.P. of the gaming mouse = ₹3,000

Rate of discount = 10%

Rate of GST = 18%

Discount = 10% of ₹3,000

$$\begin{aligned}= \frac{10}{100} \times 3000 \\[1em] = \frac{30000}{100} \\[1em] = ₹300\end{aligned}$$

S.P. of the gaming mouse = M.P. - Discount

= ₹3,000 - ₹300

= ₹2,700

Now GST = 18% of ₹2,700

$$\begin{aligned}= \frac{18}{100} \times 2700 \\[1em] = \frac{48600}{100} \\[1em] = ₹486\end{aligned}$$

Amount paid by the customer = S.P. + GST

= ₹2,700 + ₹486

= ₹3,186

Hence, the amount paid by the customer is ₹3,186.

Given:

Monthly subscription charge = ₹200 per month

Annual subscription charge = ₹2,000 per annum

Cost of the monthly plan for one year = 12 x ₹200 = ₹2,400

Cost of the annual plan for one year = ₹2,000

Discount = ₹2,400 - ₹2,000 = ₹400

Discount percentage = $\frac{\text{Discount}}{\text{Cost of monthly plan for one year}} \times 100$%

= $\frac{400}{2400} \times 100$%

= $\frac{40000}{2400}$%

= $\frac{50}{3}$%

= $16\frac{2}{3}$%

Hence, the discount on the Annual subscription plan is $\mathbf{16\frac{2}{3}}$%.

Given:

Sum of the two numbers = $\frac{31}{25}$ of the first number

Let the first number be $x$ and the second number be $y$.

According to the given condition,

$$\begin{aligned}\Rightarrow x + y = \frac{31}{25} \times x \\[1em] \Rightarrow y = \frac{31x}{25} - x \\[1em] \Rightarrow y = \frac{31x - 25x}{25} \\[1em] \Rightarrow y = \frac{6x}{25}\end{aligned}$$

Required percentage = $\frac{y}{x} \times 100$%

= $\frac{\frac{6x}{25}}{x} \times 100$%

= $\frac{6x}{25} \times \frac{1}{x} \times 100$%

= $\frac{600}{25}$%

= 24%

Hence, the second number is 24% of the first number.

Given:

Ratio of boys and girls = 3 : 2

Percentage of boys who wear spectacles = 20%

Percentage of girls who wear spectacles = 25%

Let the number of boys be $3x$ and the number of girls be $2x$.

Total number of students = $3x + 2x = 5x$

Number of boys who wear spectacles = 20% of $3x$

$$\begin{aligned}= \frac{20}{100} \times 3x \\[1em] = \frac{60x}{100} \\[1em] = \frac{3x}{5}\end{aligned}$$

Number of girls who wear spectacles = 25% of $2x$

$$\begin{aligned}= \frac{25}{100} \times 2x \\[1em] = \frac{50x}{100} \\[1em] = \frac{x}{2}\\[1em] \text{Total number of students who wear spectacles} = \frac{3x}{5} + \frac{x}{2} \\[1em] = \frac{6x + 5x}{10} \\[1em] = \frac{11x}{10}\\[1em] \text{Number of students who do not wear spectacles} = 5x - \frac{11x}{10} \\[1em] = \frac{50x - 11x}{10} \\[1em] = \frac{39x}{10}\end{aligned}$$

Percentage of students who do not wear spectacles = $\frac{\frac{39x}{10}}{5x} \times 100$%

= $\frac{39x}{10} \times \frac{1}{5x} \times 100$%

= $\frac{3900}{50}$%

= 78%

Hence, 78% of the students do not wear spectacles.

Given:

First lot bought at the rate of 11 candies for ₹10

Second lot bought at the rate of 9 candies for ₹10

Selling rate of the whole lot = ₹1 per candy

As the LCM of 11 and 9 is 99, let 99 candies of each type be bought.

C.P. of 99 candies of the first type = $\frac{99}{11} \times 10$

= 9 x 10

= ₹90

C.P. of 99 candies of the second type = $\frac{99}{9} \times 10$

= 11 x 10

= ₹110

Total number of candies bought = 99 + 99 = 198

Total C.P. of 198 candies = ₹90 + ₹110 = ₹200

S.P. of 198 candies at ₹1 per candy = 198 x ₹1 = ₹198

As S.P. < C.P., there is a loss.

Loss = C.P. - S.P.

= ₹200 - ₹198

= ₹2

Loss percentage = $\frac{\text{Loss}}{\text{C.P.}} \times 100$%

= $\frac{2}{200} \times 100$%

= $\frac{200}{200}$%

= 1%

Hence, there is a loss of 1% in the whole transaction.