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Percent and Percentage Questions and Answers



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Q76. A candidate passes his exam with 515 marks, having obtained $3\%$ above the minimum. If he had obtained 710 marks what percentage would have been above the minimum?


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$\Rightarrow$ Let minimum marks be = $x$

$\therefore 3\%$ above minimum marks = $x + \dfrac{3}{100} \times x$

$\Rightarrow x + \dfrac{3x}{100} = \dfrac{100x + 3x}{100} = \dfrac{103x}{100}$

$\Rightarrow$ Given minimum marks = 515

So we have:

$\Rightarrow \dfrac{103x}{100} = 515$

$\Rightarrow x = 515 \times \dfrac{100}{103} = 5 \times 100 = 500$

When marks obtained is 710, then:

$\Rightarrow$ Mraks above minimum = $710 - 500 = 210$

$\therefore$ Required percentage = $\dfrac{210}{500} \times 100$

$\Rightarrow \dfrac{210}{500} \times 100 = \dfrac{210}{5} = 42\%$ Ans


Q77. A candidate who gets $35\%$ marks in an examination fails by 40 marks but another candidate who gets $43\%$ marks gets 24 marks more than minimum pass marks. Find pass percentage of marks.


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$\Rightarrow$ Let maximum marks be = $x$

For $1^{st}$ student:

$\Rightarrow$ Marks obtained = $35\%$

$\Rightarrow$ Marks obtained below pass marks = 40 marks

$\therefore$ Pass marks = $x \text{ of } \dfrac{35}{100} + 40 = \dfrac{35x}{100} + 40 = \dfrac{7x}{20} + 40$

For $2^{nd}$ student:

$\Rightarrow$ Marks obtained = $43\%$

$\Rightarrow$ Marks obtained above pass marks = 24 marks

$\therefore$ Pass marks = $x \text{ of } \dfrac{43}{100} - 24 = \dfrac{43x}{100} - 24$

So we have the equation:

$\Rightarrow \dfrac{7x}{20} + 40 = \dfrac{43x}{100} - 24$

$\Rightarrow 40 + 24 = \dfrac{43x}{100} - \dfrac{7x}{20}$

$\Rightarrow 64 = \dfrac{43x - 35x}{100}$

$\Rightarrow \dfrac{8x}{100} = 64$

$\Rightarrow \dfrac{2x}{25} = 64$

$\Rightarrow x = 64 \times \dfrac{25}{2} = 32 \times 25 = 800$

$\therefore$ Maximum marks = 800

$\Rightarrow$ And minimum marks = $(\dfrac{35}{100} \times 800) + 40$

$\Rightarrow (35 \times 8) + 40 = 280 + 40 = 320$

$\therefore$ Required pass percentage = $\dfrac{320}{800} \times 100 = \dfrac{320}{8} = 40\%$ Ans


Q78. After 30 kgs of water was evaporated from a solution of salt and water which had $15\%$ of salt, the reamining solution had $20\%$ of salt. Find the weight of original solution.


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Case 1:

$\Rightarrow$ Let the weight of original solution be = $x$ kg

$\therefore$ Weight of salt in it = $\dfrac{15}{100} \text{ of } x = \dfrac{3x}{20}$

Case 2:

After 30 kg of water evaporated:

$\Rightarrow$ Weight of remaining solution = $(x - 30)$

$\Rightarrow$ Weight of salt in remaining solution = $20\% \text{ of } (x - 30)$

In both cases, the weight of salt will be the same:

$\Rightarrow \dfrac{3x}{20} = \dfrac{20}{100} \times (x - 30)$

$\Rightarrow \dfrac{3x}{20} = \dfrac{1}{5} \times (x - 30)$

$\Rightarrow \dfrac{3x}{20} = \dfrac{x}{5} - \dfrac{30}{5}$

$\Rightarrow \dfrac{3x}{20} = \dfrac{x}{5} - 6$

$\Rightarrow \dfrac{3x}{20} - \dfrac{x}{5} = - 6$

$\Rightarrow \dfrac{3x - 4x}{20} = - 6$

$\Rightarrow \dfrac{x}{20} = 6$

$\Rightarrow x = 6 \times 20 = 120$ kg Ans


Q79. A sample of 50 litres of glycerin is adultrated to the extent of $20\%$. Find how much pure glycerin should be added to bring down the impurity level to $5\%$?


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$\Rightarrow$ Amount of glycerin adultrated = $\dfrac{20}{100} \times 50 = 2 \times 5 = 10$ litres

$\Rightarrow$ Let final amount of glycerin be = $x$ litres

$\Rightarrow$ Impurity level is to be brought down by = $5\%$

$\therefore$ Final amount of glycerin = $\dfrac{5}{100} \times x = 10$

$\Rightarrow \dfrac{1}{20} \times x = 10$

$\Rightarrow x = 10 \times 20 = 200$ litres

$\therefore$ Required amount of glycerin = $200 - 50 = 150$ litres Ans


Q80. 5 litres of water is evaporated from 30 litres of salt solution which contains $16\%$ of salt. Find salt percentage in the remaining solution.


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$\Rightarrow$ Total amount of solution = 30 kgs

$\therefore \dfrac{16}{100} \times 30 = \dfrac{4}{25} \times 30 = \dfrac{4}{5} \times 6 = \dfrac{24}{5}$

$\Rightarrow$ Amount of solution evaporated = 5 litres

$\Rightarrow$ Remaining solution = $30 - 5 = 25$

$\therefore$ Remaining percentage of salt in solution = $\dfrac{\dfrac{24}{5}}{25}$

$\Rightarrow \dfrac{24}{25 \times 5} \times 100 = \dfrac{24}{125} \times 100 = \dfrac{24}{5} \times 4 = \dfrac{96}{5} = 19.2\%$ Ans


Q81. How many litres of $20\%$ alchohol solution should be added to 40 litres of $50\%$ alcohol solution to make a $30\%$ solution?


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$\Rightarrow$ Let $x$ be the quantity of $20\%$ alcohol solution to be added to 40 litres of a $50\%$ alcohol solution.

$\Rightarrow$ And, let $y$ be the quantity of the final $30\%$ solution

So, we have = $x + 40 = y$

Similarly, we have = $y = x + 40$

$\therefore 20\% \text{ of } x + 50\% \text{ of } 40 = 30\% \text{ of } y$

Now, on substituting $y$ by its value of $x + 40$, we have:

$\Rightarrow 20\% \text{ of } x + 50\% \text{ of } 40 = 30\% \text{ of } (x + 40)$

$\Rightarrow \dfrac{20x}{100} + \dfrac{50 \times 40}{100} = \dfrac{30}{100} \times (x + 40)$

$\Rightarrow \dfrac{20x}{100} + \dfrac{50 \times 40}{100} = \dfrac{30x}{100} + \dfrac{30 \times 40}{100}$

$\Rightarrow \dfrac{20x}{100} + \dfrac{2000}{100} = \dfrac{30x}{100} + \dfrac{1200}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow 20x + 2000 = 30x + 1200$

$\Rightarrow 20x - 30x = 1200 - 2000 $

$\Rightarrow -10x = -800 $

$\Rightarrow 10x = 800 $

$\Rightarrow x = \dfrac{800}{10} = 80$

$\therefore$ Required litres of $20\%$ alcohol solution = 80 litres Ans


Q82. John wants to make a 100 ml of $5\%$ alcohol solution mixing a quantity of a $2\%$ alcohol solution with a $7\%$ alcohol solution. What are the quantities of each solution he has to use?


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$\Rightarrow$ Let $x$ be the quantity of $2\%$ alcohol solution

$\Rightarrow$ And, $y$ be the quantity of $7\%$ alcohol solution

$\therefore$ We have = $x + y = 100$

$\Rightarrow$ Similarly $y = 100 - x$

So, we have the following equation:

$\Rightarrow (2\% \text{ of } x) + (7\% \text{ of } y) = 5\% \text{ of } 100$

Now, on substituting $y$ by its value of $100 - x$, we have:

$\Rightarrow \dfrac{2}{100} \text{ of } x + \dfrac{7}{100} \text{ of } (100 - x) = \dfrac{5}{100} \text{ of } 100$

$\Rightarrow \dfrac{2x}{100} + \dfrac{700}{100} - \dfrac{7x}{100} = \dfrac{500}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow 2x + 700 - 7x = 500$

$\Rightarrow 2x - 7x = 500 - 700$

$\Rightarrow -5x = -200$

$\Rightarrow 5x = 200$

$\Rightarrow x = \dfrac{200}{5} = 40$

$\therefore y = 100 - 40 = 60$ ml each of both solution Ans


Q83. Sterling silver is $92.5\%$ of pure silver. How many grams of sterling silver must be mixed to a $90\%$ silver alloy to obtain a 500 gram of $91\%$ silver alloy?


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$\Rightarrow$ Let quantity of sterling silver be = $x$

$\Rightarrow$ And, let quantity of silver alloy to be obtained be = $y$

$\therefore$ We have = $x + y = 500$

$\Rightarrow$ Similarly, $y = 500 - x$

So, we have the following equation:

$\Rightarrow (92.5\% \text{ of } x) + (90\% \text{ of } y) = 91\% \text{ of } 500$

Now, on substituting $y$ by its value of $500 - x$, we have:

$\Rightarrow (\dfrac{92.5}{100} \times x) + (\dfrac{90}{100} \times (500 - x)) = \dfrac{91}{100} \times 500$

$\Rightarrow \dfrac{92.5x}{100} + \dfrac{90 \times 500}{100} - \dfrac{ 90x}{100} = \dfrac{91 \times 500}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow 92.5x + 45,000 - 90x = 45,500$

$\Rightarrow 92.5x - 90x = 45,500 - 45,000$

$\Rightarrow 2.5x = 500$

$\Rightarrow \dfrac{25x}{10} = 500$ (Converting into fraction)

$\Rightarrow x = 500 \times \dfrac{10}{25} = 500 \times \dfrac{2}{5} = 100 \times 2 = 200$ grams Ans


Q84. How manny kilograms of pure water is to be added to 100 kilograms of a $30\%$ saline solution to make it a $10\%$ saline solution?


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$\Rightarrow$ Let weight of water be = $x$

$\Rightarrow$ And, let weight of salt water be = $y$

$\therefore$ We have = $y = x + 100$

So, we have the following equation:

$\Rightarrow x + 30\% \text{ of } 100 = 10\% \text{ of } y$

Now, on substituting $y$ by its value of $x + 100$, we have:

$\Rightarrow x + 30\% \text{ of } 100 = 10\% \text{ of } (x + 100)$

$\Rightarrow x + \dfrac{30}{100} \times 100 = \dfrac{10}{100} \times (x + 100)$

$\Rightarrow x + \dfrac{30 \times 100}{100} = \dfrac{10x}{100} + \dfrac{10 \times 100}{100}$

Assuming that $x$ contains no salt, we eliminateits value from he equation

$\Rightarrow \dfrac{3000}{100} = \dfrac{10x}{100} + \dfrac{1000}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow \dfrac{3000}{100} - \dfrac{1000}{100} = \dfrac{10x}{100}$

$\Rightarrow 10x = 3000 - 1000 = 2000$

$\Rightarrow x = \dfrac{2000}{10} = 200$ kg Ans


Q85. A 50 ml after shave lotion at $30\%$ alcohol is mixed with 30 ml of pure water. Find the percentage of alcohol in the new solution?


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$\Rightarrow$ Quantity of final mixture is given by = 50 ml $+$ 30 ml = 80 ml

Now, the amount of alcohol in the final solution is equal to the amouont of alcohol in pure water (which is zero) plus the amount of alcohol in the final solution.

$\Rightarrow$ Let percentage of alcohol in final solution be = $x$

$\therefore 0 + 30\% \text{ of } 50 = x\% \text{ of } 80$

$\Rightarrow \dfrac{30}{100} \times 50 = \dfrac{x}{100} \times 80$

$\Rightarrow \dfrac{30 \times 50}{100} = \dfrac{80x}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow 1500 = 80x$

$\Rightarrow x = \dfrac{1500}{80} = \dfrac{150}{8} = 18.75\%$ Ans


Q86. You add $x$ ml of a $25\%$ alcohol solution to a 200 ml of a $10\%$ alcohol solution to obtain another solution. Find the amount of alcohol in the final solution in terms of $x$. Find:

(i) Find the ratio, in terms of $x$, of the alcohol in the final solution to the total amount of the solution. (ii) Also, what do you think will happen if the value of $x$ is very large? (iii) Find $x$ so that the final solution has a percentage of $15\%$?


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(i) Amount of alcohol in 200 ml solution = $\dfrac{10}{100} \times 200 = 20 \text{ ml}$

$\Rightarrow$ Amount of alcohol in $x$ ml solution = $\dfrac{25}{100} \times x = \dfrac{25x}{100} = 0.25x \text{ ml}$

$\therefore$ Ratio of alcohol in final solution to the total amount:

$\Rightarrow \dfrac{20 + 0.25x}{x + 200}$ Ans

(ii) Now, if $x$ becomes very large in the above formula, then the ratio becomes close to 0. This means that if you increase the amount of $x$ in $25\%$ solution, it will dominate and the final solution will be very close to $25\%$ solution.

(iii) In order to have a $15\%$ percentage, we need to have:

$\Rightarrow 10\% \text{ of } 200 + 0.25x = 15\%(x + 200)$

$\Rightarrow \dfrac{10 \times 200}{100} + 0.25x = \dfrac{15x}{100} + (\dfrac{15 \times 200}{100})$

$\Rightarrow \dfrac{2000}{100} + \dfrac{25x}{100} = \dfrac{15x}{100} + \dfrac{3000}{100}$

Now, simplify by multiplying all terms by 100, we have:

$\Rightarrow 2000 + 25x = 15x + 3000$

$\Rightarrow 25x - 15x = 3000 - 2000$

$\Rightarrow 10x = 1000$

$\Rightarrow x = \dfrac{1000}{10} = 100$ ml Ans


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