percentage — BSSC Quantitative Aptitude

Formula:
Required Percentage Less = $\left(\frac{R}{100 + R}\right) \times 100\%$

Step-by-Step Solution:

• Let $B$'s income be $\text{Rs. } 100$.

• Then $A$'s income = $100 + 25 = \text{Rs. } 125$.

• Difference = $125 - 100 = \text{Rs. } 25$.

• Percentage by which $B$ is less than $A$ = $\left(\frac{25}{125}\right) \times 100 = 20\%$.

Exam Trick:
Using fraction conversion: $25\% = \frac{1}{4}$. Since $A = 4 + 1 = 5$ parts and $B = 4$ parts, $B$ is $1$ part less out of $5$ parts. Required percentage = $\frac{1}{5} \times 100 = 20\%$.

Formula:
When a value is increased by $x\%$ and then decreased by $x\%$, there is always a net decrease given by $\frac{x^2}{100}\%$.

Step-by-Step Solution:

• Here $x = 15$.

• Net percentage change = $-\frac{15^2}{100}\% = -\frac{225}{100}\% = -2.25\%$.

• The negative sign indicates a decrease of $2.25\%$.

Exam Trick:
Net percentage formula $a + b + \frac{ab}{100}$: Here $a = +15$ and $b = -15$.
Net change = $15 - 15 + \frac{(15)(-15)}{100} = -2.25\%$, which means a $2.25\%$ decrease.

Formula:
$\text{Expenditure} = \text{Price} \times \text{Consumption}$

Step-by-Step Solution:

• Let initial Price = $100$ and initial Consumption = $100$.

• Initial Expenditure = $100 \times 100 = 10000$.

• New Price = $120$.

• New Expenditure = $10000 + 8\% \text{ of } 10000 = 10800$.

• New Consumption = $\frac{10800}{120} = 90$.

• Percentage reduction in consumption = $\frac{100 - 90}{100} \times 100 = 10\%$.

Exam Trick:
Ratio method: Initial price $100 \rightarrow$ New price $120$.
Target expenditure $108$. Reduction needed in price ratio = $120 \rightarrow 108$, which is a decrease of $12$ on $120$.
Percentage reduction = $\frac{12}{120} \times 100 = 10\%$.

Formula:
$\text{Income} = \text{Expenditure} + \text{Savings}$

Step-by-Step Solution:

• Let initial Income = $100$.

• Initial Savings = $20$, initial Expenditure = $80$.

• New Income = $100 + 15\% = 115$.

• New Expenditure = $80 + 10\% \text{ of } 80 = 80 + 8 = 88$.

• New Savings = $115 - 88 = 27$.

• Increase in savings = $27 - 20 = 7$.

• Percentage increase in savings = $\frac{7}{20} \times 100 = 35\%$.

Exam Trick:
Alligation / Weighted Average Method:
Savings change = $S\%$, Expenditure change = $10\%$, Total Income change = $15\%$.
Ratio of Expenditure : Savings = $80 : 20 = 4 : 1$.
Weighted change: $15 = \frac{4 \times 10 + 1 \times S}{4 + 1} \implies 15 \times 5 = 40 + S \implies S = 35\%$.

Formula:
$\text{Winning Margin} = (\%\text{ Winner} - \%\text{ Loser}) \times \text{Valid Votes}$

Step-by-Step Solution:

• Let total enrolled voters = $x$.

• Votes cast = $0.90x$.

• Valid votes = $0.90 \times 0.90x = 0.81x$.

• Winner gets $54\%$ of valid votes, Loser gets $46\%$ of valid votes.

• Difference = $54\% - 46\% = 8\%$ of valid votes.

• $8\% \text{ of } 0.81x = 1620$.

• $\frac{8}{100} \times 0.81x = 1620 \implies 0.0648x = 1620$.

• $x = \frac{1620}{0.0648} = 25000$.

Exam Trick:
Chain rule: Total voters $x \times \frac{9}{10} \times \frac{9}{10} \times \frac{8}{100} = 1620$.
$x \times \frac{648}{100000} = 1620 \implies x = \frac{1620 \times 100000}{648} = 25000$.

Formula:
$\text{Difference in Percentage} = \frac{\text{Sum of deficit and surplus}}{\text{Maximum Marks}} \times 100\%$

Step-by-Step Solution:

• Difference in marks percentage = $40\% - 30\% = 10\%$.

• Difference in actual marks = $15 + 35 = 50$ marks.

• $10\%$ of Maximum Marks = $50$.

• Maximum Marks = $\frac{50}{10} \times 100 = 500$.

• Passing marks = $30\% \text{ of } 500 + 15 = 150 + 15 = 165$.

• Passing percentage = $\frac{165}{500} \times 100 = 35\%$.

Exam Trick:
$10\% \rightarrow 50 \implies 100\% = 500$ (Max Marks).
Since $50$ marks = $10\%$, $15$ marks = $\frac{15}{50} \times 10\% = 3\%$.
Passing percentage = $30\% + 3\% = 35\%$.

Formula:
$P_{\text{present}} = P_{\text{initial}} \times \left(1 + \frac{r_1}{100}\right) \times \left(1 - \frac{r_2}{100}\right) \times \left(1 + \frac{r_3}{100}\right)$

Step-by-Step Solution:

• Let initial population = $P$.

• Growth factors: $10\% = \frac{11}{10}$, $-20\% = \frac{4}{5}$, $+25\% = \frac{5}{4}$.

• $P \times \frac{11}{10} \times \frac{4}{5} \times \frac{5}{4} = 1,65,000$.

• $P \times \frac{11}{10} = 1,65,000$.

• $P = 1,65,000 \times \frac{10}{11} = 15,000 \times 10 = 1,50,000$.

Exam Trick:
Cancel out common fractions: $\frac{4}{5} \times \frac{5}{4} = 1$.
So the net multiplier over 3 years is simply $\frac{11}{10}$.
$P \times \frac{11}{10} = 165000 \implies P = 150000$.

Formula:
Since water quantity remains constant: $\text{Initial Solution} \times \%\text{ Water}_{1} = \text{New Solution} \times \%\text{ Water}_{2}$

Step-by-Step Solution:

• Initial Water percentage = $100\% - 20\% = 80\%$.

• Initial quantity of water = $60 \times 80\% = 48\text{ liters}$.

• In the new solution, acid concentration = $40\% \implies$ Water concentration = $60\%$.

• Since water quantity does not change, $60\% \text{ of New Solution} = 48\text{ liters}$.

• New Solution = $\frac{48}{0.60} = 80\text{ liters}$.

• Acid added = New Solution - Initial Solution = $80 - 60 = 20\text{ liters}$.

Exam Trick:
Equating non-changing component (Water):
$60 \times 80\% = N \times 60\% \implies N = 80\text{ liters}$.
Added Acid = $80 - 60 = 20\text{ liters}$.

Formula:
$\text{Total Percentage} = n(A) + n(B) - n(A \cap B) + \text{Failed in Both}$

Step-by-Step Solution:

• Percentage of students failing in Math = $100\% - 65\% = 35\%$.

• Percentage of students failing in English = $100\% - 55\% = 45\%$.

• Failed in both = $20\%$.

• Percentage failing in at least one subject = $35\% + 45\% - 20\% = 60\%$.

• Percentage passing in both subjects = $100\% - 60\% = 40\%$.

• Given $40\% \text{ of Total} = 240 \implies \text{Total} = \frac{240}{40} \times 100 = 600$.

Exam Trick:
Passing in at least one subject = $100\% - 20\% = 80\%$.
By Venn diagram formula: $65\% + 55\% - P(\text{Both}) = 80\% \implies P(\text{Both}) = 120\% - 80\% = 40\%$.
$40\% \rightarrow 240 \implies 100\% \rightarrow 600$.

Formula:
$\text{Increase in Income Tax} = \text{Decrease in Net Income}$
$\text{Gross Income} = \text{Income Tax} + \text{Net Income}$

Step-by-Step Solution:

• Change in Tax = $19\% \text{ of Tax}$.

• Change in Net Income = $1\% \text{ of Net Income}$.

• $19\% \times \text{Tax} = 1\% \times \text{Net Income} \implies \frac{\text{Tax}}{\text{Net Income}} = \frac{1}{19}$.

• Gross Income = $\text{Tax} + \text{Net Income} = 1 + 19 = 20$.

• Initial Tax Rate = $\left(\frac{\text{Tax}}{\text{Gross Income}}\right) \times 100 = \frac{1}{20} \times 100 = 5\%$.

Exam Trick:
Direct Tax Rate Formula = $\frac{\text{Net Income Change}}{\text{Tax Change} + \text{Net Income Change}} \times 100\%$.
$\text{Tax Rate} = \frac{1}{19 + 1} \times 100 = \frac{1}{20} \times 100 = 5\%$.