Percentage — RRB NTPC Quantitative Aptitude

• Given: A number increased by $16 \frac{2}{3}\%$ becomes 4956.

• Basic Method (Solution):
Step 1: Let the number be $x$. $x + 16.66\% \times x = 4956$
Step 2: $x(1 + \frac{1}{6}) = 4956 \rightarrow \frac{7}{6}x = 4956$
Step 3: $x = \frac{4956 \times 6}{7} = 4248$

• Shortcut Method: $16 \frac{2}{3}\% = \frac{1}{6}$. The original number is 6 units. Adding 1 unit makes it 7 units. Given 7 units = 4956, so 1 unit = $\frac{4956}{7} = 708$. The original number is 6 units = $6 \times 708 = 4248$. Quick option check: The answer must be a multiple of 6, ending in 8.

• Final Answer: 4248

• Given: Number + 930 = $444 \frac{4}{9}\%$ of the number.

• Basic Method (Solution):
Step 1: $x + 930 = 4.4444 x$
Step 2: $3.4444 x = 930$
Step 3: $x = \frac{930}{\frac{31}{9}} = 270$

• Shortcut Method: Convert the percentage to a fraction: $444 \frac{4}{9}\% = 400\% + 44 \frac{4}{9}\% = 4 + \frac{4}{9} = \frac{40}{9}$. This means if the original number is 9 units, it becomes 40 units. The addition is $40 - 9 = 31$ units. Given that 31 units = 930, we find 1 unit = 30. The original number is 9 units = $9 \times 30 = 270$.

• Final Answer: 270

• Given: $A = B + 25\%$

• Basic Method (Solution):
Step 1: Let $B = 100$, so $A = 125$.
Step 2: Difference $= 125 - 100 = 25$.
Step 3: Percentage less $= \frac{25}{A} \times 100 = \frac{25}{125} \times 100 = 20\%$.

• Shortcut Method: Use fraction equivalents. $25\% = \frac{1}{4}$. If B is 4, A is $4+1=5$. The gap is 1. B is less than A by $\frac{1}{5}$. Since $\frac{1}{5} = 20\%$, B's income is 20% less.

• Final Answer: 20%

• Given: $P = 30\%$ of $Q$, $Q = 20\%$ of $R$.

• Basic Method (Solution):
Step 1: $P = 0.3Q$ and $Q = 0.2R$.
Step 2: Substitute Q into the first equation: $P = 0.3(0.2R) = 0.06R$.
Step 3: $0.06 = 6\%$.

• Shortcut Method: Successive percentage multiplication. $P$ as a percentage of $R$ is $30\% \times 20\%$. As fractions, $\frac{3}{10} \times \frac{1}{5} = \frac{3}{50} = \frac{6}{100} = 6\%$.

• Final Answer: 6%

• Given: $80\% \times 50\% \times 90$

• Basic Method (Solution):
Step 1: $x = \frac{80}{100} \times \frac{50}{100} \times 90$
Step 2: $x = 0.8 \times 0.5 \times 90 = 0.4 \times 90 = 36$.

• Shortcut Method: $50\%$ means half. Half of 90 is 45. $80\%$ is $\frac{4}{5}$. So, $\frac{4}{5} \times 45 = 4 \times 9 = 36$.

• Final Answer: 36

• Given: $x = \frac{\frac{3}{20}}{\frac{12}{25}} \times 100$

• Basic Method (Solution):
Step 1: $x = \frac{3}{20} \times \frac{25}{12} \times 100$
Step 2: $x = \frac{75}{240} \times 100$
Step 3: $x = \frac{5}{16} \times 100 = 31.25\%$

• Shortcut Method: Compare using ratio. Ratio = $\frac{3}{20} : \frac{12}{25} = 75 : 240 = 5 : 16$. The base is 16. $\frac{1}{16}$ is $6.25\%$, so $\frac{5}{16} = 5 \times 6.25\% = 31.25\%$.

• Final Answer: 31.25%

• Given: Price reduction $= 20\%$, Extra quantity $= 1$ dozen, Fixed amount $= ₹50$.

• Basic Method (Solution):
Step 1: Let the original price be $p$ per dozen. New price $= 0.8p$.
Step 2: $\frac{50}{0.8p} - \frac{50}{p} = 1 \rightarrow \frac{62.5}{p} - \frac{50}{p} = 1$
Step 3: $\frac{12.5}{p} = 1 \rightarrow p = 12.5$. Reduced price $= 0.8 \times 12.5 = 10$.

• Shortcut Method: 20% of ₹50 = ₹10. This ₹10 reduction is exactly what allows the person to buy the extra 1 dozen mangoes. Thus, the reduced price of 1 dozen mangoes is strictly ₹10. (Alternatively, Price ratio 5:4 -> Consumption ratio 4:5. Difference 1 unit = 1 dozen, so new consumption = 5 dozens. $50 / 5 = ₹10$).

• Final Answer: ₹10

• Given: $L_{new} = L + 15\%$, $B_{new} = B - 10\%$.

• Basic Method (Solution):
Step 1: Let $L=100, B=100, Area = 10000$.
Step 2: $L_{new} = 115, B_{new} = 90$. $Area_{new} = 115 \times 90 = 10350$.
Step 3: Change = $10350 - 10000 = 350$. Percentage change $= \frac{350}{10000} \times 100 = 3.5\%$.

• Shortcut Method: Use successive percentage formula for Area = L x B: Net change $= a + b + \frac{ab}{100}$. Here $a = +15, b = -10$. Net $= 15 - 10 + \frac{(15)(-10)}{100} = 5 - 1.5 = 3.5\%$ increase.

• Final Answer: 3.5% increase

• Given: Side of square $a$ increases by $10\%$.

• Basic Method (Solution):
Step 1: Original perimeter $P_1 = 4a$.
Step 2: New side $a_{new} = 1.1a$. New perimeter $P_2 = 4(1.1a) = 4.4a$.
Step 3: Increase $= 4.4a - 4a = 0.4a$. Percentage increase $= \frac{0.4a}{4a} \times 100 = 10\%$.

• Shortcut Method: Perimeter is a linear (one-dimensional) measurement. Any percentage change in a one-dimensional property (side, radius, diameter) directly translates to the identical percentage change in any other one-dimensional property (perimeter, circumference, diagonal). No calculation needed.

• Final Answer: 10%

• Given: Price $+20\%$, Expenditure $+5\%$, Original Consumption $= 72$ kg.

• Basic Method (Solution):
Step 1: $Price_1 \times Cons_1 = Exp_1$. Let $P_1 = 100, E_1 = 100$. $Cons_1 = \frac{100}{100} = 1$ unit $= 72$ kg.
Step 2: $P_2 = 120, E_2 = 105$. $Cons_2 = \frac{E_2}{P_2} = \frac{105}{120} = \frac{7}{8}$ units.
Step 3: New consumption $= \frac{7}{8} \times 72 = 63$ kg.

• Shortcut Method: Ratio method. $Consumption = \frac{Expenditure}{Price}$. Price ratio $100 : 120 = 5 : 6$. Expenditure ratio $100 : 105 = 20 : 21$. Consumption ratio $= \frac{20}{5} : \frac{21}{6} = 4 : 3.5 = 8 : 7$. Since 8 units represent 72 kg (multiplier 9), 7 units represent $7 \times 9 = 63$ kg.

• Final Answer: 63 kg