Which is larger: $16.66\% \text{ of } 24$ OR $24\% \text{ of } 16.66$?

Answer: Equal. $X\% \text{ of } Y = Y\% \text{ of } X$. Multiplication is commutative. $16.66 \times 24 / 100$ is same as $24 \times 16.66 / 100$.

If you multiply a number by $4/5$ instead of $5/4$, what is the % error?

Answer: 36%. Let number = 20 (LCM of 4,5). Correct: $20 \times 5/4 = 25$. Wrong: $20 \times 4/5 = 16$. Error = $25-16=9$. % Error = $(9/25) \times 100 = 36\%$.

Price of sugar increases by 20%. A family increases its expenditure by only 10%. Find the % change in consumption.

Answer: 8.33% Decrease. Exp = Price × Cons. $110 = 120 \times Cons$. $Cons = 110/120 = 11/12$. Change: $1/12$ Decrease $\to 8.33\%$.

Population of a town is 10,000. Males increased by 10% and Females by 20%. Total population became 11,200. Find initial number of males.

Answer: 8000. Total Increase: $1200 / 10000 = 12\%$. Alligation: M(10) | F(20) $\to$ Mean(12). Ratio = $(20-12) : (12-10) = 8:2 = 4:1$. Males = $4/5 \times 10000 = 8000$.

A student scored 30% marks and failed by 5 marks. Another scored 40% marks and got 10 more than passing marks. Find Max Marks.

Answer: 150. Diff in %: $40\% - 30\% = 10\%$. Diff in Marks: $10 - (-5) = 15$. $10\% = 15 \to 100\% = 150$.

The length of a cuboid is increased by 20%, breadth by 30%, and height is decreased by 50%. Volume change?

Answer: 22% Decrease. Volume $V = L \times B \times H$. Using Ratio: $L(5 \to 6)$, $B(10 \to 13)$, $H(2 \to 1)$. Initial: $5 \times 10 \times 2 = 100$. Final: $6 \times 13 \times 1 = 78$. Change: $100 \to 78 = 22\%$ Decrease.

A man spends 75% of his income. Income increased by 20% and expenditure increased by 10%. % Increase in Savings?

Answer: 50%. I(100), Exp(75), Sav(25). New I=120. New Exp = 110% of 75 = 82.5. New Sav = $120 - 82.5 = 37.5$. Inc: $37.5 - 25 = 12.5$. % Inc: $12.5/25 = 50\%$.

In an election, 20% votes were invalid. Winner got 55% of VALID votes and won by 3600 votes. Total votes cast?

Answer: 45000. Let Total = 100x. Valid = 80x. Valid Votes Winner gap: $55\% - 45\% = 10\%$. $10\% \text{ of } 80x = 3600 \to 8x = 3600 \to x = 450$. Total = $100 \times 450 = 45000$.

Entry ticket of cinema hall is Rs 250. Price reduced by X%, sales increased by 50%, total revenue increased by 20%. Find X.

Answer: 20%. Rev = Price × Sale. $120 = P_{new} \times 150$. $P_{new} = 120/150 = 4/5$. Price reduced from 1 to 4/5 $\to 1/5$ decrease = 20%.

Due to reduction of 20% in price of wheat, a man is able to buy 5 kg more for Rs 320. Original price/kg?

Answer: 16. 20% ($1/5$) price drop means consumption $\uparrow$ by $1/(5-1) = 1/4$. $1/4 \text{ of Original Cons} = 5kg \to \text{Orig Cons} = 20kg$. Orig Price = $320/20 = 16$.

The value of a machine depreciates 10% annually. If present value is 8100, what was it 2 years ago?

Answer: 10000. 10% dep $\implies 10 \to 9$. 2 years $\implies 100 \to 81$. $81 \text{ units} = 8100 \to 1 \text{ unit} = 100$. Initial = 10000.

In an exam, minimum passing is 35% for SC and 40% for General. A General candidate got 320 and failed by 80. How much did an SC candidate need to pass?

Answer: 350. General Pass = $320+80 = 400$. $40\% = 400 \to 1\% = 10$. SC Pass = $35\% = 35 \times 10 = 350$.

Radius of circle $\uparrow 10%$. Area $\uparrow$ by?

Answer: 21%. $10 + 10 + 100/100 = 21%$.

If 60% of A = 3/4 of B, then A:B is?

Answer: 5:4. $3/5 A = 3/4 B$. $A/B = (3/4) \times (5/3) = 5/4$.

Salesman Commission: 5% on sales up to 10k, 4% on sales above 10k. If Total Sales 15k, find commission.

Answer: 700. $5% of 10k = 500$. Remaining sales 5k. $4% of 5k = 200$. Total = 700.

Passed Exam: 40% represents 80 marks. Max Marks?

Answer: 200. $40% = 80 \to 1% = 2 \to 100% = 200$.

Income tax rate increases from 4% to 5%. To keep net income same, gross income must increase by?

Answer: 1.05%. Let Income 100. Tax 4. Net 96. New Tax 5%. We want Net 96. So Gross - 5% of Gross = 96. $0.95 G = 96 \to G = 101.05$. Increase 1.05.

If numerator $\uparrow 20%$, denominator $\downarrow 10%$, fraction becomes 16/21. Original fraction?

Answer: 4/7. $x(1.2) / y(0.9) = 16/21$. $x/y = (16/21) \times (9/12) = 4/7$.

A is 50% larger than C and B is 25% large than C. A is what % larger than B?

Answer: 20%. $C=100. A=150. B=125$. Diff = 25. Base B=125. $25/125 = 1/5 = 20\%$.

Man spends 70% of income. Income $\uparrow 20%$. Expenditure $\uparrow 10%$. Savings $\uparrow %$?

Answer: 43.33%. I=100, E=70, S=30. New I=120, New E=77 (70+7). New S = $120-77 = 43$. Change $30 \to 43$ is 13. $13/30 \times 100 = 43.33\%$.

120 is 20% of a number. Then 120% of that number is?

Answer: 720. $20% = 120 \to 1% = 6$. $120% = 6 \times 120 = 720$.

Two numbers are 20% and 50% more than a third number. Ratio of two numbers?

Answer: 4:5. C=100. A=120, B=150. Ratio $120:150 = 4:5$.

A man's salary was reduced by 50%. Again, the reduced salary was increased by 50%. Find his loss in percentage.

Answer: 25%. 100 -> 50 -> 75. Loss = 25%. Formula: $x^2/100$ loss. $50^2/100 = 2500/100 = 25$.

Price of sugar $\uparrow 20%$. Family consumption $\downarrow 20%$. Expense change?

Answer: 4% Decrease. Always $x^2/100$ decrease. $400/100 = 4\%$.

Passing marks are 30%. S gets 200 and fails by 10 marks. Max Marks?

Answer: 700. Pass Mark = $200+10 = 210$. $30% = 210 \to 1% = 7 \to 100% = 700$.

In an election, 10% didn't vote, 11.11% votes valid were declared invalid. Winner got 75% of valid votes (vs loser) and won by 2000. Total Voters on list?

Answer: 5000. Chain Rule: $x \times 9/10 \times 8/9 = Valid$. Winner 75%, Loser 25% (of valid). Diff 50% = 1/2. $x \times (9/10) \times (8/9) \times (1/2) = 2000$. $x \times 4/10 = 2000 \to x = 5000$.

FreshGrapes contain 80% water. DryGrapes contain 10% water. If dry grapes weight 250kg, what was fresh weight?

Answer: 1125 kg. Dry Pulp = 90% of 250 = 225kg. Fresh Pulp = 20% of Fresh. $0.2 F = 225 \to F = 1125$.

Tax on commodity diminishes by 20%, consumption increases by 15%. Decrease in Revenue is?

Answer: 8%. $-20 + 15 + (-20)(15)/100 = -5 - 3 = -8\%$.

If A has 50% more than B, B has 20% less than C, C has 20% more than D. If D is 100, find A.

Answer: 144. $D=100 \to C=120$. $B = 120 \times 0.8 = 96$. $A = 96 \times 1.5 = 144$.

A student multiplied a number by 3/5 instead of 5/3. % Error?

Answer: 64%. Example again. LCM(3,5)=15. Correct $15 \times 5/3=25$. Wrong $15 \times 3/5=9$. Diff 16. Error $16/25 = 64\%$.

Home/Rajasthan 4th Grade Exam/Quantitative Aptitude/

Percentage

The heart of Arithmetic. Master Standard Reciprocals (1/n), the AB Rule, and Successive Change methods to solve questions mentally.

Expert Answer & Key Takeaways

The heart of Arithmetic. Master Standard Reciprocals (1/n), the AB Rule, and Successive Change methods to solve questions mentally.

1. Standard Reciprocals (The Speed Hack)

Memorize these to avoid division during exams.

1/2 = 50%
1/3 = 33.33%
1/4 = 25%
1/5 = 20%
1/6 = 16.66%
1/7 = 14.28%
1/8 = 12.5%
1/9 = 11.11%
1/10 = 10%
1/11 = 09.09%
1/12 = 8.33%
1/15 = 6.66%
1/16 = 6.25%
3/8 = 37.5%, 5/8 = 62.5%

Example:

Q: Calculate 14.28% of 49.

Solution: We know

14.28\% = 1/7

49 \times (1/7) = 7

2. The AB Rule (Constant Product)

A \times B = Constant

, then if A increases by

x\%

, B must decrease by

y\%

.
Rule: If A increases by 1n, B must decrease by 1n+1 to keep product constant.

Sugar Price Example: Price $\uparrow 25\% (1/4)$ . Consumption must $\downarrow 1/(4+1) = 1/5 = 20\%$ .

Example:

Q: Price of oil increases by 20% (1/5). How much should consumption reduce to keep expense same?

Solution: Increase = 15.
Decrease = 15+1 = 16.
16 = 16.66%.

3. Successive Percentage Change

When a value changes by

x\%

and then by

y\%

Formula: Net Change = $x + y + \frac{xy}{100} \%$ .
Tip: Use this for Compound Interest (2 years) and Area questions (Length/Breadth change).
For 3 Changes: Apply formula on first two, then result with third.

Example:

Q: Salary increased by 10% then 20%. Net change?

Solution:

10 + 20 + \frac{10 \times 20}{100} = 30 + 2 = 32\%

4. Population/Machine Value Formula

For continuous growth/depreciation.

After n years: $P(1 \pm \frac{R}{100})^n$
Short Trick: Use Ratio Method. If 10% increase (1/10), ratio moves $10 \to 11$ . For 2 years, $10^2 \to 11^2$ i.e., $100 \to 121$ .

Example:

Q: Population 1000 increases by 10% for 2 years.

Solution: Ratio

10:11

. 2 Years

\to 100:121

100 \to 1000 (\times 10)

121 \to 1210

5. Venn Diagrams (Set Theory)

Use for 'Both', 'Either', 'Neither' questions.

Formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
Visualization: Draw two overlapping circles. Middle part is 'Both'. Total area must be 100% (or Total - None).

Example:

Q: 60% pass Math, 50% pass English, 20% pass both. Failed in both?

Solution: Pass

\cup

60 + 50 - 20 = 90\%

.
Fail =

100 - 90 = 10\%

6. Election Problems (Voting)

Core logic: Total Votes = Valid + Invalid. Winner - Loser = Majority.

Valid Votes: Total - Invalid.
Winner's Share: Usually % of Valid Votes (read carefully).

Example:

Q: Winner gets 60% votes and wins by 2000 votes. Total votes?

Solution: Winner 60%, Loser 40%. Diff = 20%.

20\% = 2000 \to 1\% = 100 \to 100\% = 10,000

7. Examination Cases (Marks)

Passing Marks are constant.

Concept: If Student A fails by $x$ marks and Student B passes by $y$ marks, difference in % = difference in marks ( $x+y$ ).

Example:

Q: A gets 30% and fails by 10. B gets 40% and passes by 10. Max marks?

Solution: Diff in %: 10%. Diff in marks:

10+10=20

10\% = 20 \to 100\% = 200

8. Income Tax & Commission

Inverse relation between Net Income and Tax Rate.

Rule: If Tax Rate increases by $R$ and Net Income decreases by $r$ , then $Rate \times R = Net \times r$ .
Rate = $\frac{\text{Change in Net Income}}{\text{Total Income}} \times 100$ .

Example:

Q: Tax

\uparrow 19\%

, Net Income

\downarrow 1\%

. Find Tax Rate.

Solution: Total Income change proportional. Tax = 1/20.
Rate =

(1 / (19+1)) \times 100 = 5\%

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Percentage

Expert Answer & Key Takeaways

1. Standard Reciprocals (The Speed Hack)

2. The AB Rule (Constant Product)

3. Successive Percentage Change

4. Population/Machine Value Formula

5. Venn Diagrams (Set Theory)

6. Election Problems (Voting)

7. Examination Cases (Marks)

8. Income Tax & Commission

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