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 $16/25 = 64\%$.

Percentage

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. The Base 100 Logic (Philosophy)

Why do we use 100? In complex systems, 100 acts as a 'Normalizer'. Whether you are comparing the GDP of countries or the latency of two APIs, converting everything to a scale of 100 allows for instant comparison of rates. Mastery of 100 means mastery of relative scale.

2. Standard Reciprocals (The Speed Hack)

Memorize these to avoid division during exams. These are your 'Mental Pre-processors'.

1/2 = 50%
1/3 = 33.33%
1/4 = 25%
1/6 = 16.66%
1/7 = 14.28% (7 double is 14, 14 double is 28)
1/8 = 12.5%
1/9 = 11.11%
1/11 = 09.09%
1/12 = 8.33%
1/15 = 6.66%
1/16 = 6.25%

Example:

Q: A Cloud storage provider increases latency from 40ms to 49ms. Find the % increase.

Solution: Increase = 9ms. Base = 40ms.
Ratio =

9/40

.
We know

1/40 = 2.5\%

. So

9 \times 2.5 = 22.5\%

3. The AB Rule (The Equilibrium Trick)

In any equation where

Product = A \times B

(like

Revenue = Price \times Users

), if A increases, how much must B decrease to keep Revenue same?
The Ladder Rule: If A

\uparrow

by 1x, B must

\downarrow

by 1x+1.
Example: If User Acquisition Cost increases by

25\% (1/4)

, the Number of Users must decrease by

1/5 (20\%)

to keep the Marketing Budget constant.

Example:

Q: The weight of a smartphone battery increases by 20% (1/5). By what % should the density increase to keep the volume same?

Solution: Increase =

1/5

.
Required Reduction in other factor =

1/(5+1) = 1/6

1/6 = 16.66\%

4. Successive Percentage Change (The Compound Effect)

When a value changes by

x\%

and then the result changes by

y\%

. This is the 'Ladder Effect'.

Net Formula: $x + y + \frac{xy}{100} \%$ .
Modern Tip: Use this for Multi-stage discounts or Compound Growth in SaaS subscriptions.

Example:

Q: A YouTube channel grows subscribers by 10% in Month 1 and 20% in Month 2. Find net growth.

Solution:

10 + 20 + (10 \times 20)/100 = 30 + 2 = 32\%

.
Wait! Don't just add

10+20=30

. The 20% growth happened on the already increased base!

5. Venn Diagrams (Overlap Mastery)

Essential for 'Audience Overlap' or 'Test Results'.

Formula: $Total = (Only A) + (Only B) + (Both) + (Neither)$ .
The Cheat Sheet: $A \cup B = A + B - (A \cap B)$ .

Example:

Q: In a tech survey, 70% of developers use VS Code, 50% use NeoVim, and 30% use both. What % use neither?

Solution: VS Code

\cup

NeoVim =

70 + 50 - 30 = 90\%

.
Neither =

100\% - 90\% = 10\%

6. Population vs Depreciation (Exponential Logic)

Whether it's the population of Delhi or the value of a MacBook Pro.

Growth: $Final = Initial \times (1 + r)^n$
Depreciation: $Final = Initial \times (1 - r)^n$
Ratio Hack: If a MacBook value drops 10% (1/10) yearly, the ratio is $10 o 9$ . In 2 years, it is $100 o 81$ .

7. Election and Margin Logic

Winner - Loser = Margin. In complex problems, always track 'Valid Votes'. If 10% votes are invalid, your base for 100% calculation shifts to 90.

Example:

Q: Winner gets 60% of total votes and wins by 1000 votes. Find Total.

Solution: Winner 60%

o

Loser 40%. Margin = 20%.

20\% = 1000 o 100\% = 5000

8. Examination and Boundary Marks

The concept is 'Distance from the Pass Line'. If A is

-10

from Pass and B is

+20

from Pass, the distance between them is

30

marks. This gap usually corresponds to the difference in their percentage scores.

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Percentage

Expert Answer & Key Takeaways

1. The Base 100 Logic (Philosophy)

2. Standard Reciprocals (The Speed Hack)

3. The AB Rule (The Equilibrium Trick)

4. Successive Percentage Change (The Compound Effect)

5. Venn Diagrams (Overlap Mastery)

6. Population vs Depreciation (Exponential Logic)

7. Election and Margin Logic

8. Examination and Boundary Marks

Course4All Editorial Board

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