Master Percentages

Complete Curriculum: Class 6 to Competitive Exams

Course Progress:

🎯 1. Decoding the 'Percent'

The word "Percentage" literally translates to "per century" or "out of 100". It is a fraction where the denominator is permanently fixed at 100. This allows us to compare different quantities on a standard scale.

Universal Formula

$$P = \frac{\text{Value}}{\text{Total Value}} \times 100$$

🔄 2. The Big 3 Conversions

To master basic mathematics, you must fluidly convert between Percentages, Fractions, and Decimals.

  • Fraction to Percentage Multiply by 100.
    Example: $1/4 \times 100 = 25\%$
  • Percentage to Fraction Divide by 100 and simplify.
    Example: $75\% = 75/100 = 3/4$
  • Percentage to Decimal Shift the decimal point two places left.
    Example: $42.5\% = 0.425$

🔎 3. Core Basic Types

Type A: Finding x% of Y

Simply multiply the percentage as a fraction by the number.
Find 20% of 150:
$$(20 / 100) \times 150 = 30$$

Type B: Expressing X as a percentage of Y

Use the universal formula. Crucial: Make sure units match!
What percentage of 2 Hours is 40 Minutes?
First, convert 2 Hours to 120 Minutes.
$$(40 / 120) \times 100 = (1/3) \times 100 = 33.33\%$$

⚡ 1. The Golden Rule: Fraction Equivalents

Calculating $(x / 100) \times Y$ manually is too slow for higher classes. Memorizing these standard fractional equivalents is your ultimate speed shortcut.

$\frac{1}{2} = 50\%$
$\frac{1}{3} = 33.33\%$
$\frac{1}{4} = 25\%$
$\frac{1}{5} = 20\%$
$\frac{1}{6} = 16.66\%$
$\frac{1}{7} = 14.28\%$
$\frac{1}{8} = 12.5\%$
$\frac{1}{9} = 11.11\%$
$\frac{1}{10} = 10\%$
$\frac{1}{11} = 9.09\%$
$\frac{3}{8} = 37.5\%$
$\frac{5}{6} = 83.33\%$
Pro Tip: The Multiplier Effect If you memorize base fractions, you can find derivatives instantly. If $1/8 = 12.5\%$, then $3/8 = 3 \times 12.5\% = 37.5\%$.

📈 2. Percentage Multipliers (Scaling)

Instead of finding a percentage and adding/subtracting it, multiply the original number by a scaling factor.

  • 20% Increase: Multiply by $1.2$ or $\frac{120}{100}$ or $\frac{6}{5}$.
  • 15% Decrease: Multiply by $0.85$ or $\frac{85}{100}$ or $\frac{17}{20}$.

Example: A TV costs $400. Price increases by 25%. New Price = $400 \times \frac{125}{100} = 400 \times \frac{5}{4} = \$500.

🔗 3. Successive Percentage Change

When a value undergoes multiple sequential changes (like a town's population growing over two years), you cannot simply add the percentages together.

Net Effect Formula (AB Rule)

$$\text{Net Change \%} = x + y + \frac{x \times y}{100}$$

Use $+x$ for increase, and $-x$ for decrease.

🧠 Advanced Concept Shortcuts

1. The A & B Comparison Rule

If A is $x\%$ more than B, how much percent is B less than A? Instead of long formulas, use the fraction shortcut.

Shortcut: If increase is $\frac{1}{x}$, the required decrease to return to base is $\frac{1}{x+1}$. If decrease is $\frac{1}{x}$, the required increase is $\frac{1}{x-1}$.

2. Product Constancy (Price × Consumption = Exp)

If the price of sugar goes up by 25% ($\frac{1}{4}$ increase), to keep your budget exactly the same, your consumption must drop by $\frac{1}{4+1} = \frac{1}{5}$ = 20%.

3. Fresh Fruit & Dry Fruit

Core Logic: The quantity of "Pulp" (solid matter) remains constant when fruit dries. Only the water evaporates. Equate the pulp weight of fresh fruit to the pulp weight of dry fruit.

🏆 Competitive MCQ Bank

Direction: Attempt the mental shortcut first before looking at the solution.

Q1. A's salary is 16.66% more than B's. By what percentage is B's salary less than A's?

⏱️ Target: 10s
  • A) 12.5%
  • B) 14.28%
  • C) 16.66%
  • D) 20%
Fraction Shortcut (A/B Rule)

16.66% increase = $+\frac{1}{6}$.

To reverse this, the formula is $\frac{1}{x+1}$.

Therefore, decrease = $-\frac{1}{6+1} = -\frac{1}{7}$.

We know $\frac{1}{7} = 14.28\%$. Solved visually!

Q2. The price of an article is increased by 20% and then decreased by 20%. What is the net percentage change in price?

⏱️ Target: 15s
  • A) 0% (No change)
  • B) 4% Increase
  • C) 4% Decrease
  • D) 2% Decrease
The "Square" Shortcut

Golden Rule: When the exact same percentage $x$ is increased and then decreased, the result is ALWAYS a net loss of $\frac{x^2}{100}\%$.

(+20%) & (-20%) => Loss of (20² / 100)%

$$\frac{400}{100} = 4\% \text{ decrease}$$

Q3. In an election between two candidates, the winner got 60% of the valid votes and won by 1200 votes. Total valid votes are?

⏱️ Target: 30s
  • A) 4000
  • B) 5000
  • C) 6000
  • D) 7200
Margin Strategy

If Winner = $60\%$, then Loser = $40\%$ (since total is 100%).

Margin of Victory = Winner % - Loser %
Margin = 60% - 40% = 20%

We are given the margin is 1200 votes. Therefore:
$20\% \text{ of Total} = 1200$

Multiply by 5 to get 100%:
$100\% = 1200 \times 5 = 6000 \text{ votes}.$

Q4. Fresh grapes contain 80% water while dry grapes contain 10% water. If the weight of dry grapes is 50 kg, what was its total weight when it was fresh?

⏱️ Target: 45s
  • A) 200 kg
  • B) 225 kg
  • C) 250 kg
  • D) 275 kg
Constant Pulp Shortcut

Water evaporates, but the Solid Pulp remains constant.

Dry Grapes (50 kg): Contains 10% water $\implies$ 90% Pulp.
Pulp weight = $90\%$ of $50 = 45$ kg.

Fresh Grapes (F kg): Contains 80% water $\implies$ 20% Pulp.
Since pulp is constant: $20\%$ of Fresh = $45$ kg.

$$\frac{1}{5} \times F = 45 \implies F = 45 \times 5 = 225\text{ kg}$$

Q5. A student must secure 40% marks to pass. He gets 40 marks and fails by 40 marks. Find the total maximum marks.

⏱️ Target: 20s
  • A) 160
  • B) 180
  • C) 200
  • D) 220
Equivalence Line Shortcut

He scored 40 but needs 40 more to pass. Therefore, passing marks = 80.

[Scored: 40] --- (+40 needed) ---> [Pass: 80]
[  0%      ] --------------------> [ 40%    ]

If 40% = 80 marks, then 10% = 20 marks. Multiply by 10: 100% = 200 marks.