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अध्याय 5

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📖 Content — Percentage (English)

In this chapter, you will learn:

The Sale That Almost Tricked You
Picture this: It’s Diwali. A shop screams “50% off.” Next to it, another shop says “Buy one, get one free.” They look identical. But are they? One savvy shopper quickly calculates: 50% off means price becomes half. Buy one get one free also means two items for the price of one — again, price per item becomes half. For the single-item buyer, they are the same. For a bulk buyer, the second might have a hidden trap (you must buy two!). This is the subtle magic of percentage — it’s everywhere, and once you truly understand it, you turn from a confused buyer into a money-saving ninja. Today, you’ll gain that power.

You are not alone if your brain freezes at the sight of “25% of 160”. Many learners see the % symbol and panic because it feels abstract. In truth, percentage is nothing but a “fraction spoken differently.” It’s a language of comparison. Think of it this way: If I say “I scored 17 out of 20,” it’s not instantly clear how good that is. But if I say “85%”, your mind immediately cues: “Excellent!” Percentage standardizes everything into a scale of 100. It’s like measuring height in centimetres — a common ruler for every number. The moment you treat percentage as “out of 100,” the mystery dissolves.

🧠 Pause and Recall: Before reading further, speak aloud: What does “per cent” literally mean? (Answer: per hundred. “Cent” comes from Latin centum, meaning hundred — like century, centipede.)

Chunk 1: The Per-100 Machine
Imagine a magical machine. You feed it any fraction, and it transforms the denominator to exactly 100. The numerator then becomes the percentage. That’s exactly what percentage does.

In the opposite direction, to get a percentage back to its simplest fraction, you just drop the % sign and write the number over 100, then simplify.

🧠 Quick Recall: Convert 12% into a fraction in its lowest terms right now. (Solution: 12/100 = 3/25.)

Chunk 2: The Fraction↔Decimal↔Percent Triangle
This triangle is your speed weapon in exams. Once you memorise the common ones, you’ll solve problems in seconds.

Percent Fraction Decimal
1% 1/100 0.01
10% 1/10 0.1
20% 1/5 0.2
25% 1/4 0.25
50% 1/2 0.5
75% 3/4 0.75
100% 1 1.0

If you know that 12.5% = 1/8, then 12.5% of 800 instantly becomes 800 ÷ 8 = 100. No long calculations. Memorise the following golden pairs:

🧠 Recall Action: Close your eyes and recite: 25% = ? fraction, 75% = ? fraction, 10% = ? fraction. Then check.

The Weird Memory Trick (Mnemonic): The Centipede with a Percentage Hat
Picture a giant centipede wearing a top hat that has a shining “%” symbol on it. Why a centipede? Because “cent” = 100, and a centipede has about 100 legs (okay, not exactly, but your brain won’t forget the association). Whenever you see the % symbol, imagine that centipede handing you a board that reads “/100”. So 7% immediately becomes the fraction 7/100. Whenever you have to convert a percentage to a fraction, just let the centipede whisper: “Over hundred, over hundred, simplify if possible.” Next time you get stuck in an exam, simply bring that weird centipede image alive — and the formula will pop back. Weird? Absolutely. Effective? 100%.

Chunk 3: Finding Percentage of a Quantity
The phrase “X% of Y” translates to (X/100) × Y. The word “of” means multiplication.
Example: What is 30% of 240?
30% of 240 = (30/100) × 240 = (3/10) × 240 = 3 × 24 = 72.

Shortcut when Y ends with zeroes: cancel zeroes first. 20% of 350 = (20/100)×350 = (1/5)×350 = 70. You’ll feel the speed.

🧠 Recall Check: Without a calculator, find 15% of 200. (Hint: 10% = 20, 5% = 10, so 15% = 30. Or use fraction: 15/100 × 200 = 15 × 2 = 30.)

Chunk 4: Percentage Change (Increase & Decrease)
When a price goes from old to new, the percentage change is always calculated on the original quantity.
Formula: % change = (Change / Original) × 100.
If the result is positive, it’s increase; if negative, it’s decrease.

Example: A phone’s price jumped from ₹12,000 to ₹15,000. The increase is ₹3,000 on the original ₹12,000.
% increase = (3000/12000) × 100 = (1/4) × 100 = 25%.

Now try decrease: A salary is reduced from ₹50,000 to ₹40,000. Change = -10,000 on 50,000 → (-10000/50000) × 100 = -20%. So 20% decrease.

🧠 Recall Scenario: A fruit seller’s sales rose from 80 kg to 100 kg. What is the percentage increase? (Answer: (20/80)×100 = 25%.)

Chunk 5: Quick Successive Change Intuition (Foundation Visual)
If a number first increases by 10% and then the new number increases by another 20%, the total effect is not 30%. Use the formula: Net effect = A + B + (A×B)/100. So 10 + 20 + (10×20)/100 = 30 + 2 = 32% increase. For two successive discounts of 20% and 10%, the net discount = -20 -10 + (200)/100 = -30 + 2 = -28% (i.e., 28% discount). Memorise the “plus AB over 100” formula, good for multiple changes.

🧠 Apply: Find the net change when a value first increases by 25% and then decreases by 20%. (25 - 20 + (25×-20)/100 = 5 - 5 = 0%. So no change, back to original.)

Chunk 6: Checking the Trap — “Percent of What?”
The most dangerous word in percentage is “of.” Changing the base changes everything. If a student scored 60% marks in Test A (out of 50) and then 40% in Test B (out of 100), what’s the overall percentage? You cannot simply average 60% and 40%. You must find total marks obtained: 60% of 50 = 30, 40% of 100 = 40. Total marks = 70 out of 150 → (70/150)×100 = 46.67%. The average (60+40)/2 = 50% is wrong because the bases are unequal. Always find actual values when bases differ. Exams love this trap.

🧠 Think: A shirt’s price was increased by 25% and then the new price decreased by 20%. Is the final price more, less, or equal to the original? (Equal, as we saw earlier.)

Summary Table: Your Quick-Reference Cheat Sheet

Concept Key Expression Example
Percent meaning per 100 5% = 5 out of 100
% to fraction n% = n/100, simplify 40% = 2/5
Fraction to % (a/b)×100 % 3/8 = 37.5%
Decimal to % multiply by 100 0.63 = 63%
X% of Y (X/100) × Y 8% of 250 = 20
% Increase (Increase/Original)×100 20 to 30 → 50% up
% Decrease (Decrease/Original)×100 80 to 60 → 25% down
Successive change A + B + AB/100 +20, -10 → 8% inc.
Base unity check Always find actual values if bases differ Different out-of marks

🧠 Final Active Recall (Do this now, without peeking):

  1. Convert 16% to a fraction in simplest terms.
  2. What is 75% of 1200?
  3. A salary increased from ₹25,000 to ₹30,000. Find % increase.
  4. State the weird trick to remember “percent” (Centipede).
    Answers: (1) 4/25, (2) 900, (3) 20%, (4) Centipede with 100 legs wearing a % hat reminds you to put the number over 100.

Review this after 2 days: Re-read the centipede mnemonic, recite the golden fraction-percent pairs, and solve five random percentage MCQs. Science says spaced repetition makes learning permanent. See you then — strong like a percent ninja!

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