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📖 Fractions & Decimals

Chapter Outline & Goals
By the end of this chapter, you will:

Understand what fractions and decimals really mean.

Convert between fractions, decimals, and mixed numbers effortlessly.

Perform comparisons and basic operations with speed.

Avoid the trickiest traps in competitive exams.

Shocking Hook – The $300 Million Unit Mix-Up
In 1999, NASA's Mars Climate Orbiter was lost because one engineering team used pound-force seconds while another used newton seconds for thrust calculations. A tiny unit mismatch — essentially a number applied in the wrong system — destroyed a mission worth over $300 million. The lesson for you: in competitive exams too, a small slip with fractions and decimals — misreading 0.25, misplacing a decimal point, adding denominators — costs marks you cannot afford. Fractions and decimals are not just math; they are the difference between success and costly failure. Master them, and a huge portion of quantitative aptitude becomes your strength.

We Know the Pain
You've probably stared at ⅞ + 0.375 and felt your brain freeze. You're not alone. Most students treat fractions and decimals as two different monsters. The truth? They are the same number wearing different masks. Once you see the mask, the fear vanishes. This chapter won't just teach you rules; it will rewire the way you think.

1. The Real Meaning of a Fraction

A fraction is simply a division problem that hasn't been solved yet.
The top number (numerator) tells how many parts you have. The bottom number (denominator) tells how many equal parts make a whole.

Think of a chocolate bar divided into 4 equal pieces. If you pick up 3 pieces, you have ¾. That's it – nothing scary.

Types of fractions you must instantly recognise:

Proper fraction: Numerator < Denominator (⅔, ⅞) – value less than 1.

Improper fraction: Numerator ≥ Denominator (⁷⁄₄, ¹¹⁄₅) – value 1 or more.

Mixed number: Whole number + proper fraction (1¾, 3⅖).

Like fractions: Same denominator (⅖, ⅘).

Unlike fractions: Different denominators (⅔, ¾).

🧠 Active Recall Prompt: Without looking back, define proper and improper fraction in your own words. Give one example of each.

2. Decimals – Fractions in Disguise

A decimal is just a fraction whose denominator is a power of 10 (10, 100, 1000…).
0.7 means 7/10.
0.23 means 23/100.
0.005 means 5/1000.

The number of digits after the decimal point tells you the number of zeros in the denominator. This single idea makes conversion a breeze.

Terminating vs Recurring Decimals

Terminating: ends after a few digits (0.25, 1.4).

Recurring (non-terminating repeating): a digit or block repeats forever (0.333…, 0.142857 142857…).

Recurring decimals are represented with a bar above the repeating part, like 0.3̅ or 0.1̅4̅2̅8̅5̅7̅. In competitive exams, converting them to fractions is a goldmine for quick marks.

🧠 Active Recall Prompt: Write 0.125 as a fraction in its simplest form. Then check: 125/1000 = 1/8.

3. The Bilingual Trick to Compare Fractions & Decimals

Comparison is a favourite trap. Use these go-to methods:

Method 1 – Cross Multiplication (for fractions)
To compare 5/7 and 3/4:
5 × 4 = 20, 3 × 7 = 21. Since 20 < 21, 5/7 < 3/4. Done in 5 seconds.

Method 2 – Decimal Conversion
When fractions are messy, convert to decimals up to 2-3 places.
5/7 ≈ 0.714, 3/4 = 0.75 → 5/7 is smaller.

Method 3 – The "Gap" Approach
For fractions close to 1, see how far from 1 they are.
11/12 vs 14/15: 11/12 is 1/12 short, 14/15 is 1/15 short. Since 1/12 > 1/15, 11/12 < 14/15. No cross multiplication needed.

🧠 Active Recall Prompt: Which is larger: 13/14 or 15/16? Use the "gap" approach mentally. (Answer: 15/16 is larger because 1/14 > 1/16.)

4. Operations That Cost You Marks

Addition/Subtraction – The LCM Trap
To add ⅓ + ¼, do not add numerators and denominators. Find LCM of 3 and 4 = 12.
⅓ = 4/12, ¼ = 3/12, sum = 7/12.
If you ever mistakenly added 1+1 and 3+4 to get 2/7, you've just given away marks. Remember: Denominator is the name of the game; change the name before you play.

Multiplication – Simple Pizza Logic
⅔ × 4/5 means ⅔ of 4/5. Multiply straight: (2×4)/(3×5) = 8/15. "Of" means multiply.

Division – The KFC Mnemonic (Weird but Unforgettable)
Dividing fractions: Keep the first, Flip the second, Change sign to multiplication.
KFC – Keep Flip Change. Imagine Colonel Sanders holding a fraction upside down. It's silly, and you'll never forget it.
Example: ⅔ ÷ 4/5 → Keep ⅔, Flip 4/5 to 5/4, Change ÷ to × → ⅔ × 5/4 = 10/12 = ⅚.

🧠 Active Recall Prompt: Solve 2/5 ÷ 3/7 using KFC. Answer: 2/5 × 7/3 = 14/15.

5. Mixed Numbers and Improper Fractions – The MAD Transformation

To convert a mixed number like 3⅖ to an improper fraction, use MAD: Multiply, Add, Denominator stays.
Multiply whole number (3) × denominator (5) = 15. Add numerator (2) = 17. Keep denominator 5 → 17/5.
MAD: Multiply, Add, Denominator (the same). It's mad, but it works every time.

Reverse: 17/5. Divide 17 by 5 → 3 remainder 2 → 3⅖.

Decimal to Mixed Number:
3.75 = 3 + 0.75 = 3 + 75/100 = 3 + 3/4 = 3¾. Always simplify the fraction part.

🧠 Active Recall Prompt: Convert 7.2 to a mixed number in simplest form. Answer: 7 + 2/10 = 7 + 1/5 = 7⅕.

6. Recurring Decimals to Fractions – The VIP Shortcut

You'll frequently see 0.3̅, 0.2̅1̅ in exams. Memorise this:

For a pure recurring decimal like 0.a̅b̅c̅… (all digits repeat), put the repeating block over as many 9s as digits.
0.6̅ = 6/9 = 2/3.
0.2̅3̅ = 23/99.
0.1̅4̅2̅8̅5̅7̅ = 142857/999999 = 1/7 (yes, that's a known magic fraction).

For mixed recurring decimals (some digits don't repeat), use this standard trick:
Numerator = (all digits after the decimal up to the end of the first repeating block) − (the non-repeating part).
Denominator = as many 9s as repeating digits, followed by as many 0s as non-repeating digits.

Example: 0.16̅ (that is, 0.1666… — the 1 does not repeat, only the 6 does).
Numerator = 16 − 1 = 15.
Denominator = one 9 (one repeating digit) followed by one 0 (one non-repeating digit) = 90.
So 0.16̅ = 15/90 = 1/6. Check: 1/6 = 0.1666… ✓

🧠 Active Recall Prompt: Convert 0.8̅3̅ to a fraction. (Solution: 83/99, does not simplify.)

7. Decimals in Operations – The Safest Way

When adding 3.45 + 2.1, align the decimal points vertically:

3.45
+ 2.10 (put an imaginary zero)

5.55

Multiplying decimals: Ignore the decimal point, multiply as whole numbers, then put the decimal back. Count total decimal places in both numbers.
0.4 × 0.02 → 4 × 2 = 8, total 3 decimal places (one from 0.4, two from 0.02) → 0.008.

Dividing decimals: Make the divisor a whole number by shifting the decimal point right in both numbers.
0.125 ÷ 0.5 → shift one place: 1.25 ÷ 5 = 0.25.

🧠 Active Recall Prompt: Compute 0.03 × 0.004. Answer: 12 with 5 decimal places → 0.00012.

Summary Table – Your 2-Minute Revision

Weird Memory Hook to Recall Conversions
"Fraction is a pizza, decimal is its price tag."
Pizza slices (⅛) are the fraction; the price tag (₹0.125) is the decimal. The tag always has the power-of-10 denominator hidden in the number of digits after the point.

Review This After 2 Days

Revisit this chapter, re-attempt every 🧠 prompt without looking at the answer. Then solve the first 10 MCQs from the Practice section. That spaced repetition will lock these concepts into your long-term memory permanently.

You've just turned fractions and decimals from a fear into a weapon. Now move to the PYQ analysis to see how examiners weaponise them against you – and how you'll fight back.