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Fractions & Decimals

Why This Chapter Can Make or Break Your Score

Here's a fact that surprises most aspirants: in almost every SSC, Banking, and Railway exam, you will never see a question that says *"solve this fraction."* And yet, fractions and decimals quietly hide inside percentages, ratios, profit & loss, time & work, averages, and simplification — a large part of the arithmetic paper stands on this one chapter.

Think of it this way. A student who knows that 1/8 = 0.125 = 12.5% solves a percentage question in 10 seconds. A student who doesn't, reaches for long division and loses 90 seconds — and in a 25-questions-in-20-minutes game, 90 seconds is a small fortune.

This chapter is that fortune. Let's claim it.

What you will master by the end:

1. Types of fractions and how to compare them in seconds
2. Fraction ↔ Decimal ↔ Percentage conversions (the exam's favourite triangle)
3. Terminating vs recurring decimals — and converting recurring ones back to fractions
4. Operations on decimals without silly placement errors
5. The standard fraction table that toppers memorise by heart

1. What Exactly Is a Fraction?

A fraction is a way of writing a part of a whole. If a pizza is cut into 8 equal slices and you eat 3, you ate 3/8 of the pizza.

• The top number is the numerator (how many parts you have) — here, 3.
• The bottom number is the denominator (how many equal parts the whole is cut into) — here, 8.

A memory hook: "D for Denominator, D for Down." The denominator sits *down* below the line.

Types of Fractions

Mixed ↔ Improper conversion (asked constantly, directly or indirectly):

• Mixed → Improper: multiply whole × denominator, add numerator. So 3²/₅ = (3×5 + 2)/5 = 17/5.
• Improper → Mixed: divide. 17 ÷ 5 = 3 remainder 2, so 17/5 = 3²/₅.

🧠 Active Recall: Without looking up — is 7/7 proper or improper? Convert 4³/₇ into an improper fraction. (Answers: improper, since numerator = denominator; and (4×7+3)/7 = 31/7.)

2. Comparing Fractions — The Examiner's Favourite Trap

"Which is the largest: 3/7, 4/9, 5/11, 2/5?" Questions like this appear again and again, and students waste minutes finding LCMs. Don't.

Trick 1 — Cross-multiplication (for two fractions):
To compare a/b and c/d, compare a×d with c×b.
Compare 3/7 and 4/9: 3×9 = 27 vs 4×7 = 28. Since 28 > 27, 4/9 is bigger. Done in 5 seconds.

Trick 2 — Convert to decimals:
3/7 ≈ 0.428, 4/9 ≈ 0.444, 5/11 ≈ 0.454, 2/5 = 0.400. Largest = 5/11. If you know your fraction table (Section 5), this is instant.

Trick 3 — Equal numerators or denominators:

• Same numerator? The smaller denominator wins: 5/8 > 5/9 (fewer, bigger slices).
• Same denominator? The bigger numerator wins: 7/12 > 5/12.

🧠 Active Recall: Which is greater — 7/13 or 5/9? Use cross-multiplication before reading on. (7×9 = 63 vs 5×13 = 65, so 5/9 is greater.)

3. Decimals — Fractions Wearing a Different Uniform

A decimal is just a fraction whose denominator is a power of 10: 0.3 = 3/10, 0.07 = 7/100, 0.125 = 125/1000 = 1/8.

The Golden Rules of Decimal Arithmetic

Addition/Subtraction: Line up the decimal points. That's it.

  12.50
+  3.07
= 15.57

Multiplication: Multiply as whole numbers, then count total decimal places in both numbers and place the point.
0.4 × 0.25 → 4 × 25 = 100; total decimal places = 1 + 2 = 3 → 0.100 = 0.1.

Division by 10, 100, 1000…: Just shift the decimal point left.

• 45.6 ÷ 10 = 4.56
• 45.6 ÷ 100 = 0.456
• Multiplication by 10, 100… shifts it right: 4.56 × 100 = 456.

Division of decimals: Remove the decimal from the divisor by multiplying both numbers by the same power of 10.
4.5 ÷ 0.15 = 450 ÷ 15 = 30.

🧠 Active Recall: Compute 0.6 × 0.05 and 7.2 ÷ 0.08 mentally. (Answers: 0.03 and 90.)

4. Terminating vs Recurring Decimals

Divide 1 by 4: you get 0.25 — it ends. That's a terminating decimal.
Divide 1 by 3: you get 0.3333… — it never ends. That's a recurring (repeating) decimal, written as 0.3̄.

How do you know in advance? A fraction in lowest terms gives a terminating decimal only if its denominator's prime factors are just 2s and 5s (because our number system is base-10, and 10 = 2 × 5).

• 3/8 → 8 = 2³ → terminates (0.375) ✔
• 7/20 → 20 = 2² × 5 → terminates (0.35) ✔
• 5/12 → 12 = 2² × 3 → has a 3 → recurring (0.41666…) ✘

Converting Recurring Decimals to Fractions (a guaranteed exam skill)

• Pure recurring (everything repeats): put the repeating block over as many 9s as its digits.
• 0.3̄ = 3/9 = 1/3
• 0.2̄7̄ = 27/99 = 3/11
• 0.1̄2̄3̄ = 123/999 = 41/333

• Mixed recurring (some digits don't repeat):
  0.1666… = 0.16̄ → take (16 − 1)/90 = 15/90 = 1/6.
  Rule: *(whole block − non-repeating part) / (as many 9s as repeating digits, followed by as many 0s as non-repeating digits)*.

🧠 Active Recall: Will 11/40 terminate? Convert 0.4̄5̄ to a fraction. (40 = 2³ × 5, so yes — 0.275. And 45/99 = 5/11.)

5. The Fraction–Decimal–Percentage Table (Memorise This!)

This single table is the highest-yield item in the entire chapter. Toppers don't *calculate* these — they *recognise* them.

Bonus pattern: multiples come free. If 1/8 = 12.5%, then 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%. One memorised fact becomes four.

🎵 The Mnemonic: "The Pizza Family"

Imagine a family that always orders pizza:

• Mom (1/2) eats half — 50%. Easy.
• Dad (1/4) eats a quarter — 25%, exactly half of Mom.
• Big brother (1/8) eats half of Dad — 12.5%.
• Little sister (1/16) eats half of brother — 6.25%.

Each generation eats half of the previous one: 50 → 25 → 12.5 → 6.25. You'll never forget the even-denominator family again. (The odd ones — 1/3, 1/6, 1/9 — are the "recurring relatives" who never stop talking: 33.33…, 16.66…, 11.11…)

🧠 Active Recall: Close the table. What is 5/8 as a percentage? What fraction is 6.25%? (62.5%; and 1/16.)

6. Fractions of Fractions, and "Of" Means Multiply

Exam language loves the word "of": *"two-thirds of three-fourths of 480."* Translate "of" as ×:

2/3 × 3/4 × 480 = (2 × 3 × 480)/(3 × 4) = 2880/12 = 240.

Cancellation is your best friend. Never multiply first and simplify later — cancel common factors *before* multiplying. In the example above, the 3s cancel and 480/4 = 120, giving 2 × 120 = 240 instantly.

Ascending/descending order questions: convert each fraction to a decimal (using your table) and arrange. No LCM drama.

🧠 Active Recall: Find 3/5 of 2/7 of 350. (3/5 × 2/7 × 350 → 350/5 = 70, 70 × 2 = 140, 140 × 3/7 = 60. Answer: 60.)

7. Common Traps the Examiner Sets

1. 0.9 vs 0.09 vs 0.009 — misreading place value under time pressure. Always count digits after the decimal.
2. "Which is NOT equivalent to 3/4?" — the negative wording. Read the question twice.
3. Recurring decimals written without the bar — 0.33 is *not* 1/3; 0.3̄ is.
4. Adding fractions by adding tops and bottoms — 1/2 + 1/3 is not 2/5. Common denominator first: 3/6 + 2/6 = 5/6.
5. Decimal point slips in division — 0.36 ÷ 0.06 = 6, not 0.6. Shift both decimals equally.

📌 Chapter Summary Table

🔁 Review Reminder

You will forget most of this within 48 hours unless you revisit it — that's how memory works (the Ebbinghaus forgetting curve). Review this chapter after 2 days: redo the fraction table from memory, re-derive 0.16̄ = 1/6, and solve five comparison questions with cross-multiplication. Ten minutes of revision now saves an hour of re-learning before the exam.

You've just built the foundation that percentages, ratios, and profit & loss all stand on. The next chapters will feel noticeably easier — that's your reward for mastering this one. 💪