भिन्न और दशमलव

हिंदी में पढ़ें

Fractions & Decimals

Chapter Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Classify fractions (proper, improper, mixed, equivalent, like/unlike) instantly.
2. Compare and order fractions using cross-multiplication and decimal conversion.
3. Master operations on fractions and decimals — including the traps examiners set.
4. Convert between fractions, decimals, and percentages fluently.
5. Handle recurring decimals and convert them to fractions.
6. Apply fractions to real exam problems: successive changes, remainders, and approximation.

The Hook: A Rupee That Broke a Bank Exam

In a well-known banking exam, thousands of candidates lost a mark on this innocent-looking question: *"What is 0.999… (9 repeating forever) equal to?"* Most wrote "slightly less than 1." The correct answer? Exactly 1. If that surprises you, good — it means this chapter will change how you see numbers. Fractions and decimals are not "school maths." In SSC CGL Tier-II, IBPS PO Mains, and CSAT, they are the hidden engine inside percentages, ratios, profit-loss, and data interpretation. Master them here, and half of quantitative aptitude becomes faster.

1. What a Fraction Really Is

A fraction a/b is simply a division waiting to happen: *a* divided by *b*. The numerator (a) tells you *how many parts* you have; the denominator (b) tells you *into how many equal parts* the whole was cut.

• Proper fraction: numerator < denominator (3/5). Value is less than 1.
• Improper fraction: numerator ≥ denominator (7/4). Value is 1 or more.
• Mixed fraction: a whole number plus a proper fraction (1¾ = 7/4).
• Equivalent fractions: same value, different look (2/3 = 4/6 = 10/15).

🧠 Active Recall: Without looking back — is 9/9 proper or improper? Convert 17/5 into a mixed fraction.

2. Comparing Fractions — The Examiner's Favourite Trap

Which is bigger: 7/11 or 5/8? Don't convert to decimals in the exam — cross-multiply:

• 7 × 8 = 56 and 5 × 11 = 55. Since 56 > 55, 7/11 > 5/8.

Rule: for a/b vs c/d, compare a×d with c×b. The bigger product belongs to the bigger fraction.

Trick for close fractions: If two fractions have the same *gap* between numerator and denominator (like 3/4 and 7/8, gap = 1), the one with the larger numbers is bigger. So 7/8 > 3/4. But if the gap is in the *numerator* direction for improper fractions, the rule flips — always verify with cross-multiplication when in doubt.

🧠 Active Recall: Arrange in ascending order: 2/5, 3/7, 4/9. (Hint: cross-multiply pairs.)

3. Operations on Fractions

Addition/Subtraction: You cannot add apples to oranges — make denominators equal first (LCM).

• 2/3 + 3/4 = 8/12 + 9/12 = 17/12.

Multiplication: Straight across. No LCM needed.

• 2/3 × 3/4 = 6/12 = 1/2. Cancel common factors *before* multiplying — this is the single biggest time-saver in exams.

Division: Multiply by the reciprocal ("Keep–Change–Flip").

• 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9.

Memory trick (mnemonic): *"Dividing fractions? Don't ask why — just flip the second and multiply!"* Remember KCF: Keep, Change, Flip.

🧠 Active Recall: Compute 5/6 ÷ 10/9. What did you flip?

4. Decimals: Fractions Wearing a Suit

Every decimal is a fraction whose denominator is a power of 10: 0.375 = 375/1000 = 3/8.

Key skills:

• Addition/Subtraction: Align the decimal points. 3.2 + 0.45 = 3.65 (write 3.20 first).
• Multiplication: Multiply as whole numbers, then place the decimal point — total decimal places = sum of decimal places in both numbers. 0.4 × 0.25 → 4 × 25 = 100, three decimal places → 0.100 = 0.1.
• Division: Shift the decimal to make the divisor a whole number. 4.5 ÷ 0.15 = 450 ÷ 15 = 30.

Exam trap: 0.1 × 0.1 = 0.01, *not* 0.1. Multiplying two numbers less than 1 gives a *smaller* number — the opposite of what intuition from whole numbers suggests.

🧠 Active Recall: Is 0.9 × 0.9 greater or smaller than 0.9? Why?

5. Recurring Decimals → Fractions (Advanced Must-Know)

A recurring decimal repeats forever: 0.333… = 0.3̄.

Conversion rules:

• Pure recurring: 0.7̄ = 7/9; 0.1̄6̄ = 16/99. (Repeating digits over as many 9s as there are repeating digits.)
• Mixed recurring: 0.1666… = 0.16̄ = (16 − 1)/90 = 15/90 = 1/6. Formula: (whole block − non-repeating part) / (9s for repeating digits followed by 0s for non-repeating digits).

And that hook from the start: 0.9̄ = 9/9 = 1. Two different decimal representations, one number.

🧠 Active Recall: Convert 0.272727… into a fraction. (Answer pattern: two repeating digits → denominator 99.)

6. The Fraction–Decimal–Percentage Triangle

Advanced exams make you jump between all three forms. Memorise the high-frequency conversions:

Mnemonic for the eighths family: *"Eighths climb by 125"* — 1/8 = 0.125, 2/8 = 0.250, 3/8 = 0.375, 4/8 = 0.500… each step adds 0.125.

🧠 Active Recall: Close the table. What is 5/8 as a decimal? What fraction is 8.33%?

7. Advanced Applications

Successive fractions: If a worker completes 1/3 of a job, then 1/4 of the *remaining*, what fraction is left? Remaining after step 1 = 2/3. Then 1/4 of 2/3 = 1/6 is done. Total done = 1/3 + 1/6 = 1/2. Left = 1/2. Shortcut: multiply the "remainders": (1 − 1/3)(1 − 1/4) = 2/3 × 3/4 = 1/2.

Fraction of a number vs. fraction of a fraction: "1/3 of 1/2" = 1/6 — the word *of* means multiply. This tiny word causes huge errors in DI questions.

Approximation (PO Mains favourite): 0.498 × 2.003 ≈ 0.5 × 2 = 1. Learn to spot when options are far enough apart to approximate safely.

🧠 Active Recall: A tank is 5/6 full. 1/5 of the water is drained. What fraction of the tank still has water?

8. Common Exam Traps — Memorise These

1. 0.1 × 0.1 ≠ 0.1 (it's 0.01).
2. Dividing by a proper fraction *increases* the number: 6 ÷ (1/2) = 12.
3. 1/3 is not 0.33 exactly — it's 0.333…; using 0.33 in long calculations compounds error.
4. "Of" = multiply; "out of" = divide.
5. Comparing 5/9 and 6/11 by "both are a bit more than half" fails — cross-multiply (55 vs 54, so 5/9 > 6/11).

Summary Table

📌 Review this chapter after 2 days. Redo the Active Recall prompts from memory, rewrite the conversion table without looking, and re-derive 0.16̄ = 1/6. Spaced repetition is what moves these rules from "I read it" to "I can't forget it."