⭐ Sabse Important Points · एक नज़र में
- Equivalent Fractions: Fractions that represent the same value, obtained by multiplying or dividing numerator and denominator by the same non-zero number.
- Fraction to Decimal: Divide numerator by denominator; e.g., 3/4 = 0.75.
- Addition/Subtraction: To add or subtract fractions, first find a common denominator, then combine numerators.
- Decimal to Fraction: Write the decimal without the point as numerator, and denominator as 1 followed by as many zeros as decimal places; then simplify.
- Multiplication: Multiply numerators together and denominators together; simplify if possible.
- Multiplication of Decimals: Multiply as whole numbers, then place decimal point so that the number of decimal places equals the sum of decimal places in the factors.
Introduction · परिचय
Decimals and fractions are fundamental concepts in numerical ability, frequently tested in SSC MTS. This topic covers conversion between fractions and decimals, operations, and comparison. Mastering these basics ensures quick and accurate problem-solving.
Fractions: Basics and Operations
A fraction represents a part of a whole, written as a/b where a is the numerator and b is the denominator (b ≠ 0). Fractions can be proper (numerator < denominator), improper (numerator ≥ denominator), or mixed (whole number + proper fraction). Operations like addition, subtraction, multiplication, and division follow specific rules. Simplification by canceling common factors is essential.
📐 Formula / Rule Box · सूत्र याद रखो
Addition/Subtraction
a/b ± c/d = (ad ± bc) / bd
Multiplication
(a/b) × (c/d) = ac / bd
Division
(a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc
⭐ Key Points · याद रखने वाली बातें
- Equivalent Fractions: Fractions that represent the same value, obtained by multiplying or dividing numerator and denominator by the same non-zero number.
- Addition/Subtraction: To add or subtract fractions, first find a common denominator, then combine numerators.
- Multiplication: Multiply numerators together and denominators together; simplify if possible.
- Division: Multiply the first fraction by the reciprocal of the second fraction.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Simplify: (2/3) + (3/4) - (1/6)
✅ Solution: 13/12 or 1 1/12
💡 Explanation: LCM of 3,4,6 is 12. Convert: 2/3 = 8/12, 3/4 = 9/12, 1/6 = 2/12. Then 8+9-2 = 15/12 = 5/4 = 1 1/4? Wait, recalc: 8+9=17, 17-2=15, so 15/12 = 5/4 = 1 1/4. But 5/4 is 1.25, which is 1 1/4. However, 13/12 is 1 1/12. Let's correct: 8+9=17, 17-2=15, 15/12 simplifies to 5/4 = 1 1/4. So answer is 5/4 or 1 1/4. I mistakenly wrote 13/12. Correct solution: 5/4.
Decimals: Conversion and Operations
Decimals are another way to represent fractions with denominators as powers of 10. Converting fractions to decimals involves division, and decimals to fractions uses place value. Operations with decimals require aligning decimal points for addition/subtraction, and counting decimal places for multiplication/division.
📐 Formula / Rule Box · सूत्र याद रखो
Decimal to Fraction
Decimal = Numerator / (10^n), where n is the number of decimal places
| Operation | Rule |
|---|---|
| Addition/Subtraction | Align decimal points and perform operation |
| Multiplication | Multiply as whole numbers, then place decimal point (sum of decimal places) |
| Division | Shift decimal points to make divisor whole, then divide |
⭐ Key Points · याद रखने वाली बातें
- Fraction to Decimal: Divide numerator by denominator; e.g., 3/4 = 0.75.
- Decimal to Fraction: Write the decimal without the point as numerator, and denominator as 1 followed by as many zeros as decimal places; then simplify.
- Multiplication of Decimals: Multiply as whole numbers, then place decimal point so that the number of decimal places equals the sum of decimal places in the factors.
- Division of Decimals: Make the divisor a whole number by shifting decimal point, and shift the dividend's decimal point the same number of places.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Convert 0.375 to a fraction in simplest form.
✅ Solution: 3/8
💡 Explanation: 0.375 = 375/1000. Divide numerator and denominator by 125: 375÷125=3, 1000÷125=8, so 3/8.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Simplify: (0.6 × 0.6 × 0.6 + 0.4 × 0.4 × 0.4) / (0.6 × 0.6 - 0.6 × 0.4 + 0.4 × 0.4)
👀 Answer dekho
✅ Answer: 1
💡 Explanation: Use identity a^3 + b^3 = (a+b)(a^2 - ab + b^2). Here a=0.6, b=0.4. Numerator = (0.6+0.4)(0.6^2 - 0.6×0.4 + 0.4^2) = 1 × denominator. So result = 1.
Q2Which fraction is largest: 3/5, 4/7, 5/9, 6/11?
👀 Answer dekho
✅ Answer: 3/5
💡 Explanation: Convert to decimals: 3/5=0.6, 4/7≈0.571, 5/9≈0.556, 6/11≈0.545. Largest is 0.6, so 3/5.
Q3Find the value of 0.125 × 0.25 × 0.5.
👀 Answer dekho
✅ Answer: 0.015625
💡 Explanation: Multiply: 125 × 25 × 5 = 15625. Total decimal places: 3+2+1=6, so 0.015625.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to simplify fractions after operations; always reduce to lowest terms.
- Misplacing decimal points in multiplication/division; count decimal places carefully.
- Incorrectly converting mixed numbers to improper fractions; multiply whole number by denominator and add numerator.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Fraction to Decimal: Divide numerator by denominator.
- Decimal to Fraction: Use place value and simplify.
- Operations: Align decimals for addition/subtraction; count decimal places for multiplication; shift decimals for division.
- Comparison: Convert fractions to decimals or find common denominator to compare.