⭐ Sabse Important Points · एक नज़र में
- Equivalent Fractions: Multiply or divide numerator and denominator by the same non-zero number to get equivalent fractions.
- Fraction to Decimal: Divide numerator by denominator; terminating decimals have denominators with only 2 and 5 as prime factors.
- Addition/Subtraction: Find a common denominator, then add/subtract numerators.
- Decimal to Fraction: Write decimal over 10, 100, etc., based on decimal places, then simplify.
- Multiplication: Multiply numerators and denominators separately.
- Multiplication: Multiply as whole numbers, then place decimal point by counting total decimal places in factors.
Introduction · परिचय
Decimals and fractions are fundamental mathematical concepts tested in RRB NTPC. This topic covers conversion between fractions and decimals, operations, and comparison. Understanding these basics is crucial for solving arithmetic problems quickly and accurately.
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 on fractions require common denominators for addition/subtraction, while multiplication and division are straightforward.
📐 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: Multiply or divide numerator and denominator by the same non-zero number to get equivalent fractions.
- Addition/Subtraction: Find a common denominator, then add/subtract numerators.
- Multiplication: Multiply numerators and denominators separately.
- Division: Multiply by the reciprocal of the divisor.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Simplify: 2/3 + 3/4
✅ Solution: 17/12 or 1 5/12
💡 Explanation: LCM of 3 and 4 is 12. Convert: 2/3 = 8/12, 3/4 = 9/12. Sum = 17/12 = 1 5/12.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Divide: (5/6) ÷ (10/9)
✅ Solution: 3/4
💡 Explanation: Multiply by reciprocal: (5/6) × (9/10) = (5×9)/(6×10) = 45/60 = 3/4.
Decimals: Conversion and Operations
Decimals represent fractions with denominators as powers of 10. Converting fractions to decimals involves division, while decimals to fractions uses place value. Operations on decimals require aligning decimal points for addition/subtraction, and counting decimal places for multiplication/division.
📐 Formula / Rule Box · सूत्र याद रखो
Fraction to Decimal
a/b = a ÷ b
Decimal to Fraction
0.d1d2...dn = (d1d2...dn)/10^n
| Fractions | Decimals |
|---|---|
| Written as a/b | Written with decimal point |
| Can be exact or recurring | Can be terminating or recurring |
| Operations need LCM for +/− | Operations need alignment for +/− |
⭐ Key Points · याद रखने वाली बातें
- Fraction to Decimal: Divide numerator by denominator; terminating decimals have denominators with only 2 and 5 as prime factors.
- Decimal to Fraction: Write decimal over 10, 100, etc., based on decimal places, then simplify.
- Multiplication: Multiply as whole numbers, then place decimal point by counting total decimal places in factors.
- Division: Make divisor a whole number by shifting decimal, then divide.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Convert 3/8 to decimal.
✅ Solution: 0.375
💡 Explanation: 3 ÷ 8 = 0.375 (terminating because 8 = 2^3).
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Multiply: 0.6 × 0.25
✅ Solution: 0.15
💡 Explanation: 6 × 25 = 150; total decimal places = 1+2=3, so 0.150 = 0.15.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Simplify: 1/2 + 1/3 + 1/6
👀 Answer dekho
✅ Answer: 1
💡 Explanation: LCM of 2,3,6 is 6. Convert: 3/6 + 2/6 + 1/6 = 6/6 = 1.
Q2Convert 0.125 to a fraction in simplest form.
👀 Answer dekho
✅ Answer: 1/8
💡 Explanation: 0.125 = 125/1000. Divide numerator and denominator by 125: 1/8.
Q3Which is greater: 3/5 or 0.6?
👀 Answer dekho
✅ Answer: They are equal.
💡 Explanation: 3/5 = 0.6, so both represent the same value.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to simplify fractions after operations; always reduce to lowest terms.
- Misaligning decimal points in addition/subtraction; write numbers vertically with decimal points lined up.
- In division of decimals, forgetting to shift the decimal point in the dividend after shifting in the divisor.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Equivalent Fractions: Multiply/divide numerator and denominator by same number.
- Decimal to Fraction: Use place value; e.g., 0.75 = 75/100 = 3/4.
- Comparison: Convert fractions to decimals or find common denominator to compare.