⭐ Sabse Important Points · एक नज़र में
- Equivalent Fractions: Fractions that represent the same value, e.g., 1/2 = 2/4 = 3/6. Multiply or divide numerator and denominator by the same non-zero number.
- Place Value: The value of a digit depends on its position relative to the decimal point, e.g., in 3.456, 4 is in tenths place (4/10).
- Simplification: Divide numerator and denominator by their GCD to reduce a fraction to lowest terms.
- Conversion: To convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a fraction, write the decimal over a power of 10 and simplify.
- Comparison: Cross-multiply to compare fractions: a/b > c/d if a×d > b×c (for positive denominators).
- Rounding: To round to a given place, look at the next digit; if 5 or more, round up.
Introduction · परिचय
Decimals and fractions are fundamental mathematical concepts tested in RRB Group D. This topic covers conversion between fractions and decimals, operations, and comparison. Mastery is essential for quick calculations in the exam.
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 include addition, subtraction, multiplication, and division. To add or subtract, find a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal.
📐 Formula / Rule Box · सूत्र याद रखो
Addition/Subtraction
a/b ± c/d = (a×d ± b×c) / (b×d)
Multiplication
a/b × c/d = (a×c) / (b×d)
Division
a/b ÷ c/d = (a×d) / (b×c)
⭐ Key Points · याद रखने वाली बातें
- Equivalent Fractions: Fractions that represent the same value, e.g., 1/2 = 2/4 = 3/6. Multiply or divide numerator and denominator by the same non-zero number.
- Simplification: Divide numerator and denominator by their GCD to reduce a fraction to lowest terms.
- Comparison: Cross-multiply to compare fractions: a/b > c/d if a×d > b×c (for positive denominators).
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Simplify 18/24.
✅ Solution: 3/4
💡 Explanation: GCD of 18 and 24 is 6. Divide both by 6: 18÷6=3, 24÷6=4, so 18/24 = 3/4.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Add 2/3 and 3/5.
✅ Solution: 19/15 or 1 4/15
💡 Explanation: LCM of 3 and 5 is 15. Convert: 2/3 = 10/15, 3/5 = 9/15. Sum = 19/15 = 1 4/15.
Decimals: Place Value and Operations
Decimals use a decimal point to separate the whole number part from the fractional part. Each place value is a power of 10: tenths (1/10), hundredths (1/100), etc. To add or subtract decimals, align the decimal points. To multiply, multiply as whole numbers and place the decimal point so that the total number of decimal places equals the sum of decimal places in the factors. To divide, make the divisor a whole number by shifting decimal points.
📐 Formula / Rule Box · सूत्र याद रखो
Fraction to Decimal
a/b = a ÷ b
Decimal to Fraction
0.d1d2...dn = (d1d2...dn) / 10^n
⭐ Key Points · याद रखने वाली बातें
- Place Value: The value of a digit depends on its position relative to the decimal point, e.g., in 3.456, 4 is in tenths place (4/10).
- Conversion: To convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a fraction, write the decimal over a power of 10 and simplify.
- Rounding: To round to a given place, look at the next digit; if 5 or more, round up.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Multiply 0.6 and 0.25.
✅ Solution: 0.15
💡 Explanation: Multiply 6 × 25 = 150. Total decimal places = 1+2=3, so place decimal point three places from right: 0.150 = 0.15.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Convert 3/8 to a decimal.
✅ Solution: 0.375
💡 Explanation: Divide 3 by 8: 3 ÷ 8 = 0.375.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Simplify: 0.25 + 0.5 × 0.2
👀 Answer dekho
✅ Answer: 0.35
💡 Explanation: According to BODMAS, multiplication first: 0.5 × 0.2 = 0.10. Then add: 0.25 + 0.10 = 0.35.
Q2Which fraction is greater: 3/5 or 5/8?
👀 Answer dekho
✅ Answer: 5/8
💡 Explanation: Cross-multiply: 3×8=24, 5×5=25. Since 25 > 24, 5/8 > 3/5.
Q3Convert 0.125 to a fraction in simplest form.
👀 Answer dekho
✅ Answer: 1/8
💡 Explanation: 0.125 = 125/1000. Divide numerator and denominator by 125: 125÷125=1, 1000÷125=8, so 1/8.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to align decimal points when adding/subtracting decimals; always line up the decimal points.
- Incorrectly placing the decimal point after multiplication; count total decimal places in factors.
- Not simplifying fractions fully; always divide by GCD until no common factor except 1.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Fraction Addition/Subtraction: Find LCM of denominators, convert to equivalent fractions, then add/subtract numerators.
- Decimal Multiplication: Multiply as whole numbers, then place decimal point by counting total decimal places in factors.
- Comparison: Cross-multiply to compare fractions quickly without converting to decimals.