Chapter 1: 9th Class Math Notes

Chapter 1: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about real numbers.

This chapter requires knowledge of real numbers, including rational numbers, irrational numbers, rationalization methods, and some techniques to solve the variables.

Chapter 1: 9th Class Math Notes

Class 9 Math Notes – Chapter 1 Summary

This summary covers key concepts and solution methods from Chapter 1 of your 9th class mathematics notes:

Rational Numbers

• Expressing Decimals as Rational Numbers:
  • 0.4=490.4=94​, 0.37=37990.37=9937​, 0.21=21990.21=9921​.
  • Method: Multiply by 10 or 100 to eliminate the decimal, then solve for xx.
• Inserting Rational Numbers Between Two Numbers:
  • Example: Between 1331​ and 1441​, the rational numbers are 724247​ and 13484813​.
  • Formula: q1=12(a+b)q1​=21​(a+b), q2=12(q1+b)q2​=21​(q1​+b).

Properties of Real Numbers

• Key Properties:
  • Associative Property: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c).
  • Commutative Property: a+b=b+aa+b=b+a.
  • Additive Identity: a+0=aa+0=a.
  • Multiplicative Identity: a×1=aa×1=a.
  • Distributive Property: a(b+c)=ab+aca(b+c)=ab+ac.
  • Trichotomy Property: For any two real numbers aa and bb, exactly one of the following is true: a<ba<b, a=ba=b, or a>ba>b.

Simplification of Expressions

• Rationalizing the Denominator:
  • Example: 134+3=4−34+3​13​=4−3​.
  • Method: Multiply numerator and denominator by the conjugate of the denominator.
• Simplifying Exponents:
  • Example: (8116)3/4=827(1681​)3/4=278​.
  • Key Rule: (am)n=am×n(am)n=am×n.

Solving Equations and Word Problems

• Consecutive Integers:
  • Example: Three consecutive integers sum to 42. Solution: 13, 14, 15.
  • Method: Let the integers be xx, x+1x+1, x+2x+2, and solve for xx.
• Age Problems:
  • Example: Father and son’s ages sum to 72. Six years ago, the father was twice as old as the son. Solution: Son’s age six years ago was 20.
  • Method: Set up equations based on given conditions.
• Profit and Loss:
  • Example: Selling price for a 15% profit on Rs. 1520 is Rs. 1748.
  • Formula: SP=CP+ProfitSP=CP+Profit.

Compound Interest and Simple Interest

• Compound Interest:
  • Example: Rs. 3,75,000 at 14% for 1 year yields Rs. 52,500.
  • Formula: CI=P×(1+R)T−PCI=P×(1+R)T−P.
• Simple Interest:
  • Example: Rs. 3,50,000 at 7.25% for 2 years yields Rs. 50,750.
  • Formula: SI=P×R×TSI=P×R×T.

Miscellaneous Problems

• Temperature Conversion:
  • Example: 48∘C=118.4∘F48∘C=118.4∘F.
  • Formula: °F=95°C+32°F=59​°C+32.
• Finding Two Numbers:
  • Example: Two numbers sum to 68 and differ by 22. Solution: 45 and 23.
  • Method: Solve the system of equations x+y=68x+y=68 and x−y=22x−y=22.

Review Exercise

• Multiple Choice Questions:
  • Covers topics like rational vs. irrational numbers, properties of real numbers, and simplification of expressions.
  • Example: 77​ is an irrational number.
• Verification of Properties:
  • Example: Verify a(b+c)=ab+aca(b+c)=ab+ac for a=32a=23​, b=53b=35​, c=75c=57​.

9th Class Math Notes Full Book