Exercise 3.3

The World of Numbers • Class 9 Mathematics • Ganita Manjari

Practice Mode: पहले question खुद solve करें। समझ न आए तो 🇮🇳 हिंदी में समझें दबाएँ। फिर Solution देखें से पूरा calculation, visual और final answer check करें।

Topic: Rational Numbers — equivalent fractions, addition, subtraction, multiplication, division और distributive property.

Question 1

Equivalent Rational Numbers

Prove that the following rational numbers are equal: (i) 2/3 and 4/6; (ii) 5/4 and 10/8; (iii) −3/5 and −6/10; (iv) 9/3 and 3.

हिंदी में समझें:Use equivalent fractions (or cross-multiplication). (i) 2/3 = (2×2)/(3×2) = 4/6. (ii) 5/4 = (5×2)/(4×2) = 10/8. (iii) −3/5 = (−3×2)/(5×2) = −6/10. (iv) 3 = 3/1 = (3×3)/(1×3) = 9/3. Hence all given pairs are equal.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

Use equivalent fractions (or cross-multiplication). (i) 2/3 = (2×2)/(3×2) = 4/6. (ii) 5/4 = (5×2)/(4×2) = 10/8. (iii) −3/5 = (−3×2)/(5×2) = −6/10. (iv) 3 = 3/1 = (3×3)/(1×3) = 9/3. Hence all given pairs are equal.

✅ Final Answer: (i) 2/3 = 4/6 | (ii) 5/4 = 10/8 | (iii) −3/5 = −6/10 | (iv) 9/3 = 3

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 2

Addition of Rational Numbers

Find the sum: (i) 2/5 + 3/10; (ii) 7/12 + 5/8; (iii) −4/7 + 3/14.

हिंदी में समझें:For different denominators, first take the LCM. (i) LCM(5,10)=10: 2/5=4/10, so 4/10+3/10=7/10. (ii) LCM(12,8)=24: 7/12=14/24 and 5/8=15/24, so 29/24. (iii) LCM(7,14)=14: −4/7=−8/14, so −8/14+3/14=−5/14.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

For different denominators, first take the LCM. (i) LCM(5,10)=10: 2/5=4/10, so 4/10+3/10=7/10. (ii) LCM(12,8)=24: 7/12=14/24 and 5/8=15/24, so 29/24. (iii) LCM(7,14)=14: −4/7=−8/14, so −8/14+3/14=−5/14.

✅ Final Answer: (i) 7/10 | (ii) 29/24 | (iii) −5/14

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 3

Subtraction of Rational Numbers

Find the difference: (i) 5/6 − 1/4; (ii) 11/8 − 3/4; (iii) −7/9 − 2/3.

हिंदी में समझें:(i) LCM(6,4)=12: 5/6=10/12 and 1/4=3/12, so 7/12. (ii) LCM(8,4)=8: 3/4=6/8, so 11/8−6/8=5/8. (iii) LCM(9,3)=9: 2/3=6/9, so −7/9−6/9=−13/9.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

(i) LCM(6,4)=12: 5/6=10/12 and 1/4=3/12, so 7/12. (ii) LCM(8,4)=8: 3/4=6/8, so 11/8−6/8=5/8. (iii) LCM(9,3)=9: 2/3=6/9, so −7/9−6/9=−13/9.

✅ Final Answer: (i) 7/12 | (ii) 5/8 | (iii) −13/9

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 4

Multiplication of Rational Numbers

Find the product: (i) 2/3 × 3/10; (ii) 7/11 × 5/8; (iii) −4/7 × 5/14.

हिंदी में समझें:Multiply numerators and denominators, then simplify. (i) (2×3)/(3×10)=6/30=1/5. (ii) (7×5)/(11×8)=35/88. (iii) (−4×5)/(7×14)=−20/98=−10/49.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

Multiply numerators and denominators, then simplify. (i) (2×3)/(3×10)=6/30=1/5. (ii) (7×5)/(11×8)=35/88. (iii) (−4×5)/(7×14)=−20/98=−10/49.

✅ Final Answer: (i) 1/5 | (ii) 35/88 | (iii) −10/49

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 5

Division of Rational Numbers

Find the quotient: (i) 2/3 ÷ 3/10; (ii) 7/11 ÷ 5/8; (iii) −4/7 ÷ 5/14.

हिंदी में समझें:Change division into multiplication by the reciprocal. (i) 2/3 × 10/3 = 20/9. (ii) 7/11 × 8/5 = 56/55. (iii) −4/7 × 14/5 = −56/35 = −8/5.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

Change division into multiplication by the reciprocal. (i) 2/3 × 10/3 = 20/9. (ii) 7/11 × 8/5 = 56/55. (iii) −4/7 × 14/5 = −56/35 = −8/5.

✅ Final Answer: (i) 20/9 | (ii) 56/55 | (iii) −8/5

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 6

Distributive Property

Show that (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.

हिंदी में समझें:LHS: (1/2+3/4)×8/3 = (2/4+3/4)×8/3 = 5/4×8/3 = 10/3. RHS: 1/2×8/3 + 3/4×8/3 = 4/3 + 2 = 10/3. Since LHS = RHS, the distributive property is verified.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

LHS: (1/2+3/4)×8/3 = (2/4+3/4)×8/3 = 5/4×8/3 = 10/3. RHS: 1/2×8/3 + 3/4×8/3 = 4/3 + 2 = 10/3. Since LHS = RHS, the distributive property is verified.

✅ Final Answer: LHS = RHS = 10/3

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 7

Distributive Property Simplification

Simplify using the distributive property: 7/9 (6/7 − 3/4).

हिंदी में समझें:Distribute 7/9: 7/9×6/7 − 7/9×3/4 = 2/3 − 7/12. Convert to denominator 12: 8/12−7/12=1/12.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

Distribute 7/9: 7/9×6/7 − 7/9×3/4 = 2/3 − 7/12. Convert to denominator 12: 8/12−7/12=1/12.

✅ Final Answer: 1/12

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।

Question 8

Finding the Rational Number x

Find the rational number x such that 5/6 (x + 3/5) = 5/6 x + 1/2.

हिंदी में समझें:Apply the distributive property: 5/6(x+3/5)=5x/6 + (5×3)/(6×5)=5x/6+15/30=5x/6+1/2. This is exactly the RHS. Therefore the equation is true for every rational number x.

Step-by-Step Solution:

Visual: Diagram का उद्देश्य calculation के pattern को visually याद रखने में मदद करना है।

Apply the distributive property: 5/6(x+3/5)=5x/6 + (5×3)/(6×5)=5x/6+15/30=5x/6+1/2. This is exactly the RHS. Therefore the equation is true for every rational number x.

✅ Final Answer: x can be any rational number

Exam Tip: हर intermediate step लिखें और final rational number को simplest/standard form में रखें।