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NCERT Class 9 Mathematics Exercise Set 3.3 Solutions: The World of Numbers

Step-by-step working for every question of Exercise Set 3.3 in NCERT's Class 9 Mathematics book, chapter “The World of Numbers”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 3.3 Q1

   Show that each pair is equal: (i) 2/3 and 4/6, (ii) 5/4 and 10/8, (iii) −3/5 and −6/10, (iv) 9/3 and 3.

   1. Two rational numbers a/b and c/d are equal when a × d = b × c (cross-multiplication), or when they have the same lowest form.
      why?

      If a/b = c/d (b and d not zero), multiplying both sides by bd gives ad = bc — the same 'do the same to both sides' rule.

   2. (i) 2 × 6 = 12 and 3 × 4 = 12, so 2/3 = 4/6; dividing 4 and 6 by 2 also gives 2/3.
   3. (ii) 5 × 8 = 40 and 4 × 10 = 40, so 5/4 = 10/8; dividing 10 and 8 by 2 gives 5/4.
   4. (iii) (−3) × 10 = −30 and 5 × (−6) = −30, so −3/5 = −6/10; dividing −6 and 10 by 2 gives −3/5.
   5. (iv) 9 ÷ 3 = 3, so 9/3 = 3; also 9 × 1 = 3 × 3.

   Answer: Each pair is equal: (i) both are 2/3; (ii) both are 5/4; (iii) both are −3/5; (iv) both are 3.

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2. Ex 3.3 Q2

   Add: (i) 2/5 + 3/10, (ii) 7/12 + 5/8, (iii) −4/7 + 3/14.

   1. (i) The LCM of 5 and 10 is 10.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   2. (i) Convert fractions: 2/5 = 4/10 and 3/10 remains 3/10.
   3. (i) Add the numerators: 4/10 + 3/10 = 7/10.
   4. (ii) The LCM of 12 and 8 is 24.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   5. (ii) Convert fractions: 7/12 = 14/24 and 5/8 = 15/24.
   6. (ii) Add the numerators: 14/24 + 15/24 = 29/24.
   7. (iii) The LCM of 7 and 14 is 14.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   8. (iii) Convert fractions: −4/7 = −8/14 and 3/14 remains 3/14.
   9. (iii) Add the numerators: −8/14 + 3/14 = −5/14.

   Answer: (i) 7/10; (ii) 29/24; (iii) −5/14

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3. Ex 3.3 Q3

   Subtract: (i) 5/6 − 1/4, (ii) 11/8 − 3/4, (iii) −7/9 − (−2/3).

   1. (i) The LCM of 6 and 4 is 12.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   2. (i) Convert fractions: 5/6 = 10/12 and 1/4 = 3/12.
   3. (i) Subtract: 10/12 − 3/12 = 7/12.
   4. (ii) The LCM of 8 and 4 is 8.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   5. (ii) Convert fractions: 11/8 stays 11/8 and 3/4 = 6/8.
   6. (ii) Subtract: 11/8 − 6/8 = 5/8.
   7. (iii) Simplify the expression: −7/9 − (−2/3) becomes −7/9 + 2/3.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   8. (iii) The LCM of 9 and 3 is 9.
      why?

      LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.

   9. (iii) Convert fractions: −7/9 stays −7/9 and 2/3 = 6/9.
   10. (iii) Add: −7/9 + 6/9 = −1/9.

   Answer: (i) 7/12; (ii) 5/8; (iii) −1/9

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4. Ex 3.3 Q4

   Multiply: (i) 2/3 × 3/10, (ii) 7/11 × 5/8, (iii) −4/7 × 5/14.

   1. (i) Multiply the numerators 2 and 3 to get 6, and the denominators 3 and 10 to get 30.
   2. (i) Simplify the fraction 6/30 by dividing both numerator and denominator by 6 to get 1/5.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   3. (ii) Multiply the numerators 7 and 5 to get 35, and the denominators 11 and 8 to get 88.
   4. (ii) The fraction 35/88 cannot be simplified further as 35 and 88 have no common factors.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   5. (iii) Multiply the numerators −4 and 5 to get −20, and the denominators 7 and 14 to get 98.
   6. (iii) Simplify the fraction −20/98 by dividing both numerator and denominator by 2 to get −10/49.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   Answer: (i) 1/5; (ii) 35/88; (iii) −10/49

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5. Ex 3.3 Q5

   Divide: (i) 2/3 ÷ 3/10, (ii) 7/11 ÷ 5/8, (iii) −4/7 ÷ 5/14.

   1. (i) To divide 2/3 by 3/10, multiply 2/3 by the reciprocal of 3/10, which is 10/3.
   2. (i) The calculation is (2 × 10) / (3 × 3) = 20/9.
   3. (ii) To divide 7/11 by 5/8, multiply 7/11 by the reciprocal of 5/8, which is 8/5.
   4. (ii) The calculation is (7 × 8) / (11 × 5) = 56/55.
   5. (iii) To divide −4/7 by 5/14, multiply −4/7 by the reciprocal of 5/14, which is 14/5.
   6. (iii) The calculation is (−4 × 14) / (7 × 5).
   7. (iii) Simplify by dividing 14 and 7 by their common factor 7, resulting in (−4 × 2) / (1 × 5) = −8/5.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   Answer: (i) 20/9; (ii) 56/55; (iii) −8/5

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6. Ex 3.3 Q6

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

   1. Take the left-hand side (LHS) of the equation: (1/2 + 3/4) × 8/3.
   2. Find a common denominator for the fractions inside the bracket, which is 4.
   3. Convert 1/2 to 2/4 and add it to 3/4 to get 5/4.
   4. Multiply the sum 5/4 by 8/3 to get 40/12.
   5. Simplify 40/12 by dividing the numerator and denominator by 4 to get 10/3.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   6. Take the right-hand side (RHS) of the equation: 1/2 × 8/3 + 3/4 × 8/3.
   7. Multiply 1/2 by 8/3 to get 8/6, which simplifies to 4/3.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   8. Multiply 3/4 by 8/3 to get 24/12, which simplifies to 2.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   9. Add the results 4/3 and 2 by converting 2 to 6/3.
   10. Sum 4/3 and 6/3 to get 10/3.
   11. Since LHS equals RHS, the distributive property is verified.

   Answer: Both sides equal 10/3, so the two sides are equal (the distributive law).

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7. Ex 3.3 Q7

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

   1. Apply the distributive property to expand 7/9 × (6/7 − 3/4) as 7/9 × 6/7 − 7/9 × 3/4.
      why?

      Expanding multiplies every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.

   2. Simplify the first term 7/9 × 6/7 by cancelling 7 in the numerator and denominator to get 6/9, which reduces to 2/3.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   3. Simplify the second term 7/9 × 3/4 by cancelling 3 in the numerator and denominator to get 7/3 × 1/4 = 7/12.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   4. Subtract the two results: 2/3 − 7/12.
   5. Find the least common multiple of 3 and 12, which is 12, and convert 2/3 to 8/12.
   6. Perform the subtraction 8/12 − 7/12 to get 1/12.

   Answer: 1/12

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8. Ex 3.3 Q8

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

   1. Expand the left side of the equation: 5/6 × x + 5/6 × 3/5
      why?

      Expanding multiplies every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.

   2. Simplify the product 5/6 × 3/5 to get 15/30, which reduces to 1/2
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   3. The equation becomes 5/6 x + 1/2 = 5/6 x + 1/2
   4. Since both sides are identical, the equation holds for any value of x
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   5. Thus, x can be any rational number

   Answer: Since 5/6 × 3/5 = 1/2, both sides are 5/6 x + 1/2: the equation is true for every rational number x.

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