Home · Schooling · CBSE · Class 9 · Mathematics · The World of Numbers · Exercise Set 3.4

NCERT Class 9 Mathematics Exercise Set 3.4 Solutions: The World of Numbers

Step-by-step working for every question of Exercise Set 3.4 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 3.4 Q1

   Show 2/3, −5/4 and 1½ on one number line.

   1. Draw a horizontal number line with equal spacing between integers.
   2. Mark the point for 2/3 between 0 and 1, dividing the interval into three equal parts and taking the second mark.
   3. Mark the point for −5/4 at −1.25, which is one quarter of the way from −1 to −2.
   4. Mark the point for 1½ at 1.5, exactly halfway between 1 and 2.
   5. Observe the positions to confirm the order −5/4 < 2/3 < 3/2.

   Answer: −5/4 < 2/3 < 3/2

   Report an error in this solution

2. Ex 3.4 Q2

   Give three different rational numbers strictly between −1/2 and 1/4.

   1. We need to find three rational numbers strictly between −1/2 and 1/4.
   2. Convert both fractions to have a common denominator to compare them easily.
   3. −1/2 = −4/8 and 1/4 = 2/8.
   4. Now we look for fractions with denominator 8 that lie strictly between −4/8 and 2/8.
   5. The integers between −4 and 2 are −3, −2, −1, and 1.
   6. So the corresponding fractions are −3/8, −2/8, −1/8, and 1/8.
   7. We can pick any three of these, for example −3/8, −1/8, and 1/8.
   8. Check: −3/8 > −4/8 (since −3 > −4) and −3/8 < 2/8, so it is in range.
   9. Check: −1/8 > −4/8 and −1/8 < 2/8, so it is in range.
   10. Check: 1/8 > −4/8 and 1/8 < 2/8, so it is in range.

   Answer: Three rational numbers strictly between −1/2 and 1/4 are −3/8, −1/8, and 1/8.

   Report an error in this solution

3. Ex 3.4 Q3

   Simplify (−1/4) + (5/12).

   1. Write the expression as −1/4 + 5/12.
   2. Find the least common multiple of the denominators 4 and 12, which is 12.
   3. Convert −1/4 to a fraction with denominator 12 by multiplying numerator and denominator by 3, giving −3/12.
   4. Add the numerators: −3 + 5 = 2.
   5. Write the result as 2/12.
   6. Simplify 2/12 by dividing numerator and denominator by 2 to get 1/6.
      why?

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

   Answer: 1/6

   Report an error in this solution

4. Ex 3.4 Q4

   A tailor has 15¾ m of silk and one kurta needs 2¼ m. Exactly how many kurtas can he make?

   1. Convert the total silk length 15¾ m into an improper fraction: 15¾ = 63/4 m.
   2. Convert the silk needed for one kurta 2¼ m into an improper fraction: 2¼ = 9/4 m.
   3. Divide the total silk by the silk per kurta to find the number of kurtas: (63/4) ÷ (9/4).
   4. Simplify the division by multiplying by the reciprocal: (63/4) × (4/9).
      why?

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

   5. Cancel the common factor 4 and divide 63 by 9 to get 7.

   Answer: 7 kurtas

   Report an error in this solution

5. Ex 3.4 Q5

   Give three rational numbers between 3.1415 and 3.1416.

   1. We need to find three rational numbers between 3.1415 and 3.1416.
   2. To find numbers between two decimals with 4 decimal places, we can look at the 5th decimal place.
   3. The number 3.1415 is equal to 3.14150 and 3.1416 is equal to 3.14160.
   4. Any number with a 5th decimal digit between 0 and 9 (exclusive of the endpoints) will lie between them.
   5. Let's choose 3.14151, 3.14155, and 3.14159 as our three rational numbers.
   6. These numbers are all greater than 3.1415 and less than 3.1416.

   Answer: 3.14151, 3.14155, 3.14159

   Report an error in this solution

6. Ex 3.4 Q6

   Are there other ways (besides the average) to find a rational number between two rational numbers?

   1. Yes. Besides the average (a + b)/2, here are two other ways.
   2. Common denominator: write both numbers with the same, larger denominator so that their numerators are not next to each other. For 1/3 and 1/2: 1/3 = 4/12 and 1/2 = 6/12, so 5/12 lies between them.
   3. If the numerators are still next to each other, multiply the numerator and denominator by a bigger number: 1/3 = 2/6 and 1/2 = 3/6 have nothing with denominator 6 between them, but 1/3 = 8/24 and 1/2 = 12/24 give 9/24, 10/24 and 11/24.
   4. Mediant: for positive denominators with a/b < c/d, the number (a + c)/(b + d) lies between them. For 1/3 and 1/2 it gives 2/5, and 1/3 < 2/5 < 1/2 since 5 < 6 and 4 < 5 when cross-multiplied.
      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.

   5. Decimals: 1/3 = 0.333… and 1/2 = 0.5, so a terminating decimal such as 0.4 = 2/5 lies between them.

   Answer: Yes: use a common (enlarged) denominator and pick a numerator in between, use the mediant (a + c)/(b + d), or compare decimals; for 1/3 and 1/2 these give 5/12, 2/5 and 0.4. Other choices are also correct.

   Report an error in this solution

← The World of Numbers: notes, practice and all exercises