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Class 9 Maths – The World of Numbers

Exercise Set 3.5 Solutions | Ganita Manjari

Question 1

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating:

7/20 , 4/15 and 13/250

Then check your answers by expressing them as decimals.

Solution

A rational number has a terminating decimal if the denominator contains only factors 2 and/or 5.

(i) 7/20

20 = 2 × 2 × 5

Denominator contains only 2 and 5.

Therefore, 7/20 is a terminating decimal.

Now convert into decimal:

7 ÷ 20 = 0.35

Hence,

7/20 = 0.35

(ii) 4/15

15 = 3 × 5

Denominator contains factor 3.

Therefore, 4/15 is a repeating decimal.

Now divide:

4 ÷ 15 = 0.26666…

Hence,

4/15 = 0.26666…

(iii) 13/250

250 = 2 × 5 × 5 × 5

Denominator contains only 2 and 5.

Therefore, 13/250 is a terminating decimal.

Now divide:

13 ÷ 250 = 0.052

Hence,

13/250 = 0.052

Question 2

Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2/13, 3/13, 4/13 etc.? What do you notice?

Solution

1 ÷ 13 = 0.076923076923…

Repeating block:

076923

Now,

2/13 = 0.153846153846…

3/13 = 0.230769230769…

4/13 = 0.307692307692…

We notice that the repeating digits rotate in a cyclic manner.

Therefore, these decimals show cyclic properties.

Question 3

Classify the following numbers as rational or irrational.

(i) √81

Solution

√81 = 9

9 is a rational number.

Therefore, √81 is rational.

(ii) √12

Solution

√12 cannot be expressed as a fraction.

Its decimal expansion is non-terminating and non-repeating.

Therefore, √12 is irrational.

(iii) 0.33333…

Solution

0.33333… is a repeating decimal.

It can be written as:

1/3

Therefore, it is rational.

(iv) 0.123451234512345…

Solution

The block 12345 repeats continuously.

Therefore, it is a repeating decimal.

Hence, it is rational.

(v) 1.01001000100001…

Solution

The decimal does not repeat in a fixed pattern.

Therefore, it is non-terminating and non-repeating.

Hence, it is irrational.

(vi) 23.560185612239874790120

Solution

This is a terminating decimal.

All terminating decimals are rational numbers.

Therefore, it is rational.

Question 4

The number 0.9̅ means 0.99999… Using algebra, explain why 0.9̅ is exactly equal to 1.

Solution

Let:

x = 0.99999…

Multiply both sides by 10:

10x = 9.99999…

Now subtract:

10x − x = 9.99999… − 0.99999…

9x = 9

Divide both sides by 9:

x = 1

But x = 0.99999…

Therefore,

0.99999… = 1

Question 5

We have seen that the repeating block of 1/7 is a cyclic number. Try to find more numbers whose reciprocals produce decimals with repeating cyclic blocks.

Solution

Some examples are:

1/7 = 0.142857142857…

1/13 = 0.076923076923…

1/17 = 0.0588235294117647…

These reciprocals produce repeating cyclic decimal blocks.

Students preparing for Class 9 Maths Ganita Manjari should practice all exercise sets and end-of-chapter exercises regularly for better understanding of concepts and excellent exam preparation. Below is the complete chapter-wise list of exercises from Chapters 1 to 4. Students can explore all solutions, worksheets, practice papers, and important questions chapter-wise.

Chapter 1 – Orienting Yourself: The Use of Coordinates

  • Exercise Set 1.1 Solutions
  • Exercise Set 1.2 Solutions
  • End of Chapter Exercises Solutions

Chapter 2 – Introduction to Linear Polynomials

Chapter 3 – The World of Numbers

  • Exercise Set 3.1 Solutions
  • Exercise Set 3.2 Solutions
  • Exercise Set 3.3 Solutions
  • Exercise Set 3.4 Solutions
  • Exercise Set 3.5 Solutions
  • End of Chapter Exercises Solutions

Chapter 4 – Exploring Algebraic Identities

  • Exercise Set 4.1 Solutions
  • Exercise Set 4.2 Solutions
  • Exercise Set 4.3 Solutions
  • Exercise Set 4.4 Solutions
  • Exercise Set 4.5 Solutions
  • End of Chapter Exercises Solutions