Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5 Solutions: The World of Numbers

Decimals can tell us whether a number is rational or irrational. Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5 Solutions focus on this idea through terminating decimals, repeating decimals, cyclic numbers and non-repeating decimal patterns from The World of Numbers.

This exercise from The World of Numbers includes questions on 7/20, 4/15, 13/250, the repeating decimal of 1/13, rational-irrational classification, and the proof that 0.999… = 1. Students also practise how to convert repeating decimals to fractions using algebra. By the end of Ganita Manjari Class 9 Chapter 3 Exercise 3.5, they understand why terminating and repeating decimals are rational, while non-terminating, non-repeating decimals are irrational.

Key Takeaways

Terminating Decimal: A rational number terminates if the denominator has only 2 and 5 as prime factors.
Repeating Decimal: A rational number repeats if the denominator has other prime factors.
Irrational Number: A non-terminating, non-repeating decimal is irrational.
Repeating to Fraction: Algebra can convert repeating decimals into p/q form.

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5 Solutions Structure 2026

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5 Solutions

Exercise Set 3.5 comes from the section on Real Numbers: Decimals and Cyclic Patterns. The chapter explains that rational numbers have either terminating or repeating decimal expansions, while irrational numbers have non-terminating and non-repeating decimal expansions.

These Class 9 Maths rational and irrational numbers answers help students revise decimal expansion Class 9, terminating repeating decimals Class 9, cyclic numbers Class 9 and the connection between decimal forms and real numbers Class 9.

Exercise 3.5 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 explicitly performing the long divisions and expressing these rational numbers as decimals.

Solution:

A rational number in lowest terms has a terminating decimal if the denominator has only 2 or 5 as prime factors. If the denominator has any other prime factor, the decimal is repeating.

7/20

Denominator:

20 = 2² × 5

Since the denominator has only 2 and 5 as prime factors, 7/20 has a terminating decimal.

7/20 = 0.35

4/15

Denominator:

15 = 3 × 5

Since the denominator has 3 as a prime factor, 4/15 has a repeating decimal.

4/15 = 0.2666…

13/250

Denominator:

250 = 2 × 5³

Since the denominator has only 2 and 5 as prime factors, 13/250 has a terminating decimal.

13/250 = 0.052

Answer:
7/20 = 0.35, terminating decimal.
4/15 = 0.2666…, repeating decimal.
13/250 = 0.052, terminating decimal.

Exercise 3.5 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? Now compute 3/13, 4/13, etc. What do you notice?

Solution:

On long division:

1/13 = 0.076923076923…

So, the repeating block is:

076923

Now calculate some multiples:

The repeating decimals show cyclic behaviour, but in two related cycles. Some multiples use cyclic shifts of 076923, while others use cyclic shifts of 153846.

Answer: The repeating block of 1/13 is 076923. The decimals of 2/13, 3/13, 4/13, … show repeating blocks with cyclic patterns.

Exercise 3.5 Question 3

Classify the following numbers as rational or irrational. Find the explicit fractions in case they are rational.

(i) √81

Solution:

√81 = 9

Since 9 can be written as:

9 = 9/1

it is a rational number.

Answer: √81 is rational, and its fraction form is 9/1.

(ii) √12

Solution:

√12 = √(4 × 3)

√12 = 2√3

Since √3 is irrational, 2√3 is also irrational.

Answer: √12 is irrational.

(iii) 0.33333…

Solution:

Let:

x = 0.33333…

Then:

10x = 3.33333…

Subtract:

10x − x = 3.33333… − 0.33333…

9x = 3

x = 3/9

x = 1/3

Answer: 0.33333… is rational, and its fraction form is 1/3.

(iv) 0.123451234512345…

Solution:

The decimal block 12345 repeats.

So, this is a non-terminating repeating decimal. Therefore, it is rational.

Let:

x = 0.1234512345…

Since 5 digits repeat, multiply by 100000.

100000x = 12345.1234512345…

Subtract:

100000x − x = 12345.1234512345… − 0.1234512345…

99999x = 12345

x = 12345/99999

Simplify by dividing numerator and denominator by 3.

x = 4115/33333

Answer: 0.123451234512345… is rational, and its fraction form is 4115/33333.

(v) 1.01001000100001…

Solution:

The decimal does not terminate. It also does not repeat one fixed block of digits. The number of zeros keeps increasing.

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

Answer: 1.01001000100001… is irrational.

(vi) 23.560185612239874790120

Solution:

This is a terminating decimal because it has a fixed number of digits after the decimal point.

Every terminating decimal is rational.

It can be written as:

23.560185612239874790120 = 23560185612239874790120 / 1000000000000000000000

Simplifying gives:

589004640305996869753 / 25000000000000000000

Answer: 23.560185612239874790120 is rational, and one simplified fraction form is 589004640305996869753 / 25000000000000000000.

Exercise 3.5 Question 4

The number 0.9̅, which means 0.99999…, is a rational number. 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 the first equation from the second.

10x − x = 9.99999… − 0.99999…

9x = 9

Divide both sides by 9.

x = 1

But:

x = 0.99999…

Therefore:

0.99999… = 1

Answer: 0.9̅ = 1.

Exercise 3.5 Question 5

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

Solution:

A cyclic number appears when the repeating block of a reciprocal produces cyclic shifts when multiplied by certain numbers.

One example is:

1/17 = 0.0588235294117647…

The repeating block is:

0588235294117647

Multiples of this repeating block show cyclic behaviour.

Another example is:

1/19 = 0.052631578947368421…

The repeating block is:

052631578947368421

This also shows cyclic behaviour.

Answer: Examples of such numbers are 17 and 19. Their reciprocals 1/17 and 1/19 have repeating blocks with cyclic properties.

Final Answers for Exercise 3.5

Concept Used in The World of Numbers Exercise 3.5

The World of Numbers Exercise 3.5 focuses on the decimal form of rational and irrational numbers.

Important concepts include:

• Rational and irrational numbers Class 9: rational numbers can be written as fractions, while irrational numbers cannot.
• Decimal expansion Class 9: decimal form helps identify the nature of a number.
• Terminating repeating decimals Class 9: rational numbers either terminate or repeat.
• Irrational decimals: non-terminating and non-repeating decimals.
• Convert repeating decimals to fractions: use algebra by multiplying by powers of 10 and subtracting.
• Cyclic numbers Class 9: repeating blocks that shift cyclically when multiplied.
• Real numbers Class 9: rational and irrational numbers together form the real number system.

These ideas are important in Class 9 Ganita Manjari number system solutions because Chapter 3 builds the number system from natural numbers to real numbers.

About Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5

Ganita Manjari Class 9 Chapter 3 Exercise 3.5 comes after students study rational numbers, their number-line representation and irrational numbers. This exercise uses decimal expansions to separate rational numbers from irrational numbers.

These Class 9 Maths Chapter 3 Exercise 3.5 Solutions help students revise:

• terminating decimal expansions,
• repeating decimal expansions,
• prime factor test for terminating decimals,
• cyclic decimal blocks,
• rational and irrational classification,
• algebraic proof of 0.999… = 1,
• decimal representation of real numbers.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3