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.
Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.5 Solutions
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.
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:
| Fraction | Decimal Expansion |
| 1/13 | 0.076923076923… |
| 2/13 | 0.153846153846… |
| 3/13 | 0.230769230769… |
| 4/13 | 0.307692307692… |
| 5/13 | 0.384615384615… |
| 6/13 | 0.461538461538… |
| 7/13 | 0.538461538461… |
| 8/13 | 0.615384615384… |
| 9/13 | 0.692307692307… |
| 10/13 | 0.769230769230… |
| 11/13 | 0.846153846153… |
| 12/13 | 0.923076923076… |
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.
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.
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: 0.9Ì… = 1.
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.
Related Links – Class 9 Maths Chapter 3