# Number System

## Exercise 1.3 Part 2

Question 3: Express the following in the form p/q, where p and q are integers and q ≠ 0.

(i) 0.6

**Answer:** Given, 0.6 = 0.66666…….

Let, `x=0.66666…`

[Since only one digit is repeating, so multiply x with 10.]

So, `10x=10xx6.66666….`

Or, `10x=6+0.66666…`

Or, `10x=6+x`

Since, `x=0.66666…`

Or, `10x-x=6`

Or, `9x=6`

Or, `x=6/9=(3xx2)/(3xx3)=2/3`

Thus, 0.6 `=2/3`

### Alternate method:

Given, 0.6

Step: 1 - Omit the decimal and recurring symbol (repeating symbol i.e. bar)

Step: 2 – Put repeating decimal as numerator and one 9 as denominator for one repeating decimal digit.

Or, `6/9=(3xx2)/(3xx2)=2/3`

Thus, 0.6 `=2/3`

(ii) 0.47

**Answer:** Let x = 0.47

Or, `x=0.47777….`

[Since only one digit is repeating, so multiply x with 10.]

Since `10x=10xx0.477777..`

Or, `10x=4.777777…`

Or, `10x=4.3+0.4777…`

Or, `10x=4.3+x`

Since `x=0.47777…`

Or, `10x-x=4.3`

Or, `9x=4.3`

Or, `x=(4.3)/9`

Or, `x=43/90`

#### Alternate method:

Given, 0.47

Step: 1 – Take 47 as numerator and subtract 4 (non repeating decimal digit) from it.

Step: 2 - Put one 9 for one repeating decimal digit

Step: 3 - Place one zero (0) after 9 for one non repeating decimal digit as denominator.

Thus, we get

`(47-4)/90=43/90`

(iii) 0.001

**Answer:** Let x = 0.001

Or, `x=0.001001…`

Since, there are three repeating decimal digit, so multiply x with 1000

Or, `1000x=1000xx0.001001…`

Or, `1000x=1.001…`

Or, `1000x=1+0.001….`

Since `x=0.001001…`

Thus, `1000x=1+x`

Or, `1000x-x=1`

Or, `999x=1`

Or, `x=1/999`

#### Alternate method:

Given, 0.001

Step: 1- Take 001 as numerator

Step: Since there are three repeating decimal digits, so take three 9, i.e. 999 as denominator.

Thus, `0.001 = 001/999=1/999`

Question 4: Express 0.99999 .... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

**Answer:**

Since, there is only one repeating digit after decimal, thus multiply x with 10.

#### Alternate method:

Step: 1 – Take 9 as numerator.

Step: 2 – Since there is only one repeating decimal digit, so take one 9 as denominator.

After doing above steps we get,