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,
