Covert each of the following repeating decimals to fractions :
1) 0.33333..........
2) 0.77777..........
3) 0.36363636..........
4) 0.507507507507..........
5) 0.06666..........
6) 0.00151515..........
7) 0.37777..........
8) 1.21212121..........
9) 2.342342342..........
10) 2.05555..........
1. Answer :
There are only repeating digits after the decimal point in
0.33333..........
There is only one digit in the repeating pattern, that is 3.
Subtract 1 from 10, the result is 9.
Write a fraction with numerator 3 and denominator 9.
= 3/9
= 1/3
2. Answer :
There are only repeating digits after the decimal point in
0.77777..........
There is only one digit in the repeating pattern, that is 7.
Subtract 1 from 10, the result is 9.
Write a fraction with numerator 7 and denominator 9.
= 7/9
3. Answer :
There are only repeating digits after the decimal point in
0.36363636..........
There are two digits in the repeating pattern, that is 36.
Subtract 1 from 100, the result is 99.
Write a fraction with numerator 36 and denominator 99.
= 36/99
= 4/11
4. Answer :
There are only repeating digits after the decimal point in
0.507507507507..........
There are three digits in the repeating pattern, that is 507.
Subtract 1 from 1000, the result is 999.
Write a fraction with numerator 507 and denominator 999.
= 507/999
= 169/333
5. Answer :
0.06666..........
There is one digit between the decimal point and the repeating digits, that is 0.
0.06666.......... = (0.6666..........) ÷ 10 ----(1)
There are only repeating digits after the decimal point in
0.6666..........
There is only one digit in the repeating pattern, that is 6.
Subtract 1 from 10, the result is 9.
Write a fraction with numerator 6 and denominator 9 for 0.6666.......... in (1).
0.06666.......... = (6/9) ÷ 10
= (2/3) ÷ (10/1)
= (2/3) ⋅ (1/10)
= (2 ⋅ 1)/(3 ⋅ 10)
= 2/30
= 1/15
6. Answer :
0.00151515..........
There are two digits between the decimal point and the repeating digits, that is 00.
0.00151515.......... = (0.151515..........) ÷ 100 ----(1)
There are only repeating digits after the decimal point in
0.151515..........
There are two digits in the repeating pattern, that is 15.
Subtract 1 from 100, the result is 99.
Write a fraction with numerator 15 and denominator 99 for 0.151515.......... in (1).
0.00151515.......... = (15/99) ÷ 100
= (5/33) ÷ (100/1)
= (5/33) ⋅ (1/100)
= (5 ⋅ 1)/(33 ⋅ 100)
= 5/3300
= 1/660
7. Answer :
0.37777..........
There is one digit between the decimal point and the repeating digits, that is 3.
0.37777.......... = 0.3 + 0.07777..........
In 0.07777.........., there is one digit between the decimal point and the repeating digits, that is 0.
0.37777.......... = 0.3 + (0.7777..........) ÷ 10 ----(1)
There are only repeating digits after the decimal point in
0.7777..........
There is only one digit in the repeating pattern, that is 7.
Subtract 1 from 10, the result is 9.
Write a fraction with numerator 7 and denominator 9 for 0.7777.......... in (1).
0.39999.......... = 0.3 + (7/9) ÷ 10
= 3/10 + 7/90
= 27/90 + 7/90
= (27 + 7)/90
= 34/90
= 17/45
8. Answer :
1.21212121.......... = 1 + 0.21212121.......... ----(1)
There are only repeating digits after the decimal point in
0.21212121..........
There are two digits in the repeating pattern, that is 21.
Subtract 1 from 100, the result is 99.
Write a fraction with numerator 21 and denominator 99 for 0.21212121.......... in (1).
1.21212121.......... = 1 + 21/99
= 1 + 7/33
= 33/33 + 7/33
= (33 + 7)/33
= 40/33
9. Answer :
2.342342342.......... = 2 + 0.342342342.......... ----(1)
There are only repeating digits after the decimal point in
0.342342342..........
There are three digits in the repeating pattern, that is 342.
Subtract 1 from 1000, the result is 999.
Write a fraction with numerator 342 and denominator 999 for 0.342342342.......... in (1).
2.342342342.......... = 2 + 342/999
= 2 + 38/111
= 222/111 + 38/111
= (222 + 38)/111
= 260/111
10. Answer :
2.05555.......... = 2 + 0.05555.......... ----(1)
In 0.05555.........., there is one digit between the decimal point and the repeating digits, that is 0.
2.05555.......... = 2 + (0.5555..........) ÷ 10 ----(1)
There are only repeating digits after the decimal point in
0.5555..........
There is only one digit in the repeating pattern, that is 5.
Subtract 1 from 10, the result is 9.
Write a fraction with numerator 5 and denominator 9 for 0.5555.......... in (1).
2.05555.......... = 2 + (5/9) ÷ 10
= 2 + 5/90
= 2 + 1/18
= 36/18 + 1/18
= (36 + 1)/18
= 37/18
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