Covert the following recurring decimals to fractions :
1) 2.7343434..............
2) 32.03256256256..........
3) 0.01232222........
4) 2.03323232..........
5) 0.252525..........
6) 3.3333..........
7) 1.023562562562..........
8) 0.6666..........
1. Answer :
Let x = 2.7343434..............
The repeated pattern is 34.
There is one digit between the decimal point and 1^{st} repeated pattern, that is 7.
Multiply the given decimal number by 10.
10x = 27.343434..............-----(1)
There are three digits between the decimal point and the 2^{nd} repeated pattern, that is 734.
Multiply the given decimal number by 1000.
1000x = 2734.3434..............-----(2)
Subtract (1) from (2).
90x = 2707
x = 2707/90
2.7343434.............. = 2707/90
2. Answer :
Let x = 32.03256256256.............
The repeated pattern is 256.
There are two digits between the decimal point and 1^{st} repeated pattern, that is 03.
Multiply the given decimal number by 100.
100x = 3203.256256256...............-----(1)
There are five digits between the decimal point and the 2^{nd} repeated pattern, that is 03256.
Multiply the given decimal number by 100000.
100000x = 3203256.256256256...............-----(2)
Subtract (1) from (2).
99900x = 3200053
x = 3200053/99900
32.03256256256.......... = 3200053/99900
3. Answer :
Let x = 0.01232222.............
The repeated pattern is 2.
There are four digits between the decimal point and 1^{st} repeated pattern, that is 0123.
Multiply the given decimal number by 10000.
10000x = 123.2222...............-----(1)
There are five digits between the decimal point and the 2^{nd} repeated pattern, that is 01232.
Multiply the given decimal number by 100000.
100000x = 1232.2222...............-----(2)
Subtract (1) from (2).
90000x = 1109
x = 1109/90000
0.01232222........... = 1109/90000
4. Answer :
Let x = 2.03323232.............
The repeated pattern is 32.
There are two digits between the decimal point and 1^{st} repeated pattern, that is 03.
Multiply the given decimal number by 100.
100x = 203.323232...............-----(1)
There are four digits between the decimal point and the 2^{nd} repeated pattern, that is 0332.
Multiply the given decimal number by 10000.
10000x = 20332.323232...............-----(2)
Subtract (1) from (2).
9900x = 20129
x = 9900/20129
2.03323232.......... = 9900/20129
5. Answer :
Let x = 0.252525.............
The repeated pattern is 25.
There is no digit between the decimal point and 1^{st }repeated pattern.
x = 0.252525...............-----(1)
There are two digits between the decimal point and the 2^{nd} repeated pattern, that is 25.
Multiply the given decimal number by 100.
100x = 25.252525...............-----(2)
Subtract (1) from (2).
99x = 25
x = 25/99
0.252525.......... = 25/99
6. Answer :
Let x = 3.3333.............
The repeated pattern is 3.
There is no digit between the decimal point and 1^{st }repeated pattern.
x = 3.3333...............-----(1)
There is one digit between the decimal point and the 2^{nd} repeated pattern, that is 3.
Multiply the given decimal number by 10.
10x = 33.3333...............-----(2)
Subtract (1) from (2).
9x = 30
x = 30/9
x = 10/3
3.3333.............. = 10/3
7. Answer :
Let x = 1.023562562562.............
The repeated pattern is 562.
There are three digits between the decimal point and 1^{st} repeated pattern, that is 023.
Multiply the given decimal number by 1000.
1000x = 1023.562562562...............-----(1)
There are six digits between the decimal point and the 2^{nd} repeated pattern, that is 023562.
Multiply the given decimal number by 1000000.
1000000x = 1023562.562562562...............-----(2)
Subtract (1) from (2).
999000x = 1022538
x = 1022539/999000
1.023562562562.......... = 1022539/999000
8. Answer :
Let x = 0.6666.............
The repeated pattern is 6.
There is no digit between the decimal point and 1^{st }repeated pattern.
x = 0.6666...............-----(1)
There is one digit between the decimal point and the 2^{nd} repeated pattern, that is 6.
Multiply the given decimal number by 10.
10x = 6.6666...............-----(2)
Subtract (1) from (2).
9x = 6
x = 6/9
x = 2/3
0.6666.......... = 2/3
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