recurring decimal

[sanju09]

The value of the recurring decimal number 0.6313131... is equal to

A. 65/106
B. 125/198
C. 627/990
D. 691/1124
E. 737/1200

[Baldini]

Answer is B.
So you by any chance know a shortcut for this type of exercise?
thx

[bluementor]

0.631313131....
= 0.6 + 0.0303030303.. + 0.001010101...
= 0.6 + 3/99 + 0.001010101...
= 0.6 + 3/99 + (0.01010101...)/10
= 0.6 + 3/99 + (1/99)/10
=0.6 + 3/99 + (1/990)
=3/5 + 31/990
=(3(198) + 31)/990
= 625/990
=125/198

Choose B.

-BM-

[sanju09]

Answer is B.
So you by any chance know a shortcut for this type of exercise?
thx

Sure Baldini, if the repeating pattern of digits is set to start right after the decimal point, then this part will be equivalent to the integer formed by pattern divided by as many 9s as are the digits in the numerator; for example 0.861861861... will always be equal to 861/999; 3.057575757... = 3 + 0.057575757... = 3 + (1/10) (0.57575757...) = 3 + (1/10) (57/99) = 3 + (1/10) (19/33) = 3 + (19/330) = 1009/330.

The present number

0.6313131... = 0.6 + 0.0313131... = 6/10 + (1/10) (0.313131...)

= 3/5 + (1/10) (31/99) = 3/5 + 31/990 = [(198*3) + 31]/990

= (594 + 31)/990 = 625/990 = 125/198

[Baldini]

thanks a lot.