PERIOD OF DECIMAL

In the decimal expansion of the rational numbers, the number of repeating decimals is called the length of the period of decimals.

For example,

(i)  25/4  =  3.571428571428......... has the length of the period of decimal  =  6. 

(ii)  27/110  =  3.4545......... has the length of the period of decimal  =  2.

Problem : 

Express the rational number 1/27 in recurring decimal form by using the recurring decimal expansion of 1/3. Hence write 59/27 in recurring decimal form.

Solution : 

We know that 1/3  =  0.33..............

Therefore, 

1/27  =  (1/9) x (1/3)

1/27  =  (1/9) x 0.33..............

1/27  =  0.037037..............

Also, 

59/27  =  2 5/27

59/27  =  2 + 5/27

59/27  =  2 + (5 x 1/27)

59/27  =  2 + (5 x 0.037037..............)

59/27  =  2 + 0.185185..............

59/27  =  2.185185..............

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