The learner has already discovered that some common fractions cannot be
changed to exact decimal fractions, as--
1/3 =.33333 on to infinity.
2/3 =.66666 on to infinity.
7/33 =.212121, etc.
These decimals are known as _Circulates_, _Recurring_ or _Circulating_
decimals.
The part which recurs is called the _Repetend_.
This is marked by putting a dot over the first and last figures of it.
For instance, if we write the 21 in the last case above, this way: 2̊1̊,
it indicates that, if written out, the result would be 21212121, etc.,
on to infinity.
Where a circulating decimal occurs in work, it is best to reduce it to a
common fraction. If need be, it may be expressed in the result as a
circulate to any number of decimal places.
To change a pure circulate to a common fraction.
RULE.--Omit the (.) and write the figures of the repetend for the
numerator, and as many 9’s for the denominator as there are places in
the repetend.
EXAMPLES: Change the pure circulates .3̊, .2̊7̊, .1̊42857̊, to common
fractions.
(3 1)
.3̊, (- = -) _Ans._ 1/3.
(9 3)
(27 3)
.2̊7̊, (-- = --) _Ans._ 3/11
(99 11)
(142857 1)
.1̊42857̊, (------ = -) _Ans._ 1/7
(999999 7)
To change a mixed circulate to a common fraction.
RULE.--From the whole decimal subtract the finite part, and make the
remainder the numerator. For the denominator, write as many 9’s as there
are figures in the repetend, and annex as many 0’s as there are finite
places.
EXAMPLE: Change the mixed circulates .16̊ and .416̊ to common fractions.
15 1
16 - 1 = 15, --- = -. _Ans._ 1/6.
90 6
375 5
416 - 41 = 375, --- = -- _Ans._ 5/12.
900 12
To add, subtract, multiply and divide circulates, reduce them to common
fractions, then apply the respective rules.