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82
GAŅIITASĀRASAŃGRAHA.

In a mango tree; ${\displaystyle {\tfrac {1}{9}}}$ the remainder as multiplied by that same (${\displaystyle {\tfrac {1}{9}}}$ part of that same remainder), as also (the remaining) fourteen (peacocks) were found in a grove of tamāla trees. How many are they (in all) ?

60. One-twelfth part of a pillar, as multiplied by ${\displaystyle {\tfrac {1}{30}}}$ part thereof, was to be found underwater; ${\displaystyle {\tfrac {1}{20}}}$ of the remainder, as multiplied by ${\displaystyle {\tfrac {3}{16}}}$ thereof was found (buried) in the mire (below); and 20 hastas of the pillar were found in the air (above the water). O friend, you give out the measure of the length of the pillar.

Here ends the Bhāgassamvarga vareity.

The rule relating to the Aṃśavarga variety (of miscellaneous problems on fractions), characterised by the subtraction or addition (of known quantities):--

61. (Take) the half of the denominator (of tho specified fractional part of the unknown collective quantity), as divided by its own (related) numerator, and as increased or diminished by the (given) known quantity which is subtracted from or added to (the specified fractional part of the unknown collective quantity). The square root of the square of this (resulting quantity), as diminished by the square of (the above known) quantity to be subtracted or to be added and (also) by tho known remainder (of the collective quantity), when added to or subtracted from the square root (of the square quantity mentioned above) and then divided by the (specified) fractional part (of the unknown collective quantity), gives the (required) value (of the unknown collective quantity).

Examples of the minus variety.

62.[62] (A number) of buffaloes (equivalent to) the square of ${\displaystyle {\tfrac {1}{8}}}$ (of the whole herd) minus 1 is sporting in the forest. The

62.^ Algebraically, ${\displaystyle s={\Big \{}\pm {\sqrt {\left({\tfrac {n}{2m}}\pm d\right)^{2}-d^{2}-a}}+\left({\tfrac {m}{2m}}\pm d\right){\Big \}}\div {\tfrac {m}{n}}}$. This value is obtained from the equation ${\displaystyle z-\left({\tfrac {m}{n}}\mp d\right)^{2}-x=0}$, where d is the given known quantity.