# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२७७

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81
CHAPTER IV--MISCELLANEOUS PROBLEMS (ON FRACTIONS).

56. Four times the square root of ${\displaystyle {\tfrac {1}{2}}}$ the number of a collection of boars went to a forest wherein tigers were at play; 8 times the square root of ${\displaystyle {\tfrac {1}{12}}}$, of the remainder (of the collection) went to a mountain; and 9 times the square root of ${\displaystyle {\tfrac {1}{2}}}$ of the (further) remainder (left thereafter) went to the bank of a rivor; and bears equivalent in (numerical) measure to 56 were seen (ultimately) to remain (where they were) in the forest. (Give out the (numerical) measure of (all) those (boars).

Thus ends the Amśamūla variety.

The rule relating to the Bhāgasamvarya variety (of miscellaneous problems on fractions):--

57. From the (simplified) denominator (of the specified compound fractional part of the unknown collective quantity), divided by its own (related) numerator, (also simplified), subtract four times the given known part (of the quantity), then multiply this (resulting difference) by that same (simplified) denominator (dealt with the above). The square root (of this product) is to be added to as well as subtracted from that (same) denominator (so dealt with); (then) the half (of either) of these (two quantities resulting as sum or difference is the unknown) collective quantity (required to be found out).

Examples in illustration thereof.

58. A cultivator obtained (first) ${\displaystyle {\tfrac {1}{8}}}$ of a heap of paddy as multiplied by ${\displaystyle {\tfrac {1}{10}}}$ (of that same heap); and (then) he had 24 vāhas (left in addition). Give out what the measure of the heap is.

59. One-sixteenth part of a collection of peacocks as multiplied by itself, (i.e., by the same ${\displaystyle {\tfrac {1}{16}}}$ a part of the collection), was found

56. The word śārdūlavikrīḍita in this stanza means 'tigers at play', and at the same time happens to be the name of the metre in which the stanza is composed

57. Algebraically stated ${\displaystyle x={\tfrac {{\tfrac {nq}{mp}}\pm {\sqrt {\left({\tfrac {nq}{mp}}-4a\right){\tfrac {nq}{mp}}}}}{2}}}$; and this value of x may easily be obtained from the equation ${\displaystyle x-{\tfrac {m}{n}}x\times {\tfrac {p}{q}}x-a=0}$, where ${\displaystyle {\tfrac {m}{n}}}$ and ${\displaystyle {\tfrac {p}{q}}}$ are the fractions contemplated in the rule.