पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२६७

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CHAPTER IV --MISCELLANEOUS PROBLEMS (ON FRACTIONS ).

Dviragraśēşamūla and Amśmūla, and then Bhāgābhyāsa', then Amśavarga, Mūlamiśra and Bhinnadŗśya.

The rule relating to the Bhāga and the Śēșa varieties therein, (i.e., in miscellaneous problems on fractions).

4. In the operation relating to the Bhāga' variety, the (required) result is obtained by dividing the given quantity by one as diminished by the (known) fractions. In the operation relating to the śēșa variety, (the required result) is the given quantity divided by the product of (the quantities obtained respectively by) subtracting the (known) fractions from one.

Examples in the Bhāga variety.

5. Of a pillar, ${\tfrac {1}{8}}$ part was seen by me to be (buried) under the ground, ${\tfrac {1}{3}}$ in water. ${\tfrac {1}{4}}$ in moss, and 7 hastas (thereof was free) in the air. What is the length of the) pillar?

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In the Bhāgābhyāsa or Bhāgasamvarga variety, the numerical value is given of the portion remaining after removing from the whole the product or products of certain fractional parts of the whole taken two by two.

The Amśvarga variety consists of problems wherein the numerical value is given of the remainder after removing from the whole the square of a fractional part thereof, this fractional part being at the same time increased or decreased by a given number.

The Mūlamiśra variety consists of problems wherein is given the numerical value of the sum of the square root of the whole when added to the square root of the whole as increased or diminished by a given number of things.

In the Bhinnadŗśya variety, a fractional part of the whole as multiplied by another fractional part thereof is removed from it, and the remaining portion is expressed as a fraction of the whole . Here it will be seen that unlike in the other varieties the numerical value of the last remaining portion is not actually given, but is expressed as a fraction of the whole.

4. Algebraically, the rule relating to the Bhāga' variety is $x={\tfrac {1}{1-b}}$ where x is the unknown collective quantity to be found out, a is the dŗśya or agra, and b is the bhāga or the fractional part or the sum of the fractional parts given.

It is obvious that this is derivable from the equation $x-bx=a$.

The rule relating to the Śeşa variety, when algebraically expressed, comes to ${\tfrac {a}{(1-b_{1})(1-b_{2})(1-b_{3})\times \&c}}$ where b₁,b₂,b₃,&c are fractional parts of the successive remainders. This formula also is derivable from the equation $x-b_{1}x-b_{2}(x-b_{1}x)-b_{3}\{x-b_{1}x-b_{2}(x-b_{1}x)\}-\&c=a$