CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 71 Dviragrasēsamula and Amamula, and then Bhagabhyasa, then Amsar arga, Mulamusra and Bhinnad, sya. The rule relating to the Bhaga and the Sesa varieties therein, (2.e., in miscellaneous problems on fractions). 4. In the operation relating to the Bhaga 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 sesa variety, (the required result) is the given quantity divided by the product of (the quantities obtained respectively by) sub- tracting the (known) fractions from one. Examples in the Bhaga variety. 5 Of a pillar, part was seen by me to be (buried) under the ground, in water, in moss, and 7 hastas (thereof was free) in length of the) pillar? the air. What is the In the Bhagabhyasa or Dhaqasamvarya variety, the numerical value is given of the portion iemaining after 1emoving from the whole the product or products of certain fi actional parts of the whole taken two by two. The Amsavarga 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 fiactional part being at the same time increased or decreased by a given number The Mulamiśra vanety consists of problems wherein is given the numerical value of the sum of the square 10ot of the whole when added to the square 100t of the whole as incicased or diminished by a given number of things. In the Bhinnadrsua vailety, a fractional part of the whole as multiplied by another fractional part thereof is removed from it, and the remaining poition is expressed as a fiaction 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. a 4. Algebraically, the rule relating to the Bhaya variety is , where a is the unknown collective quantity to be found out, a is the drsya or agra, and b is the bhaga or the fractional part or the sum of the fractional parts given. It is obvious that this is derivable from the equation - ba a The rule relating to tho Se variety, when algebraically expressed, comes to where bi, ba, ba, &c., are fractional parts of the (1-₁) (1-b₂) (1-ba) x&c. successive remainders. This formula also is deivable from the equation -b₁-4a (-₁)-b³ { x-b₁x-b₁ (x-b₁a) } -&c.=a.
