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CHAPTER VI- MIXED PROBLEMS.
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are to be divided (ouo after another) by an optionally chosen number (and the remainders again are to be divided by an optionally chosen number, this process being repeated) over and over again. The given (mixed) quantities of the different things are to be (suocessively) diminished by the corresponding quotients in the aboven procoss. (In this manner the numerical values of tho various things in the mixed sums are arrived at). The optionally chosen divisors in the above processes of continued division) combined with the optionally chosen multiplier is also that multiplier constitute (respectively) the prices of a singlo thing in oach of the varieties of the given different things).
Choose any optional number, way y, and multiply with it there total timbers: we get 42, 44, 46. Subtract those from 72, the price of the rewportive brain. The remainders are 31, 29, and 27. These are to be divided by another optionally ohusen number, way 8. The quotients aro 3,3,, and the remainders are 7, 6 and 3. These renuinders are ngain divided by third optionally chonon number way 2. The quotients are 3, 2, 1, and the remaindern aro 1, 1, 1. Then laat remainders are in their turn divided by fourth optionally chowa nuinber which is 1 hore. The quotiontie 1, 1, 1 with no remaindern. The quotionta derived in relation to the first total number ure to wubtracted from it. Thus we ket 21-(3 + 3 + 1) = 11; thin number and the quotinta 3, 3, 1 represent the number of fruits of the different morts in the tirnt heap. Similarly wu ket in the sound group 16, 3, 2, 1, and in the third group 1, 8, 1, 1, as the number of the different kotin of fruita.
The prices are the firmt chosen multiplier, viz., 2, and its hin with the other optionally chonon multipliers. Thus we get 2, ? + Mor 10,2 + 2 or 4, and 2 + 1 ur 3, 140 the prio of who the four different kinds of fruit in orler.
The principlo underlying this rethod will be clear from the following algebraical representation : -
az + by + ca + du = p ... ... .. . ... ... 1
Let w =w. Stultiplying II by, we have . (a + b + +
d in Subtracting 111 from I, wu get a (+-) + b (y - ) + c(-)
... ... IV Dividing IV by I-, W got in the quotient, and (y - ) + c(-) AR the remainder, where -Dinn muitable integer.
Similarly we proceed till the end.
Thus it will bo acen that the KUCCA Dively choron divinor , y , and --, when combined with X, give the valon of the various prices, by itself being the price of the first thing; and in the successive quotinen a, b, c, along with - (a + b + c) are the nurobers measuring the various kinds of think.
It may be noted that, in this role, the number of divisions to be carried out in ono less than the number of the kinds of things given, and that there should be no remainder left in the last division.
