320 MATHEMATICS. 



and writing this sum over the same denominator ; forming what is 

 called an improper fraction. 



Decimal Fractions, are those in which the denominator is always 

 one-tenth, one-hundredth, one-thousandth, or other decimal part of 

 a unit ; so that by simply writing the numerator, with a point on 

 the left side of it, called the decimal point, the denominator need not 

 be written at all. Thus, T % is written, .5 ; and Jfa is the same as 

 .54 ; the denominator always consisting of the figure 1, with as many 

 cyphers on its right as there are decimal places, that is, figures on 

 the right of the decimal point. To convert a vulgar fraction into a 

 decimal, we annex cyphers to the right of the numerator, and then 

 divide it by the denominator ; observing that the quotient, or decimal 

 sought, must have as many figures as we annexed cyphers ; and sup- 

 plying any deficiency in the quotient by cyphers on its left. Deci- 

 mals are added, and subtracted, in the same manner as whole num- 

 bers ; placing them with the decimal points always one under the 

 other, and beginning on the right ; since decimals and whole num- 

 bers, together, form one continued series in tenfold proportion. De- 

 cimals are also multiplied, and divided, in the same way as whole 

 numbers ; only, the product must contain as many decimal places as 

 there are in both the factors ; and the quotient, as many as there are 

 in tne dividend more than in the divisor. The deficiency, if there 

 be any, must, in either case, be supplied by cyphers on the left. 



4. Proportion, signifies a certain definite relation of several 

 quantities. Four numbers are said to be in Arithmetical proportion, 

 when the first is as much greater or less than the second, as the third 

 is greater or less than the fourth : as in the example, 2 4 : : 18 20. 

 But four numbers are in Geometrical proportion, when the first is 

 as many times greater or less than the second, as the third is greater 

 or less than the fourth : as in the example, 20 : 4 : : 500 : 100. The 

 first term divided by the second is called the ratio of the antecedents, 

 and the third term divided by the fourth, is the ratio of the conse- 

 quents: and these two ratios are equal. It also follows that the pro- 

 duct of the two middle, or mean terms, is equal to the product of 

 the two extremes : and the product of the means, divided by one ex- 

 treme, gives the other extreme for a quotient. 



The Rule of Three, is the process in which we have three num- 

 bers given, and seek to find a fourth, which shall complete the 

 geometrical proportion. Of the three given numbers, one will ne- 

 cessarily be of the same kind as the fourth, or answer sought ; and 

 this may occupy either the second or third place. If the answer 

 sought, ought to be greater than this, then the greater of the other 

 two terms should be placed last of these two ; both being reduced to 

 the same denomination : but if the answer ought to be less than that 

 term which is of the same kind with it, then the lesser of the other 

 two terms should be placed last of those two, and the greater of them 

 will be the first. The question being thus stated, multiply the second 

 and third terms together, and divide their product by the first ; and 

 the quotient will be the fourth term, or answer sought. A Compound 

 Proportion, including the solution of problems by what is called the 

 Double Rule of Three, is merely a connection of two or more simple 



