denominator of each fraction by the denominators of all the rest, divided respectively by their greatest common measure; and the fractions will be reduced to a common denominator in lower terms* than they would have been by proceeding according to the former rule. α b

Thus

mx my

с

reduced to a common de

,

mz

*(6). To add together simple Algebraical Fractions. 31. If the fractions to be added have a common denominator their sum is found by adding the numerators together and retaining the common denominator. Thus,

32. If the fractions have not a common denominator, they must be transformed to others of the same value which have a common denominator, (by Art. 29), and then the addition may take place as before.

Thus,

*To obtain them in the lowest terms, each must be reduced to another of equal value, with the denominator which is the least common multiple of all the denominators.

*(7.) To multiply simple Algebraical Fractions.

33. To multiply a fraction by any quantity, multiply the numerator by that quantity and retain the denominator.

divided be e times as great as before, and the divisor the same, the quotient must be c times as great.

nu

34. The product of two fractions is found by multiplying the numerators together for a new merator, and the denominators for a new denominator.

α

a C

Let and be the two fractions: then X =

For if

b

b

d

b

α

d

ac

bd

y, by multiplying the equal

quantities and by b, a = ba (Art. 28), in the

same manner C = dy; therefore, by the same axiom, ac = bday; dividing these equal quantities, ac and

bday by bd, we have

* (8).

35.

To divide simple Algebraical Fractions.

To divide a fraction by any quantity, multiply the denominator by that quantity, and retain the numerator.

it was before, and

be divided being a cth part of what the divisor the same.

36. To divide a quantity by any fraction, multiply the quantity by the reciprocal of the fraction. (Art. 8.)

If we divide c by we obtain

a

α

bc

For if

b

a

and x =

b'

b

a

b

Algebraical definition of Proportion.

37. Four quantities are said to be proportionals, when the first is the same multiple, part, or parts of the second, that the third is of the fourth.

Thus the four quantities 8, 12, 6, 9, are proportionals; for 8 is of 12, and 6 is of 9.

In this case; and generally a, b, c, d are proportionals if

saying a is to b, as c to d; and thus represented, a b c : d.

The terms a and d are called the extremes, and b and c the means.

α

The fraction is called the ratio of a to b.

b

*(10). Algebraical consequences of Proportion.

if a be equal to b, c is equal 2 b

d

to d, and if a be less than b, c is less than d, and if

a be greater than b, c is greater than d.

12

39. When four quantities are proportionals, the product of the extremes is equal to the product of the

means.

Let a, b, c, d be the four quantities; then, since

both sides by bd, ad = bc.

; and by multiplying

::

Any three terms in a proportion a b c d being given, the fourth may be determined from the equation ad bc.

=

40. If the first be to the second as the second to the third, the product of the extremes is equal to the square of the mean.

For (Art. 39) if a x :: x : b, ab = x2.

41. If the product of two quantities be equal to the product of two others, the four are proportionals, making the terms of one product the means, and the terms of the other the extremes.

Let xy = ab, then dividing by ay,

or, x: a :: by.

42. If a b :: cd, and c d e f, then will abe : f.

a

Because

b

=

C

and d

d f

therefore

=

; or

b

f

43.

a b ẹ f.

If four quantities be proportionals, they are also proportionals when taken inversely.

If a b :: c: d, then b: a :: d : C. For

quantities, or taking their reciprocals,

that is, bad: c.

b

a

44. If four quantities be proportionals they are proportionals when taken alternately.

If a b c d, then a c :: b: d.

Because the quantities are proportionals,

a C

b

=

or a C :: b : d.

d

45. Unless the four quantities are of the same kind, the alternation cannot take place, because this operation supposes the first to be some multiple, part or parts, of the third.

One line may have to another line the same ratio that one weight has to another weight, but a line has no relation in respect of magnitude to a weight. In cases of this kind, if the four quantities be represented by numbers or other quantities which are similar, the alternation may take place, and the conclusions drawn. from it will be just.

46. If a

a

b c d, then componendo,

a + b b c + d : d.

с

For = ; therefore +1= + 1;
b b

a

с

b

d