Algebra for Schools

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 Excerpt: ...they? 4. The sum of two numbers is 100, and their ratio is the same as the ratio of one of them to 100. Find the value of the numbers within one-tenth. 5. One number is 12 greater than the square of another, and their ratio is 7. Find the numbers. THE LAWS OF FRACTIONS. 212. Theorems which are of use in this chapter and the succeeding one are proved by the use of the three fundamental laws of Algebra. 213. Just as the sign--means the reverse of addition, addition undone, so the sign-f-means the reverse of multiplication, multiplication undone. The sign--may be read the negative of, and the sign-4-may be read the reciprocal of. 214. The law of association for + and--signs implies such identities as these: a--b--c--d--e=(a+J)--(c--d--e)=a--(b--c)--(d--e) See 86. In the aame way the law of association for-f-and X implies: aXbs-cXd-r-e=(aXb)-i-(c-T-dXe) = aXb-i-c)x(d-i-e) In fact, THE ASSOCIATIVE AND DISTRIBUTIVE LAWS ARE, SO FAR AS FORM GOES, THE SAME FOR X AND-=AS THEY ARE FOR + AND--. 215. Theorem I. Any fraction may have its numerator and denominator multiplied or divided by the same number without altering its value. Let j-be any fraction and let m be any number. Then am. /i =--= am-r-om = a X m-f-(o X m)-bm = a X m-i-b-i-m the distributive law = a-$-b X m-T-m the commutative law b am _ a bm b 216. Theorem II. Fractions are multiplied by multiplying their numerators together for a new numerator and their denominators for a new denominator. a c Let v-and-=-be anv two fractions. b d J.1-X-T = a-i-bXc-r-d = aXc-b-i-d b d = aXc+bXd) =-j a c _ ac ' b X d=bd 217. Theorem III. The rule for dividing by a fraction is: Invert the divisor and then multiply. Let x be any number and--any fraction. x-hj-= x- -(as-b) = x-i-aXb =...