10. If 32 men build a wall 36 feet long, 8 feet high, and 4 feet wide in 4 days; in what time will 48 men build a wall 864 feet long, 6 feet high, and 3 feet wide ? Ans. 36 days. OF VULGAR FRACTIONS. (Before proceeding farther let the pupil study carefully from g 56 to the end of g 59.) $ 111. There are five kinds of Vulgar Fractions, Proper, Improper, Simple, Compound, and Mixed. A proper fraction is one in which the numerator is less than the denominator; as }, }, }, &c. The value of every proper fraction is less than 1, $ 59. An improper fraction is one in which the numerator is equal to, or exceeds the denominator; as ļ, $, }, &c. Such fractions are called improper fractions because they are equal to or exceed unity. When the numerator is equal to the denominator the value of the fraction is 1: in every other case the value of an improper fraction is greater than 1. A simple fraction is a single expression; as }, }, }, }, &c. A simple fraction may be either proper or improper. A compound fraction is a fraction of a fraction, or several fractions connected together with the word of between them; as of }, off of 3, &c. A mixed number is made up of a whole number and a fraction, as 31, 121, 161, &c. The whole numbers are sometimes called integers. $ 112. The numerator and denominator of a fraction, taken together, are called the terms of the fraction. Hence, every fraction has two terms. 9 113. A whole number may be expressed fractionally, without altering its value, by writing 1 below it for a denominator: thus, 3 may be written t, 5 may be written , 6 may be written f, &c.; when so written they are read 3 ones, 5 ones, 6 ones, &c. § 114. Since the denominator of a fraction shows into how many equal parts the unit has been divi. ded, and the numerator how many of those equal parts are taken in the fraction $ 57, the fraction of will express that the unit has been divided into ģ equal parts and that 5 of those parts are taken. Each part of the unit will be expressed by 1 and 5 of those parts by * *5: hence =* *5. In the same way m=11x7, that is 1 is equal to da repeated 7 times. Again, ix3=1*73=4 X21=11. Thus in multiplying the numerator of a fraction by any num. ber we increase the number of parts expressed by the fraction as many times as there are units in the mul. tiplier. We may therefore write PROPOSITION I. If the numerator of a fraction be multiplied by any number, the denominator remain. ing unchanged, the value of the fraction will be increased as many times as there are units in the multiplier. $ 115. If we take any fraction, as y, and divide the numerator by 2, it will change the fraction to : if we divide it by 3, we shall have , and if we divide it by 4, we shall have š. Now in each of the fractions , $ and the number of equal parts ex• pressed, is as many times less than 12, as there were units in the divisor; and as the same may be shown of all fractions, we write, PROPOSITION II. If the numerator of a fraction be diri any number, the denominator remaining he value of the fraction will be dimin. times as there are units in the divisor. § 116. If we take any fraction, as , and multiply the denominator by 4, it will change the fraction to po, Now in this last fraction the unit or single thing is divided into 20 equal parts, while in the first it is divided into 5 parts. We consider that the 20 equal parts in the 2d case, have been obtained by dividing each of the parts of the first fraction into 4 equal parts: therefore, 1 part in the first fraction, is equal to 4 parts in the second. But the number of parts expressed in each fraction is the same ; hence the second fraction is & of the first : and as the same may be shown of any fraction, we write, PROPOSITION III. If the denominator of a fraction be multiplied by any number, the numerator remaining unchanged, the value of the fraction will be di. minished as many times as there are units in the multiplier. $ 117. If we take any fraction as ms, and divide the numerafor by 6, it will change the fraction to f. Now in the second fraction the unit or single thing is divided into 2 parts, and if we divide each of these parts into 6 parts, we shall obtain the 12 parts into which the unit is divided in the first fraction : therefore, 1 part in the second fraction is equal to 6 parts in the first. But the number of parts expressed in each fraction is the same; hence, the second fraction is 6 times greater than the first, and as the same may be shown for any fraction, we write, PROPOSITION IV. If the denominator of a fraction be divided by any number, the numerator remaining unchanged, the value of the fraction will be increased as many times as there are units in the divisor. Ø 118. It appears from Prop. I. that if the nume. rator of a fraction be multiplied by any number tha value of the fraction will be increased as many *: as there are units in the multiplier. It also appears from Prop. III. that if the denominator of a fraction be multiplied by any number, the value of the fraction will be diminished as many times as there are units in the multiplier. Therefore, when the numerator and denominator of a fraction are multiplied by the same number, the increase from multiplying the numerator, will be just equal to the decrease from multiplying the denominator ; hence we have, PROPOSITION V. If the numerator and denominator of a fraction be multiplied by the same number, the value of the fraction will remain unchanged. Thus, š=***=ly and =*=. § 119. It appears from Prop. II, that if the numerator of a fraction be divided by any number, the value of the fraction will be diminished as many times as there are units in the divisor. It also appears from Prop. IV, that if the denominator of a fraction be divided by any number, the value of the fraction will be increased as many times as there are units in the divisor. Therefore, when the numerator and denominator of a fraction are divided by the same number, the decrease from dividing the numerator will be just equal to the increase from dividing the denominator: hence we have, PROPOSITION VI. If the numerator and denominator of a fraction be divided by the same number, the value of the fraction will remain unchanged. Thus, 3=by dividing by 4: and 1=1 by dividing by 2. Ø 120. Any number greater than unity that will divide two or more numbers without a remainder is called their common divisor : and the greatest nuiber that will so divide them, is called their greatest common divisor. Take, for example, the two numbers 142 and 994. The greatest common divisor cannot be greater than the least number 142. This number will divide itself :-let us see if it will also divide 994. The number 142 exactly divides 142)142(1 itself, giving a quotient of 1; it also 142 divides 994 giving a quotient of 7. Therefore, 142 is the greatest com 142)99407 mon divisor. 994 The numbers 2 and 71 are common divisors of the two numbers since either of them will divide both of the numbers without a remainder. Two numbers may have several common divisors, but they have only one greatest common divisor. Ex. 2. Take the two numbers 72 and 90. 72) 90(1 least number 72, is the 72 greatest common divisor. great. c. div. 18)72(4 After dividing we find a 72 remainder of 18. Now if 18 will divide 72, it will also divide 90, for 90=72+18, and 18 will be contained once more in 90=72+ 18 than in 72: but 18 divides 72 without a remainder : therefore, 18 is the common divi. sor: hence we see that the common divisor of two numbers must also be a common divisor between the least number and their remainder after division. But 18 is the greatest common divisor; for, the greatest common divisor must be contained at least once more in 90 than in 72: hence, the greatest common divisor cannot be greater than the difference between the two numbers, which in this case is 18. Therefore, we have $ 121. PROPOSITION VII. The greatest common di. visor of two numbers is obtained by dividing the greater by the less ; then dividing the divisor by the |