8. How many cubick feet in a load of wood, & feet long, 3 feet wide, and 2 feet 8 inches high? Ans. 64 feet. 9. How many solid feet of timber in a stick which is 40 feet 8 inches long, 2 feet 6 inches wide, and 2 feet 4 inches thick? Ans. 237 feet 2 inches. QUESTIONS ON DUODECIMALS. What is the use of Duodecimals? A. It is a rule by which artificers and workmen find the contents of their work. How do you multiply? A. First by the units of the multiplier, and then by the parts of a unit. When length and breadth are given, how do you find the area? A. By multiplying the length by the breadth. When length, breadth, and thickness are given, how do you find the solidity? A. By multiplying the length, breadth, and thickness together. VULGAR FRACTIONS. Having briefly treated of Vulgar Fractions immediately after the Compound Rules, and given some general definitions, and a few such problems as were necessary to give a limited idea, only, of Fractions, the learner is, therefore, requested to read again those general definitions on page 119. Vulgar Fractions are either proper, improper, compound, or mixed. 3 5 &c. 1. A proper or simple fraction is when the numerator is less than the denominator, as 1, 3, 5, 2. An improper fraction, is when the numerator is equal to, or greater than the denominator, as 5,8,12, 8,12, &c. 3. A compound fraction, is a fraction of a fraction, connected by the word of, as of 3, 4 of 11, of §, &c. 4. A mixed number consists of a whole number and a frắction, as 33, 24, 87, &c. A whole number may be expressed like a fraction by drawing a line under it, and placing 1 directly below for a denominator, as 9, and 15-15, &c. CASE I-To reduce a fraction to its lowest terms. RULE.-Divide the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients again in the same manner; and so on, till it appears there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms. EXAMPLES. 1. Reduce to its lowest terms. Ans. 6) Ans. DEM.-Dividing the numerator and denominator both by the same number, never alters the value of a fraction; thus, is equal to, and is equal to, consequently, is equal to ; the operation only alters the terms of the fraction, not its value. the fraction 4.8 CASE II.--To reduce a mixed number to its equivalent improper fraction. RULE.-Multiply the integer, or whole number, by the denominator of the fraction, and to the product add the numerator; then set that sum above the denominator for the fraction required. 5 the numerator added. 125 new numerator. 8 denominator we have a new fraction, equal proof plainly shows. Proof. 15 Ans. 13 DEM.-It is plain, when we multiply the whole number by the denominator of the fraction and add the numerator, that the sum expresses eighths, and placing the denominator below, in value to the mixed number, as the 2. Reduce 123 to an improper fraction. 3. Reduce 54 to an improper fraction. Reduce 188 to an improper fraction. 5. Reduce 1991 to an improper or equivalent 6. Reduce 89% to an improper fraction. CASE II-To reduce an improper fraction to a whole or mixed number. RULE.-Divide the numerator by the denominator, and the tient will be the answer sought in a whole or mixed number. quo EXAMPLES. 1. Reduce 144 to a whole or mixed number. Ans. 12. 12)144 12 Ans. This being the 2. Reduce 3. Reduce DEM.-The numerator showing how many parts the fraction contains, and the denominator showing how many of those parts it requires to make a unit, the rule is obvious from the principles of Division, reverse of the preceding Case, they prove each other. to its equivalent number. 2 to its equivalent number. Ans. 33, Ans. 44 17 4. Reduce 1 to its equivalent number. 21 Ans. 183. 5. Reduce 222 to its equivalent whole or mixed number. 12 Ans. 199 CASE IV. To reduce a whole number to an equivalent fraction, having a given denominator. RULE.-Multiply the whole number by the given denominator; then set the product over the said denominator, and it will be the frac tion required. EXAMPLES. 1. Reduce 8 to a fraction whose denominator shall be 6. Here 8×6=48; then 4 is the Ans. for 48+6-8, Proof. DEM. Multiplication and Division, being equally used, the result must evidently be the fraction proposed; and that the fraction 4 is equal to 8, is also evident from the proof. 2. Reduce 14 to a fraction whose denominator shall be 12. Ans. 16. 3. Reduce 125 to a fraction whose denominator shall be 7. Ans. 875 4. Reduce 29 to a fraction whose denominator shall be 15. 15 Ans. 435 CASE V. To reduce a compound fraction to a simple or improper fraction. RULE.-Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, and they will form the fraction required. NOTE.When part of a compound fraction is a whole or mixed number, it must first be reduced to an equivalent or improper fraction by one of the preceding cases. EXAMPLES. 1. Reduce of to a simple fraction. 1x1=} the Ans. Ans. . DEM. The truth of this rule is evident, because of 1 is, then of must be of 1 When a compound fraction consists of more than two single ones, having reduced two of them as above, the resulting fraction and a third will be the same as a compound fraction of two parts; and so on, to the last of all. 4. Reduce of of 33 to a single fraction. Ans. 15%. Ans.. 5. Reduce of § of 3 to a single fraction. CASE VI. To reduce fractions of different denominators, to equivalent fractions having a common denominator. RULE.-Multiply each numerator by all the denominators except its own, for the new numerator; and multiply all the denominators together for a common denominator. NOTE.-When any part of the given sum consists of mixed numbers or compound fractions, they must be first reduced to improper or single fractions. EXAMPLES. 1. Reduce, and to equivalent fractions having a common denominator. 1x4x8=32 the new numerator for for 2 for 2×4×8-64 the common denominator. Therefore, the new equivalent fractions are 3, 4, and the answer sought. 동욱, DEM.-The reason of this rule is plain, because it is evidently multiplying each numerator and its denominator by the same numbers, and consequently the value of the fraction is not altered, for by reducing our new fractions to their lowest terms, we would again have our given fractions. 2. Reduce and to fractions of a common denominator. Ans. 3, 3. 3. Reduce, 23, and 4, to a common denominator. 35 120 Ans. 35, 38, 4. Reduce,, and to a common denominator. Ans. 735, 560 720, and 168. 8402 846,740, 840. 5. Reduce,, and to a common denominator. 100 126 70. Ans. 140, 140, 140. NOTE. It will often be found more convenient, to reduce fractions to their lowest terms, before reducing them to a common denominator. CASE VII. To find the value of a fraction in the infe riour denominations of the integer. RULE.-Multiply the numerator by the parts in the next inferiour denomination. and divide the product by the denominator. Then if any thing remains, multiply it by the parts in the next inferiour denomination, and divide by the denominator as before, and so on, as far as necessary; then the quotients, placed in order, will be the value of the fraction required. EXAMPLES. 1. Find the value of of a pound sterling. 7 20 40)140(3s. 6d. Ans. 120 20 12 240 240 Ans. 3s. 6d. DEM. The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, it is plain, that the operation here, is only continuing the division in the inferiour denominations of the integer. Ans. 3 roods 20 poles. of a dollar. 6. Find the value of Ans. 60 cents. Ans. 4fur. 22rds 4yds. 2ft. lin. 24b. c. CASE VIII. To reduce a fraction from one denomination to another, retaining the same value. RULE.-Consider how many of the less denomination make one of the greater; then multiply the numerator by that number, if the re duction be to an inferiour denomination, but multiply the denomina tor, if to a superiour denomination. EXAMPLES. 1. Reduce of a pound to the fraction of a penny. 12) -3-X20×12-129-90, the answer. Ans. o. DEM.-The reason of this rule is obvious, for it is the same as the rule of Reduction in whole numbers from one denomination to anotner. |