12. A company of 15 men bought 4509a. 1r. 20po. of land, and paid an equal proportion; how many acres was each man's share? Ans. 300a. 2r. 20po. 13. A merchant bought 8 pieces of cloth, containing 498yd. 2qr.; how many yards were there in each piece? Ans. 62yd. 1gr. Ina, 14. A paper maker had 969ba. 4bun. 1r. of paper in 4 boxes; how many bales were there in each box? Ans. 242ba. 2bun. Or. 15q. EXPLANATIONS. You have now, I trust, become fully acquainted with the working of figures in the fundamental rules of Arithmetick, both Simple and Compound, that is, of WHOLE NUMBERS, Or INTEGERS. I shall now treat of PARTS of these whole numbers, or integers, or, as they are generally called, FRACTIONS. FRACTION is, therefore, used to describe a part, merely, of any thing that may be the subject of consideration. Thus, if we speak of certain weights, as, three ounces and a quarter, (that is, a quarter of an ounce,) this quarter being but a part, is called a fraction; and if we speak of seven pounds and a quarter, then this "quarter" is also a fraction: only observe, that, meaning, as it would, a quarter of a pound, so it would be called a fraction of a pound, while the other means a fraction of an ounce. Thus, PARTS of any thing, whether of weights, of measures, of money, or of periods of time; parts of every size, as, halves, quarters, thirds, one fourth, three fourths, four fifths, seven eighths; or, in short, any other conceivable quantity, either small or large, as, a thousandth part, or, as nine hundred and ninetynine such parts, or any portion short of whole, is a fraction; and the treatment, or the working of these parts of numbers, is called the working of fractions; while, in order to distinguish them from these FRACTIONS, the numbers of which we have heretofore treated, are called INTEGERS, or WHOLE NUMBERS. FRACTIONS. Q. What are FRACTIONS? A. Fractions, or broken numbers, are the parts of a whole number, or integer, as, parts of a pound, yard, mile, &c. Q. How many kinds of fractions are there? VULGAR FRACTIONS. Q. What is a Vulgar Fraction? A. A Vulgar Fraction is a part of a unit, or integer, expressed, or represented, by two numbers, one placed directly above the other, with a line between them; thus, a signifies three fourths of one, and signifies two thirds of one, &c. Q. What is the number above the line called? Q. Why is it called the numerator? A. Because it shows the number of parts the fraction contains. Q. What is the number below the line called? Q. Why is it called the denominator? A. Because it shows the quantity of these parts, or it shows into how many parts a unit, or whole number, is divided Vulgar Fractions are either proper, improper, compound, or mixed. Q. What is a proper or simple Vulgar Fraction? 2 A. A proper, or simple fraction, is when the numerator is less than the denominator, as, 2, 3, 4, &c. Q. What is an improper fraction? A. An improper fraction is when the numerator exceeds the denominator, as, 4, 4, 12, &c. Q. What is a compound fraction? A. A compound fraction is a fraction of a fraction, connected or coupled by the word of, as, of 12, of 4 of 3, &c. Q. What is a mixed number? A. A mixed number is composed of a whole number and a fraction, as, 44, 16, &c. A whole number may be expressed like a fraction, by drawing a line under it, and placing 1 below for a denominator, as, 7—7, and 14—4, &c. RULE. Q. How must you reduce, or abbreviate fractions to their lowest terms? A. The numerator and denominator of the given fraction must be divided by any number that will divide them without a remainder, and the quotient again in the same manner, and so on, till it that there is no number greater than 1 that will divide them, and the last quotient will express the given fraction in its lowest or least term. EXAMPLES For Theoretical Exercise on a Slate. appears 1. Reduce 24 to its lowest terms. Ans. 1. EXPLANATIONS You must first set down the fraction, and divide the numerator by 6, and say, 6 in 24 four times; set down the 4 for the 4) 6)24-4=1 Ans. or thus, 12)24=2=1 Ans. 4 numerator of a new fraction: you must then divide the denominator by 6, and say, 6 in 48 eight times; set down the 8, for the denominator, under the 4, the new numerator. You must then divide the new fraction by 4, which will give, the lowest fraction, which is the same in value of 24. You will readily perceive, that you do not alter the value of the fraction by this operation, for the numerator of the quotient bears the same proportion to the denominator of the quotient in each place, that the numerator of the dividend bears to the denominator of the dividend, as 24 is the half of 48, 4 is the half of 8, and 1 is the half of 2. You must also bear in mind, that it does not make any difference what number you take for a divisor, if it will divide the terms of the fraction without a remainder. You will readily perceive, that dividing the numerator and denominator both by the same number, does not, in any case, alter the value of a fraction. Thus, 24 is equal to 4, and 4 is equal to, and, therefore, 8 is equal to 24. Thus you see, that the operation only alters the terms of the fraction, and not its value. 2. Reduce 216 to its lowest terms. 288 80 Ans. 3. Ans. 1. Ans.. Ans. 1. Ans. Ans. §. Ans. 1. Ans. 18. 20 10. Abbreviate 784 as much as possible. Ans. 14. 952 RULE. Q. How do you find the value or quantity of a fraction in the known parts of the integer, that is, in the inferiour denomination of the integer as to coin, weight, measure, &c.? A. The numerator must be multiplied by the common parts of the integer, and the product must be divided by the denominator; and if there be a remainder, it must be reduced to the next inferiour denomination, and the product divided again, as before, till it is reduced to the lowest denomination, or till there is no more remainder. EXAMPLES For Theoretical Exercise on a State. 1. What is the value of of a pound sterling? Ans. 8s. EXPLANATIONS. £ બર 8 s. In this example, you must multiply the 2, the numerator, by 20, because twenty shillings make a pound, and, also, because shilling is the next inferiour denomination. below pound, in sterling money. When there is a remainder of shillings, you must 5)40 multiply it by 12, because twelve pence make one shilling, and pence are the next inferiour denomination; and if there be a remainder of pence, multiply it by 4, because four farthings make one penny. The denominator, 5, shows that a pound is divided into five parts; and the numerator, 2, shows how many of those parts the fraction contains. You multiply the 2 by 20, because it is two parts of twenty, of which you wish to find the amount, and 5 is the amount of each of the 2 parts. If you wish to obtain the amount of only one part, you would divide by 5, without multiplying the 2 by 20, so you must increase the 20 as many times as the numerator expresses. Again, it is very obvious, that if £1, or 20s., is divided into 5 parts, that one part must be 4 shillings, because four shillings is the fifth part of £1, or 20; and the numerator, in this example, expresses two parts, consequently, the value of the fraction is 8 shillings, as two times 4 shillings make 8 shillings. 2. What is the value of a of a day? Ans. 16h. |