ADDITION OF VULGAR FRACTIONS. If the fractions have a common denominator; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. * If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators, as before. As to the mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. * Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and therefore cannot be incorporated with one another any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals. Whence the reason of the Rule is manifest, both for Addition and Subtraction. When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on. Note.-2. Taking any two fractions whatever, and 33, for example, after reducing them to a common denominator, we judge whether they are equal or unequal, by observing whether the products 35 × 11, and 7 × 55, which constitute the new numerators, are equal or unequal. If, therefore, we have two equal products 35 × 11=7 × 55, we may compose from them two equal fractions, as 3571, or =55. If, then, we take two equal fractions, such as 7 and 35, we shall have 35 × 11=7 × 55; taking from each of these 7 × 11, there will remain (35 — 7) × 11 = (55—11) ×7, whence 35-7 we have =71, or 28=7. .55-11 In like manner, if the terms of were respectively added to those of, we should have 35X 7 55×11-88=71. α с way + a + b α с Or, generally, if. it in a similar may Hence, when two fractions are of equal value, the fraction formed by taking the sum (or the difference) of their numerators respectively, and of their denominators respectively, is a fraction equal in value to each of the original fractions. This proposition will be found useful in the doctrine of proportions. = 13, the Answer. 3. To add § and 71⁄2 and of +7+ of 2 = § + 5 + 1 = ; + % + ?= 9783. 4. To add and together. Ans. 14. PREPARE the fractions the same as for Addition, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of the fractions sought. 7. What is the difference between § of a pound, and 3 of of a shilling? Ans. 1s or 10s 7d 13q. 8. What is the difference between 3 of 51 of a pound, and 3 of a shilling? Ans. 30377 or 11 8s 11d. 2100 MULTIPLICATION OF VULGAR FRACTIONS. * REDUCE mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required. it * Multiplication of any thing by a fraction, implies the taking some part or parts of the thing; may therefore be truly expressed by a compound fraction; which is resolved by multiplying together the numerators and the denominators. Note. A Fraction is best multiplied by an integer, by dividing the denominator by it; but if it will not exactly divide, then multiply the numerator by it. 2. Required the continued product of 3, 34, 5, and of 3. Ans. 28. Ans. T 3. Required the product of 2 and §. 4. Required the product of and §. 5. Required the product of 3, 3, 13. 6. Required the product of, 3, and 3. 7. Required the product of 3, 3, and 4. 8. Required the product of, and 3 of 9. 9. Required the product of 6, and 3 of 5. 10. Required the product of 3 of 3, and § of 34. 11. Required the product of 32 and 4}}. DIVISION OF VULGAR FRACTIONS. * PREPARE the fractions as before in Multiplication: then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide: but if not, invert the terms of the divisor, and multiply the dividend by it, as in Multiplication. MAKE the necessary preparations as before directed (p. 34, 35.); then multiply continually together the second and third terms, and the first with its parts inverted as in Division, for the answer †. * Division being the reverse of Multiplication, the reason of the rule is evident. Note. A Fraction is best divided by an integer, by dividing the numerator by it; but if it will not exactly divide, then multiply the denominator by it. This is only multiplying the 2d and 3d terms together, and dividing the product by the first, as in the Rule of Three in whole numbers. EXAMPLES. 1. If of a yard of velvet cost of a pound sterling; what will of a yard cost? 3. If of a ship be worth 2737 2s 6d; what are 1⁄2 of her worth? Ans. 227 12s 1d. 4. What is the purchase of 12301 bank-stock, at 108§ per cent.? Ans. 13367 1s 9d. 5. What is the interest of 2737 15s for a year, at 34 per cent. ? Ans. Sl 17s 114d. 6. If of a ship be worth 737 is 3d; what part of her is worth 2507 10s? Ans.. 7. What length must be cut off a board that is 7 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad? 8. What quantity of shalloon that is of a yard wide, will cloth, that is 2 yards wide? 9. If the penny loaf weigh 6 oz. when the price of wheat what ought it to weigh when the wheat is 8s 6d the bushel? 10. How much in length, of a piece of land that is 11 make an acre of land, or as much as 40 poles in length and 4 Ans. 18 inches, line 9 yards of Ans. 313 yds. 5s the bushel; Ans. 4 oz. is poles broad, will in breadth? Ans. 13 poles. 11. If a courier perform a certain journey in 35 days, travelling 133 hours a day; how long would he be in performing the same, travelling only 11 hours a day? Ans. 40915 days. 12. A regiment of soldiers, consisting of 976 men, are to be new clothed ; each coat to contain 2 yards of cloth that is 1ğ yard wide, and lined with shalloon yard wide; how many yards of shalloon will line them? Ans. 4531 yds 1 qr 29 nails. DECIMAL FRACTIONS. 100000 A DECIMAL FRACTION is that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus, is 4, and ✯ is ·24, and is 074, and 124 is 00124; where ciphers are prefixed to make up as many places as there are ciphers in the denominator, when there is a deficiency in the figures. Thus, the understood denominator of a decimal is always either ten, or some power of ten; whence its name. A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point. Thus, 3.25 is the same as 32, or 325. Ciphers on the right-hand of decimals make no alteration in their value; for •4, or ‘40, or '400 are decimals having all the same value, each being =†, or }. But when they are placed on the left-hand, they decrease the value in a tenfold proportion: Thus, 4 is, or 4 tenths; but 04 is only 1, or 4 hundredths, and 004 is only 100, or 4 thousandths. In decimals, as well as in whole numbers, the values of the places increase towards the left-hand, and decrease towards the right, both in the same ten-fold proportion; as in the following Scale or Table of Notation. ADDITION OF DECIMALS. SET the numbers under each other according to the value of their places, as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right-hand, add up all the columns of numbers as in integers and point off as many places for decimals, as are in the greatest number of decimal places in any of the lines that are added; or, place the point directly below all the other points. EXAMPLES. 1. Add together 29.0146, and 3146·5, and 2109, and '62417, and 14:16. 29.0146 3146.5 2109' •62417 14.16 5299-29877 the Sum. 2. What is the sum of 276, 39.213, 72014′9, 417, and 5032? Ans. 77779 113. 3. What is the sum of 7530, 16·201, 3·0142, 957·13, 6·72119 and 03014? Ans. 8513 09653. 4. What is the sum of 312.09, 3.5711, 7195.6, 71-498, 9739 215, 179, and ⚫0027 ? Ans. 17500 9768. SUBTRACTION OF DECIMALS. PLACE the numbers under each other according to the value of their places, as in the last Rule. Then, beginning at the right-hand, subtract as in whole numbers, and point off the decimals as in Addition. EXAMPLES. 1. Find the difference between 91.73 and 2.138. 91.73 Ans.89 592 the Difference. |