invert the terms of the divisor, and multiply the dividend by it, as in Multiplication. 12. It is required to divide of by 4 of 73. Ans. Tr RULE OF THREE IN VULGAR FRACTIONS. MAKE the necessary preparations as before directed; then multiply continually together, the second and third terms, and the first with its parts inverted as in Division, for the answer*. EXAMPLES. 1. If of a yard of velvet cost of a pound sterling; what will of a yard cost? 2. What will 3 == 6s 8d, Answer. oz. of silver cost, at 6s 4d an ounce ? Ans. 1 1s 44d. 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, VOL. I. 10 3. If of a ship be worth 2731 2s 6d; what are of her worth? Ans. 2271 12s 1d. 4. What is the purchase of 12301 bank-stock, at 108 per cent. ? Ans. 13361 1s 9d. 5. What is the interest of 2731 15s for a year, at 31 per cent. ? Ans. 87 17s 111d. 6. If of a ship be worth 731 1s 3d; what part of her is worth 2507 10s? Ans. . 7. What length must be cut off a board that is 73 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad? Ans. 18 inches. 8. What quantity of shalloon that is of a yard wide, will line 91 yards of cloth, that is 2 yards wide? Ans. 31 yds. 9. If the penny loaf weigh 6 oz. when the price of wheat is 5s the bushel; what ought it to weigh when the wheat is 8s 6d the bushel ? Ans. 4 oz. 10. How much in length, of a piece of land that is 11}} 'poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth? 11. If a courier perform a certain travelling 13 hours a day; how long forming the same, travelling only 11 Ans. 13 poles. journey in 35 days, would he be in perhours a day? 12. A regiment of soldiers, consisting be new clothed; each coat to contain 2 is 13 yard wide, and lined with shalloon many yards of shalloon will line them? Ans. 401 days. of 976 men, are to yards of cloth that yard wide: how Ans. 4531 yds 1 qr 29 nails. DECIMAL FRACTIONS. 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 lefthand. Thus, is 4, and is 24, and is .074, and I is 00124; where ciphers are prefixed to make up as many places as are ciphers in the denominator, when there is a deficiency in the figures. 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 3, or 135. 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 3. But when they are placed on the left-hand, they decrease the value in a ten-fold proportion: Thus, 4 is, or 4 tenths; but 04 is only T or 4 hundredths, and 004 is only, 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 tenfold proportion; as in the following Scale or Table of Notation. 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. To add together 29-0146, and 3146.5, and 2109, and -62417, and 14-16. 29.0146 3140-5 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. 17500-9718. 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. To find the difference between 91.73 and 2.138. 91.73 Ans. 89.592 the Difference. 2. Find the diff. between 1·9185 and 2·73. Ans. 0.8115. Ans. 209-90858. Ans. 2713.084. MULTIPLICATION OF DECIMALS. * PLACE the factors, and multiply them together the same as if they were whole numbers.-Then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers. *The rule will be evident from this example:-Let it be required to multiply 12 by 361; these numbers are equivalent to 1 and 301 the product of which is 4332 = ·04432, by the nature of Notation, which consists of as many places as there are ciphers, that is, of as many places as there are in both numbers. And in like manner for any other numbers. To multiply Decimals by 1 with any Number of Ciphers, as by 10, or 100, or 1000, &c. THIS is done by only removing the decimal point so many places farther to the right-hand, as there are ciphers in the multiplier; and subjoining ciphers if need be. EXAMPLES. 1. The product of 51.3 and 1000 is 51300. 2. The product of 2.714 and 100 is 8. The product of 916 and 1000 is CONTRACTION II. To contract the Operation so as to retain only as many Deci mals in the Product as may be thought necessary, when the Product would naturally contain several more Places. SET the unit's place of the multiplier under the figure of the multiplicand whose place is the same as is to be retained for the last in the product; and dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in.-Then, in multiplying, reject all the figures that are more to the right-hand than each multiplying figure, and set down the products, so that their right-hand |