9. If the penny loaf weigh 67 oz, when the price of wheat is 5s the bushel; what ought it to weigh when the wheat is 8s 6d the bushel? Ans. 477 oz. 10. How much in length, of a piece of land that is 114 poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth ? Ans. 1345 poles. -11. If a courier perform a certain journey in 354 days, travelling 135 hours a dav; how long would he be in performing the same, 'travelling only 11 hours a day? Ans. 40% days. 12. A regiment of soldiers, consisting of 976 men, are to be new cloathed; each coat to contain 25 yards of cloth that is 1 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. I 24 TO0050 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, 45 is 4, and foto is 24, and to is '074, and is .00124 ; where ciphers are prefixed to make up as many places as are ciphers in the denominator, when there is a deficiency of 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 do, or 25. 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 = to, or '. But when they are placed on the left-hand, they decrease the value in a ten-fold proportion: Thus, '4 is to, or 4 tenths; but .04 is only rés, or 4 hundredths, and .004 is only totoo, or 4 thousandths. The 1st place of decimals, counted from the left-hand towards the right, is called the place of primes, or 10ths; the 2d is the place of seconds, or 100ths; the 3d is the place of thirds, or 1000ths; and so on. For, 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 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, like 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 3146.5 i 2109 •62417 14:16 5299.29877 the Sum. Ex. 2. What is the sum of 276, 39.213, 72014.9, 417, and 5032 ? 3. What is the sum of 7530, 16-201, 3.0142, 957.13, 6.72119 and 03014. 4. What is the sym of 312.09, 3.5711, 71956, 71.498, 9739.215, 179, and .0027? F 2 SUBTRACTION SUBTRACTION or 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. 3. To subtract 4.90142 from 214.81. Ans. 209.90858. 4. Find the diff. between 2714 and .916. 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 too and ; the product of which is 4332 = .04332, 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. EXAMPLES. 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. CONTRACTION II. To Contract the Operation, so as to retain only as many Decimals in the Product as may be thought Necessary, when the Product would naturally contain several more Places. Set the units place of the multiplier under that 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 figures may may fall in a column straight below each other, but observing to increase the first figure of every line with what would arise from the figures omitted, in this manner, namely ! from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure. EXAMPLES 1. To multiply 27:14986 by 92.41035, so as to retain only four places of decimals in the product. Contracted Way. Common Way. 27.14986 27.14986 53014.29 92:41035 24434874 542997 81 131574930 81 +4958 27141986 108599 44 54299712 24434874 2508.9280 2508-9280|650510 2. Multiply 480.14936 by 2.72416, retaining only four decimals in the product. 3. Multiply 2490•3048 by •573286, retaining only five decimals in the product. 4. Multiply 325•701428 by •7218393, retaining only three decimals in the product. DIVISION OF DECIMALS. Divide as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor*. * The reason of this. Rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend, is equal to those in the divisor and quotient, taken together, by the nature of Multiplication ; and consequently the quotient itself niust contain as many as the dividend exceeds the divisor. Another |