2. To find the sum of 376-25 +86-125637-4725 + 6·5+ 41·02 + 358.865. Ans. 1506-2325. 3. Required the sum of 3-5+47-25 + 2·0073 + 92701 + 1·5. Ans. 981-2673. 4. Required the sum of 276 + 54321 + 112 + 0·65 + 12·5 + 0463. Ans. 455-5173. SUBTRACTION OF DECIMALS. RULE-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 difference between 1.9185 and 2.73. Ans. 0-8115. Ans. 209-90858. Ans. 2713-084. MULTIPLICATION OF DECIMALS. RULE.*-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 and the product of which is 13888504332 by the nature of Notation, which consists of as many places as there are ciphers, that is, as many places as there are in both numbers. Aud in like manner for any other number 2. Multiply 79-347 by 23.15. Ans. 1836 88305 CONTRACTION L Ans. 520773512. Ans. 00178986144. To multiply decimals by 1 with any number of ciphers, as 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. 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 than each multiplying figure; and set down the products, so that their right hand figures 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 1 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. 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. RULE.-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.* When the places of the quotient are not so many as the rule requires, let the defect be supplied by prefixing ciphers. When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required. WHEN the divisor is an integer, with any number of ciphers annexed; cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before.† 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 must contain as many as the dividend exceeds the divisor. + This is no more than dividing both divisor and dividend by the same number, either IC, or 100, or 1000, &c., according to the number of ciphers cut off; which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same number of places in both the divisor and dividend, either to the right or left, whether they have ciphers or not. CONTRACTION II. HENCE, if the divisor be 1 with ciphers, as 10, or 100, or 1000, &c.; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left as the divisor has ciphers; prefixing ciphers if need be. WHEN there are many figures in the divisor; or only a certain number of decimals are necessary to be retained in the quotient, then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder be a new dividend; and for every such dividend, leave cut one figure more on the right hand side of the divisor; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication. Note. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction. EXAMPLES. 1. Divide 2508-92806 by 92-41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures. 2. Divide 4109-2351 by 230-109, so that the quotient may contain only four decimals. 3. Divide 37-10438 by 5713-96, that the quotient may contain only five decimals. 4. Divide 913-08 by 2137-2, that the quotient may contain only three decimals. REDUCTION OF DECIMALS. CASE I To reduce a vulgar fraction to its equivalent decimal. RULE. Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required. To find the value of a decimal in terms of the inferior denominations. RULE.-Multiply the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder, to the right hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations separated on the left hand, will make up the answer. Note. This operation is the same as Reduction Descending in whole numbers. EXAMPLES. 1. Required to find the value of 775 pounds sterling. To reduce integers or decimals to equivalent decimals of higher denominations. RULE. Divide by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers. |