Note In the last answer, we have five places for decimals; put, as the 5 in the fifth place is only Too of a unit, it will be found sufficiently exact for most practical purposes, to extend the decimals to only three or four places. To know whether you have obtained an equal decunal, change the decimal into a common fraction by placing its proper denominator under it, and reduce the fraction to its lowest terms. If it produces the same common fraction again it is right; thus, taking the two foregoing oxamples,,4-3. Again,,09375-183586=33. From these illustrations we derive the following RULE. 1. How do you proceed to reduce a common fraction to its equal decimal? A. Annex ciphers to the numerator, and divide by the denominator. 1. How long do you continue to annex ciphers and divide? 4 Till there is no remainder, or until a decimal is obtained sufliciently exact for the purpose required. I. Hoo many figures of the quotient will be decimals? 4. As many as there are ciphers annexed. IV. Suppose that there are not figures enough in the quotient for this purpose, what is to be done? supply the deficiency. A. Prefix ciphers to More Exercises for the Slate. 3. Change 4, 4, 4, and to equal decimals. A.‚5,‚75, ,25,,04. 4. What decimal is equal to ? (5) What? (5) What (75) What? (4) Ans. 1,34. 5. What decimal is equal to Tho' (5) What? (25) What? (3) What? (175) What? (625) A.1,6 €. What decimal is equal to ? (1111) What? (444) What? (10101) What? (3333)* A.,898901. + *When decimal fractions continne to repeat the same figure, like 333, &c, in this example, they are called Repotonds, or Circulating Decimals. When only one figure repeats, it is called a single repetend; but, if two or more figures repeat, it is called a compound repetend: thus, ,333 &c. is a single repeten, 010101, &c. a compound repetend. When other decimals come before circulating decimals, as„in,8333, the decimal is called a mixed repetend. It is the common practice, instead of writing the repeating figures several times, to place a dot over the repeating figure in a single repotend, thus, (7) 13* LVIII. To reduce Compound Numbers to Decimals of the highest Denomination. Reduce 15 s. 6d. to the decimal of a pound OPERATION. 12)6, 0 d. 20)15, 5 s. = In this example, 6 d. of a shilling, and, reduced to a decimal by ¶ LVIÏ., is equal to 5 of a, shilling, which, joined with 15 s., makes15, 58. In the same manner, 15,5 s.÷20 s.—,775 £, Ans. Is written i; also over the first and last repeating figure of a compound repetend; thus, for,030303, &c. we write,,03. The value of any repetend, notwithstanding it repeats one figure or moie an infinite number of times, coming nearer and nearer to a unit each time, though never reaching it, may be easily determined by common fractions; as will appear from what follows. 18. By reducing to a decimal, we have a quotient consisting of,1111, &c., that the repetend,,1; †, then, is the value of the repetend 1, the value of ,333, &c; that is, the repetend 3 must be three times as much; that is, ‹‚4—† ;‚5—3; and ‚9—8—1 whole. and Hence, we have the following RULE for changing a single repetend to its equal common fraction,-Make the given repetend a numerator, writing 9 underneath for a denominator, and it is done. What is the value of‚ì? Of‚1⁄2? Of‚À? Of,7? Of,8? of,d? A. z,f,t By changing to a decimal, we shall have,,010101, that is, the repetend ‚01. Thon, the repetend,04, being 4 times as much, must be, and‚36 must in 3§, also,45=†† if he reduced to a decimal, it produces,001. Then the decimal,004, being 4 times as much, is gĝg, and,036. This principle will be rue for any number of places. Hence we derive the following RULE for reducing a circulating decimal to a common fraction,-Make the given repetend a numerator, and the uznominator will be as many Is as there are figures in the repetend Change Change,Î8 to a common fraction. 4. In the following example, viz. change ,83 to a common fraction, the repeating figure is 3 that is, §. and 8 then instead of being & of Hence we derive the following RULE. 1. How must the several denominations be placed? A. One above another, the highest at the bottom." II. How do you divide? A. Begin at the top, and divide as in Reduction; that is, shillings by shillings, ounces by ounces &c., annexing ciphers. III. How long do you continue to do so? A. Till the denon inations are reduced to the decimal required. More Exercises for the Slate. 2. Reduce 7 s. 6 d. 3 qrs. to the decimal of a pound. A.,378125 £ 3. Reduce 5 s. to the decimal of a pound. 5. Reduce 2 qrs. 3 na. to the decimal of a yard. A. ,003125 L. A.,6875 yd. 6. Reduce 2 s. 3 d. to the decimal of a dollar. A. $,375 7. Reduce 3 qrs. 3 na. to the decimal of a yard. A. ,9375 yd 8. Reduce 8 oz. 17 pwts. to the decimal of a pound Troy. A.,7375 lb 9. Reduce 8 £, 17 s. 6d. 3 qrs. to the decimal of a pound. A. 8,878125 £. ▲ unit, is, by being in the second place, of; then g added together, thus, &+5%=38=35, 'Ans. Hence, to find the value of a mixed repetend-First find the value of the rapeating decimals, then of the other decimals, and add these results together. 2. Change,918 to a common fraction. A. 18b+580=888 = 11 Proof, 11÷12,918. 3. Change 203 to a common fraction. A. EL. To know if the result be right, change the common fraction to a decimal again. If it produces the same, the work is right. Repeating decimals may be easily multiplied, subtracted, &c. by first redzeing them to their equal common fractious." 1 LIX. To reduce Decimals of higher Denomina tions to Whole Numbers of lower Denominations. This rule is the reverse of the last. Let us take the answer to the first example. Reduce,775£ to whole numbers of lower denominations. OPERATION £,775 20 s. 15,500 12 In this example,775 £, reduced to shillings, that is, multiplied by 20, gives 15,5, (for ciphers on the right of a decimal are of no value;) then the decimal part ,5×12=6,00 6d. Ans. 15 s. 6 d. d. 6,000 Hence we derive the following RULE. 1. How do you proceed? A. Multiply the given decimal as m Reduction; that is, pounds by what makes a pound, ounces by what makes an ounce, &c. II. How many places do you point off in each product for decimals? A. As many as there are decimal places in the given decimal. III. Where will you find the answer? A. The several denominations on the left hand of the decimal points will be the answer. More Exercises for the Slate. The following examples are formed by taking the answers in tho last rule; of course, the answers in this may be found in the examples of that. The examples in each are numbered so as to correspond. 2. Reduce,378125 £ to whole numbers of lower denomi nations. (For ans. see ex. No. 2, ¶ LVIII.) 3. What is the value of 25 £ of a pound? 8 What is the value of ,7375 of a pound Troy? Application of the two foregoing Rules. 1. What will 4 yards of cloth cost, in pounds, at 7 s. 6 d. a 7s. 6d., reduced to a decimai,=,375 £× 4 yds. == yard? £1,500 2. At $6 a cwt., what will 2 cwt. 2qrs. of rice cost? A. $15. A. $15,50 qr. 7 lbs. of sugar, at $11,25 a cwt. A. $68,203+ 4. What cost 6 cwt. 0 A. $46,8971 6. At $1,25 a bushel, what will 36 bu. 0 pk. 4 qts. cost? A. $45,156,25 7. At $4,75 a yard, what will 26 yds. 2 qrs. of broadcloth cost? A. $125,87). 8. At 2, 10 s. a cwt., what will 6 cwt. 3 qrs. of rice cost? 2 £, 10 s. 2,5 £, and 6 ciet. 3 grs. = 6,75 cut. ; then, 6,75× 2,5=16,875 £× 20: 17,5 X 126 d. Ans. 16 £, 17 s. 6 d. 9. What will 6 gallons, 2 qts. of brandy cost, in pounds, at 15 shillings a gallon? A. 4. 17 s. 6 d. REDUCTION OF CURRENCIES. LX. An apology may by some be deemed necessary for the mission, this work, of much inat is contained in other treatises, respe ing what is called "the currencies of the different United States The thor, however, deems it rather neces sary to apologize for introducing the subject at all. Those merely nominal currencies, originally derived from Great Britain, have long been obsolete in law, and ought to become so in practice. So long, however, as that practice continues, it may be necessary to retain a brief notice of it in elementary works. Note. It was not intended that the following Table should be exact in every particular to a mill, but enough so, to correspond with the pecuniary calculations current among men of business; and, as such, it will be committed to memory more easily. The design of the Table is not that it should be learned by rote, but by actual, calculations from a few data; thus-as far. is of a cent, then 2 farthings are 3. Again, as 3 d. is 4 cents, and 3s. are 50 cents, then 3 s. 3 d. are 54 cents. It would be well for the teacher to direct the attention of the pupil to this object by explanations Repeat the |