barrels will 3,52275 bushels fill? (105) Will,4026 of a bushel? (12) Will 120,780 bushels? (36) Will 63,745 bushels? (19) Will 40,260 bushels? (12) A. 68,17 barrels. 11. What is the quotient of 1561,275 divided by 24,3? (6425) By 48,6? (32125) By 12,15? (1285) By 6,075? (257) Ans. 481,875. 12. What is the quotient of ,264 divided by ,2? (132) By,4? (66) By ,02? (132) By,04? (66) By ,002? (132) By ,004? (66) Ans. 219,78. REDUCTION OF DECIMALS. [ LVII. To change a Vulgar or Common Fraction to its equal Decimal. 1. A man divided 2 dollars equally among five men; what part of a dollar did he give each? and how much in 10ths, or decimals? In common fractions, each man evidently has of a dollar, the answer; but, to express it decimally, we proceed thus: OPERATION. Denom. 5)2,0(,4 20 Ans. 4 tenths, ,4 In this operation, we cannot divide 2 dollars, the numerator, by 5, the denominator; but, by annexing a cipher to 2, (that is, multiplying by 10,) we have 20 tenths, or dimes; then 5 in 20, 4 times; that is, 4 tenths, ,4: Hence the cominon fraction, reduced to a decimal, is,4, Ans. 2. Reduce to its equal decimal. 3 32 OPERATION. 288 120 96 In this example, by annexing one cipher 32)3,00(,09375 to 3, making 30 tenths, we find that 32 is not contained in the 10ths; consequently, a cipher must be written in the 10ths' place in the quotient. These 30 tenths may be brought into 100ths by annexing another cipher, making 300 hundredths, which contain 32, 9 times; that is, 9 hundredths. By continuing to annex ciphers for 1000ths, &c., dividing as before, we obtain ,09375, Ans. By counting the ciphers annexed to the numerator, 3, we shall find them equal to the decimal places in the quotient. 240 224 160 160 5 Note. In the last answer, we have five places for decimals; but, as the 5 in the fifth place is only Toooo 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 decimal, 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 examples,,4&=Z. Again,,09375=188780-32. 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. II. How long do you continue to annex ciphers and divide? 4. Till there is no remainder, or until a decimal is obtained sufficiently exact for the purpose required. III. How many figures of the quotient will be decimals? A. 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? A. Prefix ciphers to supply the deficiency. More Exercises for the Slate. 3. Change, 4, 4, and 's to equal decimals. A.,5,,75, ,25,,04. 4. What decimal is equal to ? (5) What? (5) What T2 ? (75) What? (4) Ans. 1,34. 5. What decimal is equal to 18? (5) What? (25) What? (5) What? (175) What? (625) A. 1,6. 6. What decimal is equal to ? (1111) What=? (4444) What? (10101) What? (3333)* A.,898901. + * When decimal fractions continue to repeat the same figure, like 333, &c., in this example, they are called Repetends, 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 repetend,,010101, &c. a compound repetend. When other decinals come before circulating decimals, as ,8 in ,8333, the desimal 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 repetend; thus, 111, &. ↑ 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., makes 15, 5s. In the same manner, 15,5 s.÷÷20 s.—,775 £, Ans. 18 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 more 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. By reducing to a decimal, we have a quotient consisting of,1111, &c., that is, the repetend,,;, 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—§ ; and,9—3—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‚2? Of‚4? Of‚7? Of,8? Of,&? A. J, Z, f, 7, 8, 8. By changing to a decimal, we shall have, ,010101, that is, the repetend ‚ØÏ. Then, the repetend‚04, being 4 times as much, must be, and‚36 must be 3§, also,45—13. If go be reduced to a decimal, it produces,001. Then the decimal ,004, being 4 times as much, is, and,036. This principle will be Arue 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 denominator will be as many 9s as there are figures in the repetend. 8 Τ Change,i8 to a common fraction. A. §=&· Τ In the following example, viz. change,83 to a common fraction, the repeating figure is 3, that is, f, and,8 is; then 3, 10stead of being † of Hence we derive the following RULE. I. How must the several denominations be placed? above another, the highest at the bottom. A. One 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 denom 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 £. A. ,6875 yd. 6. Reduce 2 s. 3 d. to the decimal of a dollar. A. ,7375 lb. 9. Reduce 8 £, 17 s. 6 d. 3 qrs. to the decimal of a pound. A. 8,878125 £. a unit, is, by being in the second place, of; then and added together, thus, 1+3=ZZ=35, Ans. Hence, to find the value of a mixed repetend-First find the value of the repeating decimals, then of the other decimals, and add these results together. 2. Change ,916 to a common fraction. A. 91 Proof, 11 12 ,916. 3. Change 203 to a common fraction. A. E. 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 redueing them to their equal common fractions. 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 d. 6,000 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. Hence we derive the following RULE. I. How do you proceed? A. Multiply the given decimal as in 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 the 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 denominations. (For ans. sce ex. No. 2, ¶ LVIII.) 3. What is the value of 25 £ of a pound? 4. What is the value of ,003125 of a pound? 7. What is the value of ,9375 of a yard? 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 decimal,=,375 £x 4 yds. = yard? £1,500 20 10,000 Ans., 1 £ 10 ■. £× |