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From these illustrations we derive the following
RULE. Q. How do you write the numbers down, and divide ? A. As in whole numbers. Q. How many figures do you point off in the quotient for decimals ?
A. Enough to make the number of decimal places in the divisor and quotient, counted together, equal to the number of decimal places in the dividend.
Q. Suppose that there are not figures enough in the quotient for this purpose, what is to be done ?
A. Supply this defect by prefixing ciphers to said quotient.
Q. What is to be done when the divisor has more decimal places than the dividend ?
A. Annex as many ciphers to the dividend as will make the decimals in both equal.
Q. What will be the value of the quotient in such cases ?
Q. When the decimal places in the divisor and dividend are equal, and the divisor is not contained in the dividend, or when there is a re mainder, how do you.proceed ?
A. Annex ciphers to the remainder, or dividend, and divide as before.
Q. What places in the dividend do these ciphers take ?
More Exercises for the Slate. 4. At $,25 a bushel, how many bushels of oats may be bought for $300,50 ? A. 1202 bushels.
5." At $,12}, or $,125 a yard, how many yards of cotton cloth may be bought for $16? A. 128 yards.
6. Bought 128 yards of tape for $,64 ; how much was it & yard? A. $,005, or 5 mills.
7. If you divide 116,5 barrels of flour equally among 5 men, how many barrels will each have ? A. 23,3 barrels.
Note.—The pupil mast continue to bear in mind, that, before he proceeds to add together the figures annexed to each question, he must prefix ciphers, when required by the rule for pointing off.
8. At $2,255 a gallon, how many gallons of rum may be bought for $28.1875 ?-125. For $56,375 ?-25. For $112,75 ?50. For $338,25 ?-150. A. 237,5 gallons. 9. If $2,25 will board one man a week, how many.
weeks can he be boarded for $1001,25 ?-445. For $500,857-2226. For $200,7?-892. For $100,35 ?-446. For $60,75?-27. A. 828,4 weeks.
10. Íf 3,355 bushels of corn will fill one barrel, how many barrels will 3,52275 bushels fill?–105. Will ,4026 of a bushel ?12. Will 120,780 bushels ?-36. Will 63,745 bushels ?-19. Will 40,200 bushels ?-12. A. 68,17 barrels.
11 Whiat is the quotient of 1561,275 divided by 24,3 ?-6425. By 48,6?-32125. By 12,15 ?-1285. By 6,075 ?-257. A. 481,875.
12. What is the quotient of ,264 divided by ,2?-132. By ,4?66. By ,02?–132. "By ,04 ?-66. By ,0022-132. By ,004 ?-66. A. 219,78.
REDUCTION OF DECIMALS.
ILVII. TO CHANGE A VULGAR OR COMMON FRAC
TION TO ITS EQUAL DECIMAL. 1. A man divided 2 dollars equally among 5 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.
In this operation, we cannot di. Numer. vide 2 dollars, the numerator, by 5, Denom. 5)2,0(,4 the denominator; but, by annex20
ing a cipher to 2, (that is, multi
plying by 10,) we have 20 tenths, Ans. 4 tenths, 1,4.
or dimes; then 5 in 20, 4 times;
that is, 4 tenths,=,4: Hence the common fraction, reduced to a decimal, is , 4, Áns, 2. Reduce to its equal decimal. OPERATION.
In this example, by annexing one 32)3,00(,09375 cipher to 3, making 30 tenths, we 288
find that 32 is not contained in the
10ths; consequently, a cipher must 120
be written in the 10ths' place in 96
the quotient. These 30 tenths may
be brought into 100ths by annex240
ing another cipher, making 300 224
hundredths, which contain 32, 9
times; that is, 9 hundredths. By 160
continuing to annex ciphers for 160
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.
Note.-In the last answer, we have five places for decimals ; but, as the 5 in the fifth place is only todo oo 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=}. Again, ,09375=183760 = From these illustrations we derive the following
RULE. Q. How do you proceed to reduce a common fraction to its equal decimal ?
A. Annex ciphers to the numerator, and divide by the denominator.
Q. How long do you continue to annex ciphers and divide ?
A. Till there is no remainder, or until a decimal is obtained sufficiently exact for the purpose required.
Q. How many figures of the quotient will be decimals ?
Q. 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 1, , 4, and 25 to equal decimals. A. ,5, ,75, ,25, ,04.
4. What decimal is equal to z ?-5. What=?-5. What =I3?-75. What= ?-4. A. 1,34.
5. What decimal is equal to Too ?-5. What = -25 What=142-5. What= ?-175. What=18?-625. A. 1,6
6. What decimal is equal to š?-1111. What=* ?-4444. What=g9?-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, 9010101, &c. a compound ropetenu.
When other decimals come before circulating decimals, as ,8 in ,8333, the decimal is called a mixed repetend.
ILVIII. TO REDUCE COMPOUND NUMBERS To Decr.
MALS OF THE HIGHEST DENOMINATION.
1. Reduce 15 s. 6 d. to the decimal of a pound.
ling, and , reduced to a decimal by
1 LVII., is equal to ,5 of a shilling, 20 ) 15,5 s.
which, joined with 15 s., makes=15,5 8. In the same manner, 15,5 8. • 208.= ,775£, Ans.
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, &c., 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 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 ,i ; $, then, is the value of the repetend i, the value of ,333,
that is, the repetend 3 must be three times as much; that is, j and Å=*; ,5 = ģ; and ,9 = = 1 whole.
Hence we have the following RULE for changing a single repetend to its equal common fraction:-Make the given repetend a numerator, writing I underneath for a denominator, and it is done.
What is the value of ,i? ofà? of 42 of 7? Of,8? Of ,6? A. , $$
By changing on to a decimal, we shall have ,010101, that is, the repotend, „ól. Then, the repetend 904, being 4 times as much, must be afg, and „36 must be 3 g, also ,45=
If yfg be reduced to a decimal, it produces ,ibi. Then the decima? ,004, boing 4 times as much, is gfg, and ,036= . This principle will be true for any number of places.
Ilence we derivo the following RULE for reducing a circulating decimal to a common fraction :-Mako the given repetend a numerator, and the denomipator will be as many 98 as there are figures in the repetend.
Change ,i8 to a common fraction. A. Moritt
In the following example, viz. Change ,83 to a common fraction, the repeating figure is 3, that is, g, and ,8 is ; then š, instead of boing f of
Hence we derive the following
Q. How must the several denominations be placed
A. One above another, the highest at the bottom.
Q. 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.
Q. How long do you continue to do so ?
A. Till the denominations 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.
A. 25£. 4. Reduce 3 farthings to the decimal of a pound.
A. ,003125.£. 5. Reduce 2 qrs. 3 na. to the decimal of a yard. 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. 6 d. 3 qrs. to the decimal of a pound.
a unit, is, by being in the second place, g of Italy; then 8 and 9% added together, thus, 1% +=35= 6, 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. 90% tooo=28= t. Proof, 11 + 12=,916.
3. Change ,203 to a common fraction. A. .
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 reduc ing them to their equal common fractions.