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DIVISION OF COMPOUND NUMBERS.
Ex. 1. Divide £17, 6s. 9d. by 4.
4 is contained in £17, 4 £
times and £l over. Set the 4)17 6 9 0 quotient 4 under the pounds,
and reduce the remainder £1 4 6
1 Ans. 8
to shillings, which added to the 6s., make 26s. 4 in 26s., 6 times and 2s. over.Set the quotient 6 under the shillings, and reduce the remainder 2s. to pence, which added to the 9d. make 33d. 4 in 33d., 8 times and Id. over. Set the 8 under the pence, reduce the ld. to farthings, and divide as before. Ans. £4, 6s. 8d. 1 far.
173, Hence, we deduce the following general
RULE FOR DIVIDING COMPOUND NUMBERS.
Begin with the highest denomination, and divide each separately. Reduce the remainder, if any, to the next lower denomination, to which add the number of that denomination contuined in the given example, and divide the sum as before. Proceed in this manner through all the denominations.
Obs. 1. Each partial quotient will be of the same denomination as that part of the dividend from which it arose.
2. When the divisor exceeds 12, and is a composite number, it is advisable to divide first by one factor and that quotient by the other. (Art. 78) If the divisor exceeds 12, but is not a composite number, long division may be einployed. (Art. 77.)
3. The process of dividing different denominations is called Compound Division.
Quest.–173. Where do you begin to divide a compound number? What is done with the remainder? Obs. Of what denomination is each partial quotient? When the divisor is a composite number, how proceed? What is the process of dividing different denominations called ?
2. Divide £274, 4s. 6d. by 21.
13 1 2 Ans. 3. Divide £635, 17s. by 31.
£ £ 31)635, 17 ( 20,1031. ,
The remainder £15 is reduced
to shillings, to which we add the 620
given shillings, making 317, and 15 rem.
divide as before. The remain20
der 7s. may be reduced to pence 317
and divided again if necessary. 310
4. Divide £7, 8s. 2d. by 3.
7. A man bought 5 cows for £23, 16s. 8d. : how much did they cost apiece?
8. A merchant sold 10 rolls of carpeting for £62, 12s. 9d. : how much was that per roll ?
9. Paid £25, 10s. 6 d. for 12 yards of broadcloth: what was that per yard ?
10. A silversmith melted up 2 lbs. 8 oz. 10 pwts. of silver, which he made into 6 spoons : what was the weight of each?
11. The weight of 8 silver tankards is 10 lbs. 5 oz. 7 pwts, 6 grs. : what is the weight of each ?
12. If a family of 8 persons consume 85 lbs. 12 oz. of meat in a month, how much is that apiece ?
13. A dairy woman packed 95 lbs. 8 oz. of butter in 10 boxes : how much did each box contain ?
16. A tailor had 76 yds. 2 qrs. 3 na. of cloth, out of which he made 8 cloaks : how much did each cloak contain ?
17. A man traveled 49 m. 8 fur, 32 r. in 11 hours : at what rate did he travel per hour ?
18. A man had 285 bu. 3 pks. 6 qts. of grain, which he wished to carry to market in 15 equal loads : how much must he carry at a load ?
19. A man had 80 A. 45 r. of land, which he laid out into 36 equal lots : how much did each lot contain ?
20. A farmer had 75 C. 92 st. of wood, which he carried to market at 63 loads : how much did he carry at a loud ?
Art. 175. When any thing is divided into equal parts, those parts, we have seen, are called Fractions ; (Art. 105;) also, that the parts take their name from the number of parts into which the thing is divided. Thus when any number or thing is divided into 10 equal parts, 1 of those parts is called one tenth; when divided into 100 equal parts, the parts are called hundredths ; when divided into 1000 equal parts, the parts are called thousandths, &c. Now if i tenth is subdivided into ten equal parts, the parts will be hundredths, for '-10= Tdo; (Art. 138 ;) if joy is subdivided into 10 equal parts, the parts will be thousandths, for do=10=10oo, &c. Hence it appears, that a tenth is ten times less than a unit; a hundredth, ten times less than a tenth; a thousandth, ten times less than a hundredth ; a ten-thousandth, ten times less than a thousandth, &c.
176. The class of fractions which arise from dividing a unit into ten equal parts, then subdividing each of
Quest.-175. What are fractions? From what do the parts take their name? 176. What are decimal fractions? Why called 'decimals ?
these parts into ten other equal parts, and so on, are called decimal fractions ; because they decrease regularly by tens, or in a ten-fold ratio. (Art. 10. Obs. 2.)
177. Each order of integers or whole numbers, it has been shown, increases in value from units towards the left in a ten-fold ratio ; (Art. 9;) and, conversely, each order must decrease from left to right in the same ratio, till we come to units place again.
178. By extending this scale of notation below units towards the right hand, it is manisest that the first place on the right of units will be ten times less in value than units place; that the second will be ten times less than the first; the third, ten times less than the second, &c.
Thus we have a series of places or orders below units, which decrease in a ten-fold ratio, and exactly correspond in value with tenths, hundredths, thousandths, &c.
179. Hence, to express Decimal Fractions, or fractions whose denominator is 10, 100, 1000, &c.
Write simply the numerator with a point (.) before it, to distinguish the fractional parts from whole numbers.
For example, to may be written thus .l; fo thus .2; i thus .3; &c.; jho may be written thus .01, putting the 1 in hundredths place; Tóm thus .05; &c. That is, tenths are written in the first place on the right of units ; hundredths in the second place; thousandths in the third place, &c.
180. The denominator of a decimal fraction is always I with as many ciphers annexed to it as there are figures in the numerator, and need not be expressed.
Obs. The point placed before decimals, is often called the Separatrix.
Quest.–177. In 'what manner do whole numbers increase and decrease ? 178. By extending this scale below units, what would be the value of the first place on the right of units? The second ?. The third ? With what do these orders correspond? 179. How are decimal fractions expressed ? 180. What is the denominator of a decimal fraction ? Obs. What is the point placed before decimals called ?
181. The names of the different orders of decimals or places below units, may be easily learned from the following
1 4 5 9 8 6
182. It will be seen from this Table that the value of each figure in decimals, as well as in whole numbers, depends upon the place it occupies, reckoning from units. Thus, if a figure stands in the first place on the right of units, it expresses tenths ; if in the second, hundredths, &c.; each successive place or order towards the right, decreasing in value in a lenfold ratio. Hence,
183. Each removal of a decimal figure one place from units towards the right, diminishes its value ten times.
Prefixing a cipher, therefore, to a decimal diminishes its value ten times; for it removes the decimal one place farther from units' place. Thus .4=10; but .04=167 and .004=1000, &c.; for the denominator to a decimal fraction is 1 with as many ciphers annexed to it as there are figures in the nunerator. (Art. 180.)
Annexing ciphers to decimals does not alter their value; for each significant figure continues to occupy the same place from units as before. Thus,.5= fó; 30.50=
or jó, by dividing the numerator and denominator by 10; (Art. 116;) and .500=2000, or is, &c.
Quest.–181. Repeat imal Table, beginning units, tenths, &c. 182. Upon what does the value of a decimal depend ? 183. What is the effect of removing a decimal one place towards ihe right? What then is the effect of prefixing cipbers to decimals? What, of annexing them ?