COMPOUND DIVISION. 306. The process of dividing numbers of different denominations, is called COMPOUND DIVISION. Ex. 1. Divide £25, 3s. 4d. 2 far. by 6. £ 6)25 Operation. 3" 10" 3 Ans. Beginning with the pounds, we find 6 is contained in £25, 4 times and 1 over. Set the 4 under the pounds, and reduce the remainder £1 to shillings, which added to the 3s. make 23s. 6 in 23s. 3 times and 5s. over. Set the 3 under the shillings, and reduce the remainder 5s. to pence, which added to the 4d. make 64d. 6 in 64d., 10 times and 4d. over. Set the 10 under the pence, reduce the 4d. to farthings, and divide as before. Ans. £4, 3s. 10d. 3 far. 307. 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 contained 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. 129.) If the divisor exceeds 12, but is not a composite number, long division may be enployed. (Art. 120. II.) 3. Compound Division is the same in principle, as Simple Division. Prejixing the remainder to the next figure of the dividend in simple division, is the same as reducing it to the next lower order or denomination, and adding the next figure to it. QUEST-306. What is Compound Division? 307. 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? Does it differ from Simple Division? 2. A man wished to divide 75 cwt. 2 qrs. 10 lbs. of beef equally among 35 families: how much could he give to each? Ans. 7 lbs. 3 oz. 6 pwts. 4. Divide 410 lbs. 4 oz. 5 pwts. 6 grs. by 8. 5. Divide 786 bu. 18 qts. by 25. 6. A farmer raised 1000 bu. 3 pks. 16 qts. of wheat on 40 acres: how much was that per acre? 7. A man bought 10 horses for £200, 15s.: how much did he give apiece? 8. Divide £87, 10s. 74d. by 18. 9. A merchant tailor put 216 yds. 3 qrs. of cloth into 20 cloaks: how much cloth did each cloak contain? 10. Divide 500 yds. 3 qrs. 2 na. by 54. 11. A man traveled 1000 miles in 12 days: at what rate did he travel per day? 12. Divide 1500 m. 2 fur. 30 r. 12 ft. by 7. 13. Divide 120 gals. 3 qts. 1 pt. by 72. 14. Divide 400 hhds. 10 gals. 2 qts. 1 pt. by 9. 15. Divide 365 d. 10 hr. 40 min. by 15. 16. Divide 111 yrs. 20 d. 13 hrs. 25 min. 10 sec. by 11. 17. Divide 45° 17′ 10′′ by 25. 18. Divide 65 signs 12° 47' by 41. 19. Divide 164 cords, 30 ft. by 17. -20. Divide 410 cords, 10 ft. 21 in. by 61. 21. If a chest of tea weighing 96 pounds cost £33, what will i pound cost? 22. If the duty on a pipe of wine is £50, 6s. 6d., what is the duty per gallon? 23. If a person spends £200 a year, what are his expenses per day? 21 308. Fractions which decrease in a tenfold ratio, or which express simply tenths, hundredths, thousandths, &c., are called DECIMAL FRACTIONS. They arise from dividing a unit into ten equal parts, then dividing each of these parts into ten other equal parts, and so on. Thus, if a unit is divided into 10 equal parts, 1 of those parts is called a tenth. (Art. 178.) If a tenth is divided into 10 equal parts, 1 of those parts will be a hundredth; for, ÷10=10. If a hundredth is divided into 10 equal parts, 1 of the parts will be a thousandth; for, T÷10=T, &c. (Art. 227.) OBS. Fractions of this class are called decimals, because they regularly de crease in a tenfold ratio. (Art. 37. Obs. 2.) Decimal fractions are said to have been invented by Lord Napier, in 1602. 309. Each order of whole numbers, we have seen, increases in value from units towards the left in a tenfold ratio; and, conversely, each order must decrease from left to right in the same ratio, till we come to units' place again. (Art. 37.) 310. By extending this scale of notation below units towards the right hand, it is manifest 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 orders below units, which decrease in a tenfold ratio, and exactly correspond in value with tenths, hundredths, thousandths, &c. (Art. 308.) QUEST.-308. What are Decimal Fractions? From what do they arise? Obs. Why called decimals? 309. In what manner do whole numbers increase and decrease? 310. 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 in value ? 311. Decimal Fractions are commonly expressed by writing the numerator with a point ( . ) before it. The point placed before decimals is called the Decimal Point, or Separatrix. Its object is to distinguish the fractional parts from whole numbers. If the numerator does not contain so many figures as there are ciphers in the denominator, the deficiency must be supplied by prefixing ciphers to it. For example, is written thus .1; thus .2; thus .3; &c. 1 in hundredths' place; is written thus .01, putting the 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. 312. The denominator of a decimal fraction is always 1 with as many ciphers annexed to it, as there are figures in the given numerator. (Art. 308.) 313. The names of the different orders of decimals, or places below units, may be easily learned from the following 314. 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 QUEST.-311. How are decimal fractions expressed? What is the point placed before decimals called? 312. What is the denominator of a decimal fraction? 313. Repeat the Decimal Table, beginning units, tenths, &c. 314. Upon what does the value of a deeimal depend? the second, hundredths, &c.; each successive place or order towards the right, decreasing in value in a tenfold ratio. Hence, 315. 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=; but .04=; and .004, &c.; for the denominator to a decimal fraction is 1 with as many ciphers annexed to it, as there are figures in the numerator. (Art. 312.) 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; so .50=, or fo, by dividing the numerator and denominator by 10; (Art. 191,) and .500=%, or F, &c. OBS. 1. It should be remembered that the units' place is always the right hand place of a whole number. The effect of annexing and prefixing ciphers to decimals, it will be perceived, is the reverse of annexing and prefixing them to whole numbers. (Art. 98.) 2. A whole number and a decimal, written together, is called a mixed number. (Art. 183.) 316. To read decimal fractions. Beginning at the left hand, read the figures as if they were whole numbers, and to the last one add the name of its order. Thus, OBS. In reading decimals as well as whole numbers, the units' place should always be made the starting point. It is advisable for the learner to apply to QUEST.-315. What is the effect of removing a decimal one place towards the right? What then is the effect of prefixing ciphers to decimals? What, of annexing them? Obs. Which is the units' place? What is a whole number and a decimal written to gether, called? 316. How are decimals read? Obs. In reading decimals, wha should s made the starting point? |