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305. Hence, we deduce the following general
RULE FOR MULTIPLYING COMPOUND NUMBERS.
Multiply each denomination separately, beginning with the lowest, and divide each product by that number which it takes of the denomination multiplied, to make one of the next higher ; set down the remainder, and carry the quotient to the next product, as in addition of compound numbers. (Art. 300.)
OBs. 1. When the multiplier is a composite number, it is advisable to multiply first by one factor and that product by the other. (Art. 97.)
2. Compound Multiplication is the same in principle as Simple Multiplication. In each we carry for that number which it takes of the order or denomination we are multiplying, to make one of the next higher order or denomination.
2. What will 28 horses cost, at £21, 3s. 74d. apiece ?
3. What cost 7 acres of land, at £35, 6s. 7d. per acre ? 4. What cost 18 barrels of flour, at £1, 6s. 8 d. per barrel ? 5. A man bought 15 loads of hay, each weighing 1 ton, 17lbs.: what was the weight of the whole ?
6. Multiply 16 tons, 3 cwt. 10 lbs. by 25. 7. Multiply 12 lbs. 3 oz. 16 pwts. by 56..
8. If i dollar weighs 17 pwts. 41 grs., how much will 96 dollars weigh?
9. Multiply 48 hhds. 15 gals. 2 qts. 1 pt. by 63. 10. Multiply 56 pipes, 1 hhd. 23 gals. by 100.
Quest.-305. Where do you begin to multiply a compound number? What is done with each product? Obs. When the multiplier is a composite number, how proceed ! Does it differ from Simple Multiplication ?
11. Bought 72 pieces of cloth, each containing 324 yards: how much did they all contain ?
12. If i cloak requires 10 yds. 3 qrs., how much will 500 cloaks require ?
13. Multiply 175 miles, 7 fur. 18 rods by 84.
24. If a ship sails 3° 24' 10" per day, how far will she sail in 60 days?
25. If 1 acre produce 45 bu. 26 qts., how much will 100 acres produce ?
26. If 1 barrel of flour requires 4 bu. 3 pks. 5 qts. of wheat, how much will 500 barrels require ?
27. What cost a chest of tea containing 17 lbs., at 6s. 10 d. per pound?
28. What is the duty on 1000 gals. of brandy, at 13s. 70. per gallon ?
29. What is the duty on 10560 lbs. of sugar, at 6d. 3 far. per pound ?
30. What is the duty on 1500 yards of broadcloth, at 6s. 91d.
per yd. ?
31. If i load of wood measures 117 ft. 110 in., how much will 40 loads of the same size measure ?
32. If 1 quarter of beef weighs 216 lbs. 7 oz., how much will 4 quarters weigh?
33. If i bushel of salt weighs 72 lbs. 10 oz., how much will 350 bushels weigh?
34. If 1 cask of oil contains 86 gals. 2 qts. 1 pt., how much will 100 casks of the same size contain ?
306. The process of dividing numbers of different denominations, is called COMPOUND DIVISION. Ex. 1. Divide £25, 3s. 4d. 2 far. by 6. Operation.
Beginning with the pounds, we find £
6 is contained in £25, 4 times and 1 6)25 3 4 11 2
Set the 4 under the pounds, 4 3 11 10 " 3 Ans. and reduce the remainder £1 to shil
lings, 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. Troceed 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 employed. (Art. 120. II.)
3. Compound Division is the same in principle, as Simple Division. Prefixing 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 ? Operation.
lbs. We divide by the factors of 35, which 7)75 " 2 " 10 are 7 and 5, and set down each result as 5)10"3" 5 above.
2 0 11 16 Ans.
3. Divide 312 lbs. 9 oz. 18 pwts. by 43.
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. 6 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. 77d. 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.
21. If a chest of tea weighing 96 pounds cost £33, what will 1 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?
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, H10=ito. If a hundredth is divided into 10 equal parts, 1 of the parts will be a thousandth ; for, ito--10=Tooo, &c. (Art. 227.)
Obs. Fractions of this class are called decimals, because they regularly decrease 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. 36.)
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 ?