13. A dairy-woman packed 95 lbs. 8 oz. of butter in 10. boxes: how much did each box contain?

14. A tailor had 76 yds. 2 qrs. 3 na. of cloth, out of which he made 8 cloaks: how much did each cloak contain?

15. A man traveled 50 m. and 32 r. in 11 hours: at what rate did he travel per hour?

16. 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?

17. A man had 80 A. 45 r. of land, which he laid out into 36 equal lots: how much did each lot contain?

18. Divide 685 bu. 2 pks. 4 qts. by 45.

19. If £85, 7s. 7d. 3 far. are divided equally among 81 persons, how much will each receive?

APPLICATIONS OF THE COMPOUND RULES.

175. A Bill, in business operations, is a written statement of items, with the price of each, and the amount of the whole. Required the amount of each of the following bills:

Put the following memoranda into the form of bills, and find the amount of each:

3. James Henry bought, July 1st, 1852, of C. B. Lawrence, 25 lbs. gunpowder, at 4s. 6d.; 36 guns, at £1, 12s. 6d.; 12 rifles, at £2, 8s.; and 45 knapsacks, at 12s. 6d. What was the amount of his bill?

4. If you buy 27 lbs. sugar, at 7d. a pound; figs, at 4s. 6d. a drum; 17 boxes of raisins, at what will be the amount of your bill?

36 drums of

6s. 7d. a box,

5. J. Dill bought 10 doz. pair silk hose, at 4s. 8d. a pair; 16 doz. thread ditto, at 3s. 41⁄2d. ; 21 doz. worsted ditto, at 4s. 61d.: what was the amount of his bill?

6. James Gordon sold 15 acres, 2 roods, and 15 rods of land, at £3, 15s. 7d. per rod: what amount did he receive?

7. Bought a piece of land 68 rods long, and 251⁄2 rods wide, at £6, 4s. 6d. per acre: what did it amount to?

8. Elisha Fanning sold a customer a quarter of veal weighing 18 lbs. 4 oz., at 81d. per pound; a quarter of mutton weighing 16 lbs. 8 oz. at 71⁄2d.; and a saddle of venison weighing 28 lbs. 4 oz. at 1s. 7d. per pound: what was the amount of the bill?

9. A drover bought 10 oxen each weighing 9 cwt. 15 lbs., at 81d. per pound: what was the amount of his bill?

10. A hardware merchant bought 43 tons, 2 qrs. 17 lbs. of iron, at 1s. 7d. per pound: what was the amount of his bill?

11. A laborer dug a cellar 62 feet long, 25 feet wide, and 81 ft. deep, at 51d. per cu. yard: what was the amount of his bill

12. Bought 50 casks of molasses each containing 58 gals. 3 qts., at 2s. 6d. per gal.; afterwards 215 gals. 2 qts. leaked out, and the remainder was sold at 3s. 4d. per gal.: what was the result of the operation?

13. Bought 2 cwt. 3 qrs. 10 lbs. of saltpetre, at 3s. 7d. a pound; 16 cwt. 2 qrs. 17 lbs. dyewood, at 4s. 6d. a pound; 5 cwt. 1 qr. 11 lbs. indigo, at 15s. 8d. a pound: what was the amount of the bill?

14. George Spencer bought of Henry Brown, 75 yards of broadcloth, at 15s. 6d. per yard; 115 yards of silk, at 7s. 6d. per yard; 263 yards of bombazine, at 4s. 7d. per yard; 325 yards of cassimere, at 11s. 8d.: what was the amount of his

SECTION VIII.

DECIMAL FRACTIONS.

ART. 176. Decimal Fractions are those which arise from dividing an integer into ten equal parts; then subdividing one of these parts into ten others, and so on, each succeeding part regularly decreasing in a ten fold ratio. Thus, if a unit is divided into 10 equal parts, 1 of these parts is a tenth. (Art. 103.) Now if 1 tenth is divided into 10 equal parts, 1 of these parts will be a hundredth; for÷10= (Art. 188.) Again, if 1 hundredth is divided into 10 equal parts, 1 of these parts will be a thousandth; for÷10=0, &c.

OBS. These fractions are called decimals, from the Latin numeral decem, ten, which indicates both their origin and ratio of decrease.

177. Each order of whole numbers, we have seen, increases in value from units towards the left in a ten-fold ratio; and, conversely, each order must decrease from left to right in the same ratio, till we come to units' place again. (Art. 9.)

178. 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 ten-fold ratio, and exactly correspond in value with tenths, hundredths, thousandths, &c., when expressed by common frac tions. Hence,

179. Decimal Fractions are commonly expressed by writing the numerator with a point (.) before it, called the separatrix. Thus, is written .1; thus .2; thus .3, &c. is writ

QUEST.-176. What are decimal fractions? Obs. Why called decimals? 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?

ten .01, putting the one in hundredths place; 15 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.

OBS. 1. 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. 2. The object of the decimal point or separatrix, is to distinguish the fractional parts from whole numbers. To prevent it from being mistaken for the point used in numeration, the decimal point should be a period (.), and the other a comma (,).

180. The denominator of a decimal fraction is always 1, with as many ciphers annexed to it as there are decimal figures in the given numerator. (Art. 176.)

181. The names of the different orders of decimals or places below units, may be easily learned from the following

DECIMAL TABLES.

→ Hundreds.

(Decimal Point.)
Tenths.

Hundredths.
Thousandths.

Ten thousandths.
Hundred thousandths.
Millionths.

Ten Millionths.

∞ Hundred millionths.
Billionths.

Ten billionths.
Hundred billionths.

Trillionths, &c.

co Units. Tens.

4

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. 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

QUEST. Obs. If the numerator does not contain so many figures as there are ciphers in the denominator, what must be done? What is the object of the deeimal point? 180. What is the denominator of a decimal fraction? 181. Re peat the Decimal Table, beginning units, tenths, &c. 182. Upon what does the value of a decimal depend? 183. What is the effect of removing a decimal figure one place to the right ?

value ten times; for it removes the decimal one place farther from units' place. Thus .4=; but .04=74, and .001=1000. 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=5; so .50-50 .500=500%, or 5, &c. (Art. 116.)

1009

or, and

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 decinals, is the reverse of annexing and prefixing them to whole numbers. (Art. 58.) 2. A whole number and a decimal written together, is called a mixed number. (Art. 108.)

184. 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.

OBS. 1. In reading decimals as well as whole numbers, the units' place should always be made the starting point. It is advisable for young pupils to apply to every figure the name of its order, or the place which it occupies, before attempting to read them. Thus beginning at units' place-units, tenths, hundredths, thousandths, &c., pointing to each figure as he pronounces the name of its order.

2. Sometimes we pronounce the word decimal when we come to the separatrix, and then read the figures as if they were whole numbers; or, simply repeat them one after another. Thus, 125.427 is read, one hundred twenty-five, decimal four hundred twenty seven; or, one hundred twenty-five, decimal four, two, seven. Read the following numbers:

QUEST.-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 together, called? 184. How are decimals read? Obs. In reading decimals, what should be made the starting point? What other method of reading decimals is mentioned ?