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 ft. of wood, which he carried to market at 63 loads: how much did he carry at a load?

SECTION VIII.

DECIMAL FRACTIONS.

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 1 tenth is subdivided into ten equal parts, the parts will be hundredths, for 10= Too; (Art. 138;) if is subdivided into 10 equal parts, the parts will be thousandths, for 10-10-1000, &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 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 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, may be written thus .1; thus .2; thus .3; &c; may be written thus .01, putting the 1 in hundredths place; 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 Sepa

ratrix.

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

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 tenfold 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=; but .04=100 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 numerator. (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; so .50= fo, or fo, by dividing the numerator and denominator by 10; (Art. 116;) and .500-1500, or 1%, &c.

50

100'

5

QUEST. 181. Repeat 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 one place towards the right? What then is the effect of prefixing ciphers to decimals? What, of annexing them?

18. Change of 1 rod to the fraction of a foot. 19. Change 197 of 1 yard to the fraction of a nail.

1187

20. Change 100000 of 1 ton to the fraction of a pound.

ADDITION OF COMPOUND NUMBERS.

1. What is the sum of £4, 8s. 6d. 2 far.; £3, 12s. 8d. 3 far.; and £8, 6s. 9d. 1 far. ?

£

4

3

8

16

Operation.
d. far.

S.

9

12

6

"

2

8 3

6 9 1

Having placed the farthings under farthings, pence under pence, &c., we add the column of farthings together, as in simple addition, and find the sum is 6, which is equal to 1d. and 2 far. over. Set the 2 far. un

der the column of farthings, and carry the 1d. to the column of pence. The sum of the pence is 24, which is equal to 2s. and nothing over. Place a cipher under the column of pence, and carry the 2s. to the column of shillings. The sum of the shillings is 29, which is equal to £1 and 9s. over. Write the 9s. under the column of shillings, and carry the £1 to the column of pounds. The sum of the pounds is 16, the whole of which is set down in the same manner as the left hand column is in simple addition. (Art. 25.) The answer is £16, 9s. Od. 2 far.

168. Hence we derive the following general

RULE FOR ADDING COMPOUND NUMBERS.

I. Write the numbers so that the same denominations shall stand under each other.

QUEST.-168. How are compound numbers written for addition? Which denomination is added first? When the sum of any column is found, what is to be done with it?

II. Beginning with the lowest denomination, find the sum of each column separately, and divide it by that number which it requires of the column added to make ONE of the next higher denomination. Set the remainder under the column, and carry the quotient to the next column.

III. Proceed in this manner with all the other denominations except the highest, whose entire sum is set down as in simple addition. (Art. 29.)

PROOF. The proof is the same as in Simple Addition. (Art. 28.)

OBS. The process of adding numbers of different denominations is called Compound Addition. It is the same as Simple Addition, except the method of carrying from one denomination to another.

2. What is the sum of £10, 6s. 7d.; £18, 12s. 10d., £5, 3s. 4d.? Ans. £34, 2s. 9d.

9. Add 7 lbs. 9 oz. 16 pwts. 10 grs.; 3 lbs. 10 oz. 8 pwts. 9 grs.; 8 lbs. 3 oz. 1 pwt. 4 grs.

10. An Englishman bought a carriage for £35, 12s.; a horse for £27, 8s. 10d.; a harness for £7, 16s. lld.: how much did he give for the whole?

QUEST. What is done with the last column? How is the operation proved? Obs. What is the process of adding compound numbers called? How does it differ from simple addition?