5. To the quotient of 12, add the quotient of 51÷31. Ans. 28.

6. What number divided by will give 10 for a quotient? Ans. 6.

7. What number multiplied by will give 10 for a product?

Ans. 16.

8. What number is that, from which if you take of itself, the remainder will be 16?

Ans. 28.

9. What number is that, to which if you add of itself, the sum will be 20?

Ans. 14.

10. A boat is worth $900; a merchant owns of it, and sells of his share: what part has he left, and what is it worth? Ans. left, worth $375.

12

of my share for Ans. $10000.

11. I own of a ship, and sell $19444: what is the whole ship worth? 12. What part of 3 cents, is of 2 cents? 13. What part of 368, is 176 ?

1

13

14. What number,+++=35? Ans. 1887. 15. What number,+ of of 49,-1?

3 1

16. From the quotient of, subtract of §÷11.

1997 2664 Ans. 8.

3 16'

17. If I walk 2044 rods in of an hour, at that rate how far will I walk in 114 hr.?

18. What part of 11 feet, is 31 inches?

Ans. 8468 rd.

Ans..

19. Two men bought a bl. of flour; one paid $31, and the other $33: what part of it should each have? Ans. One 48, the other

20. A has $2400; g of his money, +$500, is what sum has B?

55

103

of B's: Ans. $1600.

21. John Jones divided his estate among 2 sons and 3 daughters, the latter sharing equally with each other. The younger son received $2200, which was of the share of the elder, his share being 1 of the whole estate: find the share of each daughter. Ans. $13563.

XII. DECIMAL FRACTIONS.

ART. 166. A common fraction (Art. 124), is one whose denominator may be any number; as, 2, 3, 4, &c.

A decimal fraction is one whose denominator is 10, or a number of 10's multiplied together; as, 100, 1000, &c.

EXPLANATION.-A common fraction arises from dividing a unit into any number of equal parts. A decimal fraction arises from dividing a unit by 10, or some multiple of 10 by itself; and its denominator is not usually expressed in figures.

ART. 167. If a unit be divided into 10 equal parts, each part will be 1 tenth; thus, 7 of 1.

If each tenth be divided into 10 equal parts, the unit will be divided into 100 equal parts, each part being 1 hundredth; thus, of 100.

If each hundredth be divided into 10 equal parts, the unit will be divided into 1000 parts, each part being 1 thousandth of the whole; thus, 10 of 100=1000•

A comparison of the fractions, 100, 1000, &c., shows that the value of each is one-tenth of that which precedes it; that is, they decrease in value tenfold.

ART. 168. The orders of WHOLE numbers decrease from left to right tenfold; thus, 1 hundred is one-tenth of 1 thousand; 1 ten is one-tenth of 1 hundred; and 1 unit is one-tenth of 1 ten: in like manner,

The orders may be continued from the place of units toward the right, by the same law of decrease; and,

The first order on the right of units will then express tenths of units, that is tenths; the next order, tenths of tenths, or hundredths; the next, tenths of hundredths, or thousandths; and so on:

Hence, if a point (.) be placed to separate units and tenths, may be written in the order of tenths; thus, .1, as 1 (unit) is written in the order of units: and,

3

10, 10, 16, &c., may be written, .2, .3, .4, &c.

REVIEW.-166. What is a Com. Fraction? A Decimal? 167. What is 1 tenth? 1 hundredth ? 1 thousandth? What the value of each?

177

Also, Th, To, 180, &c., may be written in the order of hundredths, thus: .01, .02, .03, &c.: here,

There being no tenths, place a cipher in the order of tenths.

As the orders decrease from left to right, in the same manner as in whole numbers, Decimal Fractions may be expressed in figures, without writing their denominators.

NOTE.-3 24 205

10, 100, 1000, and .3, .24, .205 are alike decimal fractions.

ART. 169. A figure in the first decimal place or order, expresses a fraction whose denominator is a unit with one cipher annexed; thus, .26: and,

A figure in the second decimal place, expresses a fraction whose denominator is a unit with two ciphers annexed; Thus, .02: also,

100

A figure in the third place expresses a fraction whose denominator is a unit with three ciphers annexed;

2

Thus, .002 = 1000; and so on: hence,

The denominator of a Decimal Fraction is 1, with as many ciphers annexed as there are decimal places in the numerator. ART. 170. TABLE OF DECIMAL PLACES OR ORDERS.

REM. The orders called ten-thousandths, hundred-thousandths, &c., are, really, tenths of thousandths, hundredths of thousandths, &c.

ART. 171. The table shows, that the value expressed by any figure in decimals, as well as in whole numbers, depends on the place from units which it occupies :

Thus, .2 expresses 2 tenths,

.02

.002

2 hundredths,

2 thousandths, Too

But, of is 180; 10 of 180 is Too, and so on. Hence, as the figures decrease from left to right tenfold, such fractions are called decimals, from the Latin decem, meaning ten.

ANNEXING AND PREFIXING CIPHERS.

ART. 172. Since the orders decrease, from left to right, in a tenfold ratio, and each cipher prefixed to a decimal removes it one place toward the right, therefore,

1

Each cipher prefixed to a Decimal, diminishes its value ten times.

Thus, prefix one cipher to the decimal .5, it becomes .05; prefix two, it becomes .005; three, .0005: each fraction expressing one-tenth of the preceding.

As the value expressed by a Decimal depends on the place from units which it occupies, and as its place is not changed by annexing ciphers, therefore,

Annexing ciphers to a Decimal does not alter its value. Thus, .2.20.200; that is, 2=20=200.

REM.-1. Annexing ciphers to a decimal, is equivalent to multiplying both terms of a fraction by the same number, which does not alter its value. Art. 135.

2. The effect of annexing and prefixing ciphers to decimals, is the reverse of that in whole numbers.

REVIEW.-168. How do the orders of whole numbers decrease from left to right? Give an example. How may the orders be continued from the place of units toward the right?

168. What will the order next to units express? The next order? The next? How may be written without the denominator??? TOO? TOO? TOO? In writing one-hundredth, why place a cipher in

3

the order of tenths?

169. What is the denominator of the fraction, expressed by a figure in the first decimal place? In the second? In the third? What is the denominator of a decimal?

ART. 173. As ten hundredths make one tenth, and ten tenths one unit, the decimals increasing from right to left as in whole numbers, Art. 8; therefore,

Whole numbers and decimals may be written in the same line, by placing the separating point between them; thus,

2 units and 2 hundredths are written 2.02

A whole number and a decimal, written together, is a mixed decimal number.

The decimal point (.)-called a separatrix-separates whole numbers and decimals.

ART. 174. If, in UNITED STATES MONEY, the dollar be taken as the unit, dimes will be tenths, cents hundredths, and mills thousandths of a dollar; hence,

Any sum in U. S. Money may be expressed in dollars and decimal fractions of a dollar. Thus,

$2 and 15 cts., are $2, and 15 hundredths; written, $2.15 $1 and 8 mills, are $1, and 8 thousandths; written, $1.008

DECIMAL NUMERATION AND NOTATION. ART. 175. TO READ DECIMALS; First, read the decimal as a whole number, then add the name of the right hand order. Thus, .6 is read six tenths.

.06

.204

two hundred and four thousandths.

REM.-1. In reading abstract mixed decimal numbers, for example, 400.035, the word units should be added after the whole number, to distinguish it from the decimal .435 Thus, 400.035 is

REVIEW.-170. Begin with tenths, and name the places of decimals in order. REM. What is the meaning of ten-thousandths?

171. On what does the value expressed by any figure depend? How do decimals decrease from left to right?

172. How does each 0 prefixed affect the value? Why? Give examples. How does annexing ciphers affect the value? Why? Give examples. 172. REM. 1. Annexing ciphers to a decimal is equivalent to what? 2. How does annexing and prefixing ciphers, compare with the same operation in whole numbers?