required the quantity of sugar that should be received for 3§

tons.

48. When $8 are paid for 1 yards of broadcloth, how much must be given for 83 yards?

49. Gave $414 for 20

for 11 acres?

acres of land; what shall be given

MISCELLANEOUS QUESTIONS BY ANALYSIS.

1. Sold a small farm for $ 896.50; what was received for it of it? For of it?

2. Gave $17

What 37 barrels ?

For ff of it?

for 3 barrels of flour; what cost 1 barrel?

3. Sold a house for $3687; what sum was received for of it?

4. Bought 177 tons of nay for $ 1873; what is the cost of of a ton?

5. Bought a hogshead of molasses for $137; what cost of it? What cost? What cost?

6. When $37 are paid for 100 gallons of molasses, what cost of a gallon?

7. When 12 cents are paid for of a gallon of molasses, what will 481 gallons cost?

8. If of a barrel of flour cost $32, what will 6g barrels cost?

9. When $236 are paid for 11 acres, what will be paid for 2017 acres?

10. Paid in Liverpool 974£. for 3 bales of cloth; how many bales should be received for 10733£. ?

11. If 63 barrels of flour cost $48111, what will of a barrel cost?

12. If 3 pounds of coffee cost 34 cents, what sum must be paid for 74 pounds?

13. If 25 tons of hay cost $ 63, what will be the cost of 16 tons?

14. If a piece of land 3 rods square cost $171, what will be the cost of 4 square rods?

15. Paid $314 for 2cwt. of iron; required the sum to be paid for 689cwt.?

16. For 6 cords of wood, J. Holt paid $63; what sum must be paid for 18 cords?

17. Gave $243TT for 96 barrels of tar; what quantity could be purchased for $ 1000 ?

18. Paid $7888.30 for 83% acres of wild land; what sum did I pay for each acre, and what would be the cost of 7 acres?

19. Gave 132£. 12s. for 73 tons of starch; what cost 127 tons ?

20. For 17 days' work I paid $25.44; what should be paid for 89 days' labor?

21. Sold 77 bushels of apples for $7.28; what should I receive for 191 bushels?

22. Paid $4355.52 for 49 pieces of carpeting; what did 375 pieces cost?

23. If of of the cost of the Capitol at Washington was $300,000, what was the whole cost?

24. Purchased 7 must be paid for 19

thousand of boards for $135.80; what thousand?

25. My wood-pile contains 6 cords and 76 cubic feet. If I dispose of of it; what is the value of the remainder at 4 cents per cubic foot?

26. I have a field 30 rods square, and having sold 18 square rods to S. Brown, and 82 square rods to J. Smith, what part of the field remained unsold?

27. Bought 7T. 12cwt. 3qr. 18lb. of iron, and having sold 3T. 18cwt. 1qr. 20lb., what is the value of g of the remainder at 5 cents per lb. ?

28. Bought 37 tons of iron at $ 68.50 per ton, for of which. I paid in coffee at $8.50 per cwt., and for the remainder I paid cash. Required the amount of cash paid, and also the value of the coffee.

Ans. Cash, $

Value of the coffee,

29. A man having received a legacy of $7896, spent of it in speculations, and the remainder he put in the savings' bank, where it continued 15 years. It was then found that the sum deposited had doubled. Required the sum in the bank.

30. Bought a piece of broadcloth for $88, and sold of it to J. Smith, and of the remainder to O. Lake; what is the value of the part unsold?

31. A gentleman gave of his estate to his wife, of the remainder to his oldest son, and of what then remained to his daughter, who received $750; required the whole estate.

§ XX. DECIMAL FRACTIONS.

ART. 173. A DECIMAL FRACTION is a fraction whose denominator is 10, 100, 1000, &c.

ART. 174. Decimal fractions are commonly expressed by writing the numerator only with a point before it, called the decimal point or separatrix; thus,

99 100

66

.9 tenths.

.99 hundredths.

66 .999 thousandths.

1000

ART. 175. By examining the foregoing fractions, it will be seen that .9 can occupy only one place while it remains a proper fraction; .99, only two places; and 99% = .999, only three places; for, if their numerators are increased by.1, 10.01, 0.001, respectively, each fraction becomes a unit or whole number. Hence,

The first figure or place of any decimal on the right of the point is tenths, the second hundredths, the third thousandths, &c.

NOTE.When a decimal place has no significant figure, it must be filled with a cipher.

ART. 176. The denominator of .9 is 1 with one cipher annexed; the denominator of 9.99 is 1 with two ciphers annexed; the denominator of 1.999 is 1 with three ciphers annexed. Hence,

The denominator of a decimal fraction is 1 with as many ciphers annexed as the numerator has places.

ART. 177. Decimal fractions originate from dividing the unit, first, into 10 equal parts, and then each of these parts into 10 other equal parts, and so on indefinitely. Thus, 1÷10 .1; 16 ÷ 10 = 100 .01; ÷ 10

= TO
.001. Hence,

10

==

τουσ =

The unit in decimal fractions is divided into 10, 100, 1000, &c., equal parts.

QUESTIONS. Art. 173. What is a decimal fraction? — Art. 174. How are decimal fractions commonly expressed? - Art. 175. What is the first figure or place of any decimal? The second? The third ? &c. Why? What must be done when a decimal place has no significant figure to fill it? -Art. 176. What is the denominator of a decimal fraction?- Art. 177. How do decimal fractions originate?

ART. 178. If ciphers are placed on the left hand of decimal figures, they change their places, each cipher removing them one place to the right; thus, .3

.003 = Too. Hence,

=

1%, but .03 = 180, and

Ciphers placed on the left hand of decimals decrease their value in a tenfold proportion.

ART. 179. If ciphers are placed on the right hand of decimal figures, their places are not changed; thus, .3 =

1o, and

Ciphers placed on the right hand of decimals do not alter

their value.

NUMERATION OF DECIMAL FRACTIONS.

ART. 180. The relation of decimals to whole numbers and to each other, and also the names of their and places, may be learned from the following

different orders

QUESTIONS.- Art. 178. What effect have ciphers placed at the left hand

of decimals? Why? - Art. 179. What effect if placed at the right hand? Why? Art. 180. What may be learned from the table?

The preceding table consists of a whole number and decimal, which, taken together, are called a mixed number. The part on the left of the separatrix is the whole number, and that on the right the decimal. The value of the decimal is expressed in words thus: Two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninety-three billionths. And the mixed number thus: Seven millions six hundred fifty-four thousand three hundred twenty-one, and two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninety-three billionths.

From the table and explanations we have the following rule for numerating and reading decimals.

RULE. Beginning at the left hand, name the order of each figure of the given decimal, and then read it as in whole numbers, giving the name of the last order to the whole.

The pupil may write in words, or read orally, the following numbers:

NOTATION OF DECIMAL FRACTIONS.

ART. 181. By examining the decimal table, we perceive that tenths occupy the first place, hundredths the second, &c., and that each figure takes its value by its distance from the place of units; therefore, to write decimals, we have the following

RULE.

- Write the decimal figures in the place of the order denoted by their names, supplying with ciphers such places as have no significant figures.

The pupil may write in figures the following numbers: — 1. Three hundred seven and twenty-five hundredths. 2. Forty-seven and seven tenths.

QUESTIONS. Of what does it consist? What is the number called, when taken together? What is the part on the left hand of the separatrix? The part on the right? What is the value of the decimal? What is the value of the mixed number? What is the rule for numerating and reading decimals? Art. 181. Upon what does the value of a decimal figure depend? What is the rule for writing decimals?