required the quantity of sugar that should be received for 3 Ans. 116cwt.

tons.

48. When $8 are paid for 14 yards of broadcloth, how much must be given for 8 yards? Ans. $49. 49. Gave $414 for 20 acres of land; what shall be given for 11 acres? Ans. $236.

MISCELLANEOUS QUESTIONS BY ANALYSIS.

1. Sold a small farm for $896.50; what was received for of it? For of it? For 19 of it? Ans. $815. 2. Gave $17 for 3 barrels of flour; what cost 1 barrel? What 37 barrels ? Ans. $213.033'5.

it?

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

4. Bought 17 tons of hay for $1873; of a ton?

5. Bought a hogshead of molasses for it? What cost? What cost?

Ans. $3226.12. what is the cost of Ans. $7.61,253. $137; what cost of Ans. $30.521.

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

Ι

Ans. $0.212.

7. When 12 cents are paid for of a gallon of molasses, what will 48 gallons cost? Ans. $16.0119.

8. If of a barrel of flour cost $33, what will 63 barrels cost? Ans. $48111.

9. When $236 are paid for 11 acres, what will be paid for 207 acres? Ans. $414. 10. Paid in Liverpool 974£. for 3 bales of cloth; how many bales should be received for 10734&£.? Ans. 33 bales.

11. If 6g barrels of flour cost $48111, what will of a barrel cost? Ans. $3.284. 12. If 33 pounds of coffee cost 34 cents, what sum must be paid for 744 pounds? Ans: $6.901. 13. If 2 tons of hay cost $63, what will be the cost of 164 tons? Ans. $38113. 14. If a piece of land 3 rods square cost $174, what will be the cost of 4 square rods? 15. Paid $314 for 2gcwt. of iron; required the sum to be paid for 689cwt. Ans. $76809.

Ans. $73

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

Ans. $170.10.

17. Gave $243 for 96 barrels of tar; what quantity could be purchased for $1000?, Ans 394133 barrels.

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? Ans. $660.80.

19. Gave 132£. 12s. for 71⁄2 tons of starch; what cost 127 tons? Ans. 224£. 5s. 20. For 17 days' work I paid $25.44; what should be paid for 89 days' labor?

Ans. $128.64. what should I re

Ans. $19.12.

21. Sold 7 bushels of apples for $7.28; ceive for 1911 bushels? 22. Paid $4355.52 for 499 pieces of carpeting; what did 37 pieces cost?

Ans. $3294.72.

23. If of of the cost of the Capitol at Washington was $300,000, what was the whole cost? Ans. $2,000,000. 24. Purchased 7 thousand of boards for $135.80; what must be paid for 193 thousand? Ans. $359.45. 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? Ans. $23.1434.

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?

Ans. 8.

27. Bought 7T. 12cwt. 3qr. 18lb. of iron, and having sold 3T. 18cwt. 1qr. 20lb., what is the value of of the remainder at 5 cents per lb. ? Ans. $242.594.

28. Bought 37 tons of iron at $68.50 per ton, for 2 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, $633.62; Value of the coffee, $1900.87. 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.

19

Ans. $3948. 30. Bought a piece of broadcloth for $88, and sold of it to J. Smith, and of the remainder to Q. Lake; what is the value of the part unsold? Ans. $37.499. 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.

Ans. $12,000.

32. From an acre of land I sold two house-lots, each 100 feet square; what is the value of the remainder, at 8 cents per square foot? Ans. $1884.80.

§ XIX. DECIMAL FRACTIONS.

ART. 175. A DECIMAL FRACTION is a fraction whose denominator is ten, or the product of a number of tens.

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

190909%

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 = .999, only three places; for, if their numerators are increased by .001, respectively, each fraction becomes a unit or whole number. Hence,

=

.01, Tobo

=

= .1,

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.

= 100

ART. 176. The denominator of 1% .9 is 1 with one cipher annexed; the denominator of 99% .99 is 1 with two ciphers annexed; the denominator of 999 = .999 is 1 with three ciphers annexed. Hence,

1000

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; 10 = τόσ =. :.01; 10 = τούσ = .001. Hence,

=

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

QUESTIONS. - Art. 175. What is a decimal fraction? How are decimal fractions commonly expressed? 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:

Too. Hence,

3

=

1%, but .03

=

and .003

=

Ciphers placed on the left hand of decimals decrease their value in a ten-fold proportion.

ART. 179. If ciphers are placed on the right hand of decimal figures, their places are not changed; thus, .3, and .30 30 1.3. Hence,

=

100

=

=

Ciphers placed on the right hand of decimals do not alter their value.

NUMERATION OF DECIMAL FRACTIONS.

different

orders

and

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

places,

QUESTIONS.

Decimals.

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 decimal part is numerated from the left to the right, and its value 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 reading decimals:

RULE. - Read the decimal as though it were a whole number, adding the name of the right-hand order.

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 as though it were a whole number, 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 reading decimals?-Art. 181. Upon what does the value of a decimal figure depend? What is the rule for writing decimals?