7. What is the cost of laying two floors, each 16 ft. 8' by 12 ft. 6', at 18 cts. per sq. yd.? Ans. $8.33. 8. Find the price, at $24 per thousand ft., of 3 boards measuring as follows: 17 ft. 11' by 1 ft. 2', 19 ft. 4' by 1 ft. 11', and 22 ft. 8' by 1 ft. 9'. Ans. $2 343. 9. How many feet, board measure, in 6 planks 2 in. thick, each 25 ft. 9' long, 6' wide? (See Art. 213, Ex. 43, note.)

Ans. 154 ft.

ILL. EX. A plat of ground contains 65 ft. 0' 7"; its width is 6 ft. 4'; what is its length?

OPERATION.

6 ft. 4') 65 ft. 0' 7"(10 ft. 3' 3", Ans.

≈ 3"

;

6. ft. in 65 ft. = 10, 10 X 6 ft. 4' 63 ft. 4', which, subtracted from the dividend, gives a remainder of 1 ft. 8' 7" 20′ 7′′; 6 ft. in 20"-3′ 19" 3' X 6 ft. 4' 19′ 0′′, which 19"0" subtracted from 20' 7" leaves 0.0 19"; 6 ft. in 19"

ייד י1 ;ייד 1

Divide the highest the quotient

3" X 6 ft. 4'19". Hence the RULE FOR DIVISION OF Duodecimals. term in the dividend by the highest term in the divisor; will be the first term in the answer. Multiply the entire divisor by that term, and subtract the product from the dividend. Divide as before, and thus proceed till all the terms of the dividend are di vided. Should there be a remainder, it may be reduced to numbers of lower denominations and divided, or annexed to the quotient in a fractional form, having for its denominator the divisor expressed in units.

EXAMPLES.

1. Divide 54 ft. 7' 4" 6" by 4 ft. 1'.

Ans. 13 ft. 4' 6".

2. What is the width of a table, 4 feet 3' long, which contains 14 ft. 2'?

Ans. 3 ft. 4'.

3. How many feet of joist, 4 inches wide and 3 inches thick, allowing nothing for waste by sawing, can be made from a piece of timber 44 ft. 5' long, 1 ft. 3' wide, and 1 ft. 4' thick?

Ans. 888 ft. 4!

4. How many blocks of stone containing 1 ft. 11' 5" 6'" can be sawed from a block containing 11 ft. 8′ 9′′? Ans. 6 blocks. 5. What is the thickness of a block of granite, one of whose surfaces contains 75 ft. 10′ 8′′, and whose solid contents are 107 ft. 6' 1" 4""? Ans. 1 ft. 5'.

221. GENERAL REVIEW, No. 4.

1. Reduce 7 £ 3 s. 6 d. to farthings.

2. Reduce 4876 gr. to lb., etc., Troy.

3. 4 lb, 93, 7 3, 29, 8 gr. + 3 lb, 63, 23, 25, 8 gr. =?

4. 3 T. 1 cwt. 2

5. 1 m.

6 f. 16 r. 3 yd. 1 ft. 8 in. =?

6. Multiply 2 m. 30 ch. 12 1. by 8.

7. Multiply 5 y. 212 d. 10 h. 15 m. by 20. (3651 days to the

year.)

8. Divide 4 A. 3 R. 24 r. by 9.

9. In c. 1. how many feet?

10. What part of 1 A. is 3 R. 13 r. 5 ft.?

11. Reduce cu. yds. to feet and inches.

12. Reduce 8′ 53′′ to the fraction of a degree.

13. What cost 12 bu. 2 pks. of plums at $.06 a pint?

14. What cost 2 qts. 14 pts. oil at $1.12 per gallon?

15. Required the number of square feet in a garden 4 rds long and 1 rd. 15 ft. wide.

16. How many cu. ft. of space in a cellar measuring on the inside of the wall 5 yd. 1 ft. in length, 4 yds. in width, and 10 ft in depth?

17. What is the difference of time in two places whose longi tudes differ 7° 8' 4"?

18. When the difference of time is 3 h. 4 m. 6 s., what is the difference of longitude between two places?

19. How many days from Jan. 5, 1864, to March 3, 1865?

For changes, see Key.

DECIMAL FRACTIONS.

222. As by the Decimal System of representing numbers (Art. 23), each lower denomination is one tenth of the next higher, one ten being one tenth of one hundred, one unit one tenth of one ten, so one unit may be divided into ten equal parts, or tenths, one tenth into ten equal parts, or hundredths, etc. Thus we have fractional numbers descending from the unit by a scale of tens. Represented as common fractions, the denominators of these numbers are 10, 100 (102), 1000 (103), etc. Hence,

223. A Decimal Fraction is a fraction whose denominator is some integral power of ten.

224. Decimal Fractions are generally written like whole numbers; they are distinguished from whole numbers by having the decimal point placed at their left.

225. Decimal Fractions are read like whole numbers, the denomination being always given; this is determined by the place of the right hand figure in reference to the decimal point; thus, is read 5 tenths.

.5
.05 66 66

5 hundredths.

EXERCISES UPON THE TABLE.

1. Which place at the right of the decimal point is occupied by tenths? by thousandths? by millionths? by billionths? by trillionths? by hundredths? by ten-thousandths? by hundred-thousandths? by ten-millionths? by hundred-millionths? by hundred-billionths?

2. What denomination occupies the second place at the right of the point? the third? the fourth? the fifth? the sixth? the first? the seventh? the ninth? the eighth? the eleventh? the twelfth? the fifteenth?

227. To read decimal fractions, observe the following

RULE. Read the decimal fraction as if it were a whole number, giving it the denomination of the right hand figure.

Read the following, first as mixed numbers, then as improper

NOTE. The word units may be placed after the 7000 in Ex. 12, in reading it as a mixed number, to distinguish it from the 7 thousand tenthousandths in Ex. 13. Read thus in all similar cases of ambiguity.

Name the terms in the above examples, beginning at the left. Ans. (Ex. 1) 9 tenths; (Ex. 2) 4 tenths, 6 hundredths, 9 thousandths; etc.

228. To write Decimal Fractions, observe the following

RULE. Write the figures as in whole numbers, putting the decimal point so that the right hand figure shall be in the place of the denomination named in the decimal fraction, supplying vacant places, if there be any, with zeros.

EXERCISES.

Write the following in figures: —

1. Sixty-four hundredths.

2. Nine hundred forty-two thousandths.

3. Nine hundred forty-two ten-thousandths.

4. Eight thousand three hundred twenty-five ten thousandths. 5. Seventy-five hundred-thousandths.

6. Seven thousand five hundred-thousandths.

7. Fifty and four hundred eighty-two thousandths.

8. One hundred fifty-five millionths.

9. One hundred units, and fifty-five millionths.

10. Three hundred thousand eight billionths.

11. Three hundred thousand units, and eight billionths.

12. Forty million eight hundred four thousand and twentyfive, and three hundred four thousand eight hundred seventy-five hundred-millionths.

13. Seven million units, and one ten-millionth.

14. Seven million and one ten-millionths.

15. Thirty and six tenths.

16. Three hundred six tenths.

17. Three hundred seventy and

ten-thousandths.

18. Four hundred seven thousand eight hundred seventy-five and ten-billionths.

NOTE.-Zeros may be annexed or omitted at the right of a decimal fraction without altering the value of the fraction, for both numerator and denominator are thereby multiplied or divided by the same number. (Art 119, Prop. iii., iv.) Thus, .50 (5%)=.5 (†)=.500 (-50%).

FUNDAMENTAL OPERATIONS.

229. Decimal Fractions may be written and operated upon like common fractions, the same principles being applicable to both; but as they increase and decrease, like whole numbers, by a scale of tens, they can also be treated in all respects like whole numbers. Close attention must be given to placing the decimal point.