17. Of 1728, what is .75? .334? .25? .125? .20? .625 ? 18. Multiply (2.108+.0074) by (12.684.465).

19. Multiply .01837 by 1000; .00145 by 100000; .6874 by 100; 5.375 by 10; 17.056 by 10000. What is the sum of the products? 20. What is the square of .0364? Of 20.75? Of 45.25 ? 21. What is the cube of 8.045? Of .0875? Of 67.375 ?

If, as in this example, the number of figures in the quotient is less than the number of decimal places to be pointed off, supply the deficiency by prefixing ciphers.

143. RULE.-Divide as in integers, and point off from the right of the quotient as many decimal places as the number in the dividend exceed those in the divisor.

NOTES.-1. If the divisor contains more decimal places than the dividend, before dividing make them equal by annexing ciphers to the dividend. If necessary to continue the division, more ciphers may be added.

2. If, after dividing all the figures of the dividend, there is a remainder, the division may be continued by annexing ciphers (120). The ciphers thus annexed must be regarded as decimal places of the dividend.

3. To divide a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the left as there are ciphers in the divisor, prefixing ciphers to the dividend, if necessary.

EXAMPLES.

144. 1. Divide three thousand four hundred and fifty-six hundred-thousandths by seventy-two hundredths.

2. Divide six, and twenty-five hundredths by twenty-five thousandths.

Divide

3. 35.88 by .345; by 4.16. 4. .89958 by .47; by .319. 5. 12.6 by 14.4; by .125. 6. 96.3 by .20; by .25.

7. 5.27 by 1.24; by .85.

8. .0648 by .00425; by .0288. 9. .31752 by .648; by .00384. 10. .1898 by .33; by .0048. 11. 85.2451 by 4.56%; by 8.274. 12. 45.367 by .016; by 1.080.

13. Divide 17.28 by .20; by .25; by .33; by .125; by .66. 14. 321 is .1784 of what number?

15. 186 is five hundredths of what number?

16. What must 37.375 be multiplied by to produce 448.5 ? 17. What must 631.25 be divided by to produce 250 ? 18. Divide 176.824 by 100; 876.35 by 1000; 17380.5 by 10000; 2886.57 by 10; 375 by 1000000. Find the sum of the quotients. 19. $12.52 is how many hundredths of $375.60 ? 20. $273.60 is how many thousandths of $1728 ?

REVIEW EXAMPLES.

145. 1. Add 16 hundredths, 137 millionths, 48 ten-thousandths, and 2016 ten-millionths.

2. Add 16.07, 240.127, 6.044, 27.1234.

3. Reduce to a decimal.

4. Reduce .083 to a fraction. 5. From 175 take 16.0831. 6. From 375.16 take 198.8888. 7. Change .8375 to a fraction. 8. Multiply 117.084 by 7.37. 9. Divide 43.75 by .0125. 10. Divide .06 by 1.66}. 11. 1.75 is of what number? 12. What is of $175.75 ? 13. What is .33 of 187.5 ?

14. What is .333 times 1728 ?

15. $3.75 is how many hundredths of $75?

16. $86.40 is how many hundredths of $2592 ?

17. 16.56 is .05 of what number?

18. What will 17280 bricks cost at $3.25 per M.?

19. If 278 barrels of pork cost $4378.50, what is the cost of 100 barrels ?

20. What cost 12456 feet of plank at $8.75 per M. ?

21. What is the value of 5 bbls. sugar, containing 312, 304, 301, 305, 304 pounds respectively, at 93 cents per pound?

22. A miller wishes to purchase an equal quantity of wheat, corn, and rye; he pays for wheat $2.224 a bushel; for corn, 981 cents a bushel; and for rye $1.16ğ a bushel. How many bushels of each can he buy for $92776.50 ?

23. Bought 280 cords of hard wood, at $6.75, and 790 cords of soft wood, at $3.624 per cord. Also, 750 bushels of corn, at 624 cents, and 925 bushels of oats, at 37 cents per bushel. What was paid for the whole, and what was the average price of wood per cord, and of grain per bushel?

24. Bought on contract 350 reams of foolscap paper, at $3.831 per ream, 451 reams of which were returned as unsuitable, and 275 reams of letter, at $2.67 per ream, 371 reams of which were rejected. How much was paid for the remainder?

25. A merchant paid for merchandise during the year $137618.75, and sold merchandise to the amount of $146347.87. What was the gain, if the net market value of the merchandise remaining unsold was $24378 ?

26. A quartermaster has $8345 on hand, and receives $4379.62 from each of six sales of property; he turns over to quartermaster A $2875.28, and pays $120 for corn. Upon being relieved from duty, he turns over to quartermaster B one-third of the residue, and divides the remainder equally among three others, C, D, and E. What was paid over to each?

27. Merchandise on hand, Jan. 1, 1879, $46312.85; merchandise sold during the year, $317829.32; merchandise purchased in the same time, $301449.72; merchandise on hand, Dec. 31, 1879, $61378.12. What was the net gain or loss?

28. A farmer sold land for $22.50 an acre, as follows: to A, 98% acres; to B, & of the number sold to A; and to C, the number sold to A and B both. How much land was sold, how much did B and C each receive, and what was the amount realized? 29. What are the prime factors of 2791?

30. At $28.75 per thousand, how many feet of lumber should be given for 2816 pounds of sugar at 7 cts. per pound?

31. Mr. A offered to sell his horse for more than it cost him, but afterward sold it for $504, which was less than his first asking price. How much did his horse cost him?

32. In England, during the year 1875, there were 147,730,313 tons of bituminous coal produced, 535,845 persons employed, and 1244 lives lost. How many tons of coal were produced to each person employed, how many tons to each life lost, and how many persons were employed per life lost?

33. In the anthracite coal mines of Pennsylvania, during the year 1875, there were 22,000,000 tons of coal produced, 69,589 persons employed, and 238 lives lost. How many tons of coal were produced to each employé, how many to each life lost, and how many persons were employed to each life lost?

34. In the Lehigh district of Pennsylvania, in 1878, there were 3,956,588 tons of coal produced, and 51,492 kegs of powder used. How many tons of coal were produced per pound of powder used, each keg containing 25 pounds?

35. A man bequeaths of his property to his wife, to his son, to his daughter, and the remainder, which is $36375, to charitable institutions. What is the amount bequeathed to each, and the total amount?

36. If a person traveling 33 miles per hour completes a journey in 16 hours, what time would it require if he traveled 44 miles per hour?

37. If I purchase two building lots for $3750 each, and sell one for more than it cost, and the other for .334 less, what is the gain or loss on the two lots?

38. A speculator sells two farms for $6000 each; how much does he gain or lose, if he sells one for .20 more than it cost, and the other for less than it cost?

39. A gentleman after spending of all his money, and of the remainder, had $177.50 remaining; how much had he at first?

40. A merchant bought 100 yards of cloth at $3.621 per yard, and 87 yards at $4.124 per yard. At what average price per yard should he sell the whole, to realize a profit equal to of the cost? 41. If 314 bushels of corn cost $17.50, how many bushels can be bought for $616?

42. *In 1864 there were 33908 miles of railroad in operation in the United States, and in 1878, 81841 miles. What was the average annual increase of mileage?

* This is exclusive of sidings. Mr. Poor, from whose Manual the above was taken, estimates that there are 19,500 miles of railroad in double, treble, and quadruple tracks, sidings, etc., making the total length of single track equal to 101,341 miles in 1878.

DENOMINATE

NUMBERS.

DEFINITIONS.

146. A Denominate Number is a concrete number (7), and may be either simple or compound.

147. A Simple Denominate Number refers to units of the same name and value; as 7 inches, 4 pounds.

148. A Compound Denominate Number refers to units of different names, but of the same nature; as 3 feet 6 inches, 4 pounds 8 ounces.

149. Denominate numbers are used to express divisions of time, weights, measures, and moneys of different countries.

150. The scale of integers and decimals is uniform; that of most denominate numbers is varying.

The moneys of the United States, Canada, France, Italy, Spain, Germany, Norway and Sweden, Denmark, Brazil, Japan, and of some other countries, and the metric system of weights and measures, have a uniform decimal scale.

DIVISIONS OF TIME.

151. The natural divisions of time are the year and the day, the other divisions being artificial. The year is the time in which the earth makes one revolution around the sun. The day is the time in which the earth makes one revolution on its axis.

152. The Solar Day is the interval between two consecutive returns of the sun to the meridian. On account of the varying motion of the earth around the sun, the solar days are of unequal length. For civil purposes in measuring time the average of all the days in the year is taken as the unit.

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