25. What cost 45 yards of muslin at 12 cents a yard?

Ans. $5.717.

26. What cost 967 tons of coal, at $12 per ton?

Ans. $1162.

27. What cost 1965 pounds of meat, at 63 cents a pound! Ans. $13.27.

28. What cost 327 yards of cloth, at 62 cents a yard ? Ans. $204.791.

29. What cost of a barrel of flour, if 1 barrel cost $8 f Ans. $6.

30. What cost of a yard of cloth, if 2 yards cost $18?

Ans. $8.05.

31. What cost 23 tons of hay, if 3 tons cost $48?

Ans. $40.

32. If 1 yard of cloth cost $11, what will 57 yards cost? Ans. $6.50.

33. If 2 tons of hay cost $30, what will 57 tons cost? Ans $693.

34. If 1 yard of cloth cost $8.014, what will 1414 yards cost? Ans. $117. 35. A has of $8560, which is 21 times B's money; how much money has B? Ans. $2568. 36. B has 725 acres of land; 23 times B's equals 3 times C's; how much has C? Ans. 507 acres. 37. If 4 be added to both terms of the fraction §, will the value of the fraction be increased or diminished?

38. If 4 be subtracted from both terms of the fraction §, will the value of the fraction be increased or diminished?

39. Will the value of the fraction § be increased or dimin ished if 4 be added to both terms? If 4 be subtracted from both terms?

40. Mr. Mann bought a tract of land for $6500, and sold of it to Mr. Chase, and of the remainder to Mr. Colburn; what was the cost of the remainder? Ans. $7429. days, and after paying his board of his earnings, he had $12.36 daily wages? Ans. 80 cents.

41. A man worked 25 and other expenses with remaining; what were his

SUPPLEMENTARY WRITTEN EXERCISES.

To be omitted unless otherwise directed.

42. Mr. Bowman gave 12 Lushel, for butter worth 18 did he get?

bushels of potatoes, at 30 cents a cents a pound; how much butter Ans. 203 pounds.

43. James Allen bought land for $10,260, and sold it so as to gain of the cost, the gain being $3 an acre; how many acres did he buy? Ans. 180 acres. 44. Mr. Robinson bought 100 yards of double width sheeting, he sold of it to Mr. Brown and of the remainder to Mr. Jones; what is the value of the remainder at 25 a yard?

cents Ans. $11.58.

45. If at a certain time of day a pole 63 feet long casts a shadow 36 feet long, what is the length of a pole which casts a shadow 33% feet long?

Ans. 57% feet. gallons, and has a pipe

46. A cistern has a capacity of 289 discharging into it 25 gallons per hour, and there is a leak through which it loses 54 gallons per hour; how long will it take to fill the cistern? Ans. 14 hours.

47. Mr. Williams, laying in a stock of goods, invests of his money in flour, in sugar and molasses, in tea and coffee, and the remainder, $540, in sundry other groceries; what was the whole amount invested?

Ans. $1620.

INTRODUCTION TO DECIMALS.

MENTAL EXERCISES.

1. If a unit is divided into 10 equal parts, what is one of these parts called?

2. If one-tenth is divided into 10 equal parts, what is one part called? 2 parts? 3 parts?

3. If is divided into 10 equal parts, what is one part called? parts? 4 parts?

4. What part of 4 is 4 tenths? of 5 is 5 tenths? of 5 tenths is 5 hundredths? of 6 tenths is 6 hundredths?

5. What is of 3 hundredths? What part of 3 hundredths is 3 thousandths? of 6 thousandths is 6 ten-thousandths?

6. In the number 4444, the 4 units is what part of the 4 tens? the 4 tens is what part of the 4 hundreds?

7. A term in units place denotes what part of the value which it does in tens place?

8. A term in tens place denotes what part of the value which it does in hundreds place?

9. By the same law, a term to the right of units place would denote what part of its value in units place? Ans. One-tenth.

10. How may we indicate that a term is at the right of units place? Ans. By placing a dot (.) at the right of units place.

11. How will you express 2 in this manner? Ans. 2.5.

12. What does the dot between the 2 and the 5 denote? Ans. That 2 is in units place and 5 in tenths place.

13. In the expression 11.1, the 1 at the right of units denotes what part of a unit?

14. In the expression 11.11, the second term at the right of the period denotes what part of a tenth? what part of a unit?

15. How may we then write tenths, hundredths, etc., without a denominator? Ans. By writing them at the right of units.

16. What shall we call the first place at the right of units? Sec ond place? Third? Fourth? Fifth?

17. Write without a denominator 2 tenths; 3 tenths; 4 tenths; 6 tenths; 8 tenths; 3 hundredths; 5 hundredths; 1 thousandth; 3 thousandths; 6 thousandths.

18. Read the following expressions: 4.6; 25.45; 26.34; 18.05; 25.235; 36.205; 46.008.

19. These fractions arising from the successive division by 10 are called decimal fractions. The term decimal is derived from decem, meaning ten

SECTION V.

DECIMAL FRACTIONS.

195. A Decimal Fraction is a number of tenths, hun dredths, thousandths, etc.

196. A Decimal Fraction is usually expressed by plac ing a point before the numerator and omitting the denominator; thus .5 expresses; .05 expresses 1; .005 10%, etc.

197. The Symbol of a decimal is the period, called the decimal point, or separatrix. It indicates the decimal, and separates decimals and integers.

198. The places at the right of the decimal point are called decimal places. The first place to the right of the decimal point is tenths, the second place is hundredths, etc. Thus, .245 expresses 2 tenths, 4 hundredths, and 5 thousandths.

199. The method of expressing decimal fractions arises from the method of notation for integers, and is a continua tion of it. This beautiful law, as applied to integers and fractions, is exhibited in the following

NOTATION AND NUMERATION TABLE.

200. A Decimal is a decimal fraction expressed by the method of decimal notation; as, .5, .25, etc.

201. A Pure Decimal is one which consists of decimal figures only; as, .345.

202. A Mixed Decimal is one which consists of an integer and a decimal; as 4.35.

203. A Complex Decimal is one which contains a com mon fraction at the right of the decimal; as, .451.

NOTES.-1. The first treatise upon decimals was written by Stevinus, and published in 1585.

2. The decimal point, Dr. Peacock thinks, was introduced by Napier, the Inventor of logarithms, in 1617, though De Morgan says that Richard Witt made as near an approach to it as Napier.

EXERCISES IN NUMERATION.

1. Read the decimal .45.

SOLUTION. This expresses 4 tenths and 5 hundredths, or since 4 tenths equals 40 hundredths, and 40 hundredths plus 5 hundredths equal 45 hundredths, it may also be read 45 hundredths. Hence the following rules:

Rule I.-Begin at tenths, and read the terms in order towards the right, giving each term its proper denomination. Rule II.-Read the decimal as a whole number, and give it the denomination of the last term at the right.

NOTE. In the second method we may determine the denominator by numerating from the decimal point, and the numerator by numerating towards the decimal point.

1

EXERCISES IN NOTATION.

1. Express 25 thousandths in the form of a decimal.

SOLUTION.-25 thousandths equal 20 thousandths plus 5 thousandths, or 2 hundredths and 5 thousandths; hence we write the 5 in the third or thousandths place, 2 in the second or hundredths place, and fill the vacant tenths place with a cipher, and we have .025. Hence the following

rules:

Rule I.-Place the decimal point, and then write each term so that it may express its proper denomination, using ciphers when necessary.