per head; afterward bought of the whole number at $12

27. Bought 320 sheep at $2 435 at $17 per head; then sold per head, and the remainder at $2§; did I gain or lose, and how much? Ans. Lost $44.

28. If 5 be added to both terms of the fraction, will its value be increased or diminished? Ans. Increased 184 29. If 5 be added to both terms of the fraction &, will its value be increased or diminished? Ans. Diminished

30. How many times can a bottle holding of 3 of a gallon, be filled from a demijohn containing of 13 gallons?

Ans. 7.

31. Bought of 71⁄2 cords of wood for 4 of $32; how much did 1 cord cost?

32. Purchased 728 pounds of candles at 163 cents a pound; had they been purchased for 33 cents less a pound, how many pounds could have been purchased for the same money? Ans. 9531.

33. What number, divided by 13, will give a quotient of 94? Ans. 123. 34. The product of two numbers is 6, and one of them is 1846; what is the other? Ans. 3. 35. A stone mason worked 113 days, and after paying his board and other expenses with of his earnings, he had $20 left; how much did he receive a day?

36. If of 4 tons of coal cost $53, what will of 2 tons cost? Ans. $5. 37. In an orchard & of the trees are apple trees, to peach trees, and the remainder are pear trees, which are 20 more than

of the whole; how many trees in the orchard? Ans. 800. 38. A man gave 63 pounds of butter, at 12 cents a pound, for of a gallon of oil; how much was the oil worth a galIon ? Ans. 100 cents.

39. A gentleman, having 271 acres of land, sold of it, and gave of it to his son; what was the value of the remainder, at $57 per acre? Ans. $4577.

40. A horse and wagon cost $270; the horse cost 14 times as much as the wagon; what was the cost of the wagon? 41. What number taken from 2 times 12 will leave 203? Ans. 11. 42. A merchant bought a cargo of flour for $21731, and sold it for of the cost, thereby losing of a dollar per barrel; how many barrels did he purchase? Ans. 126. 43. A and B can do a piece of work in 14 days; A can do as much as B; in how many days can each do it?

Ans. A, 324 days; B, 241 days.

44. How many yards of cloth of a yard wide, are equal to 12 yards of a yard wide? Ans. 11.

45. A, B, and C can do a piece of work in 5 days; B and C can do it in 8 days; in what time can A do it?

46. A man put his money into 4 packages; in the first he put, in the second, in the third, and in the fourth the remainder, which was $24 more than of the whole; how much money had he? Ans. $720.

47. If $74 will buy 34 cords of wood, how many cords can be bought for $10? Ans. 4. 48. How many times is of of 27 contained in of of 423?

49. A boy lost when it was just at first?

of his kite string, and then added 30 feet, of its original length; what was the length Ans. 100 feet.

50. Bought of a box of candles, and having used 7 of them, sold the remainder for 1 of a dollar; a box cost at the same rate?

how much would

Ans. $549. 51. A post stands in the mud, in the water, and 21 feet above the water; what is its length?

52. A father left his eldest son of his estate, his youngest son of the remainder, and his daughter the remainder, who received $1723ğ less than the youngest son; what was the value of the estate? Ans. $2111413.

DECIMAL FRACTIONS.

143. Decimal Fractions are fractions which have for their denominator 10, 100, 1000, or 1 with any number of ciphers annexed.

NOTES.

1. The word decimal is derived from the Latin decem, which signifies ten.

2. Decimal fractions are commonly called decimals.

3. Since o

10

100, 100 = 1000, &c., the denominators of decimal fractions increase and decrease in a tenfold ratio, the same as simple numbers.

DECIMAL NOTATION AND NUMERATION.

144. Common Fractions are the common divisions of a unit into any number of equal parts, as into halves, fifths, twenty-fourths, &c.

Decimal Fractions are the decimal divisions of a unit, thus: A unit is divided into ten equal parts, called tenths; each of these tenths is divided into ten other equal parts called hundredths; each of these hundredths into ten other equal parts, called thousandths; and so on. Since the denominators of decimal fractions increase and decrease by the scale of 10, the same as simple numbers, in writing decimals the denominators may be omitted.

In simple numbers, the unit, 1, is the starting point of notation and numeration; and so also is it in decimals. We extend the scale of notation to the left of units' place in writing integers, and to the right of units' place in writing decimals. Thus, the first place at the left of units is tens, and the first place at the right of units is tenths; the second place at the left is hundreds, and the second place at the right is hundredths; the third place at the left is thousands, and the third place at the right is thousandths; and so on.

What are decimal fractions? How do they differ from common fractions? How are they written ?

The Decimal Point is a period (.), which must always be placed before or at the left hand of the decimal. Thus,

NOTE. The decimal point is also called the Separatrix.

This is a

correct name for it only when it stands between the integral and decimal parts of the same number.

And universally, the value of a figure in any decimal place is the value of the same figure in the next left hand place. The relation of decimals and integers to each other is clearly shown by the following

And

Ten thousandths "

any order of decimals by one figure less than the corresponding order of integers.

145. Since the denominator of tenths is 10, of hun

What is the decimal point? What is it sometimes called? What is the value of a figure in any decimal place?

dredths 100, of thousands 1000, and so on, a decimal may be expressed by writing the numerator only; but in this case the numerator or decimal must always contain as many decimal places as are equal to the number of ciphers in the denominator; and the denominator of a decimal will always be the unit, 1, with as many ciphers annexed as are equal to the number of figures in the decimal or numerator.

The decimal point must never be omitted.

EXAMPLES FOR PRACTICE.

1. Express in figures thirty-eight hundredths. 2. Write seven tenths.

3. Write three hundred twenty-five thousandths. 4. Write four hundredths.

5. Write sixteen thousandths.

Ans. .04.

6. Write seventy-four hundred-thousandths. Ans. .00074. 7. Write seven hundred forty-five millionths.

8. Write four thousand two hundred thirty-two ten-thousandths.

9. Write five hundred thousand millionths.

10. Read the following decimals :

NOTE. To read a decimal, we first numerate from left to right, and the name of the right hand figure is the name of the denominator. We then numerate from right to left, as in whole numbers, to read the

numerator.

146. A mixed number is a number consisting of integers and decimals; thus, 71.406 consists of the integral part, 71, and the decimal part, .406; it is read the same as 71, 71 and 406 thousandths.

EXAMPLES FOR PRACTICE.

1. Write eighteen, and twenty-seven thousandths.
2. Write four hundred, and nineteen ten-millionths.

How many decimal places must there be to express any decimal ›