-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 202? 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 2 as much as B; in how many days can each do it?

Ans. A, 32 days; B, 24 days.

44. How many yards of cloth of a yard wide, are equal to 12 yards of a yard wide? Ans. 114.

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 Ans. $720. money had he?

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 42?

49. A boy lost of his kite string, and then added 30 feet, when it was just of its original length; what was the length at first? Ans. 100 feet. 50. Bought of a box of candles, and having used of them, sold the remainder for 1 of a dollar; how much would a box cost at the same rate? Ans. $57%.

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

2. A father left his eldest son of his estate, his youngest son of the remainder, and his daughter the remainder, who received $1723 less than the youngest son; what was the value of the estate? Ans. $2111433.

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.

10

3. Since o 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

Thousandths

Ten thousandths "

any order of decimals by one figure less than the corre

sponding 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 ?

be

dredths 100, of thousands 1000, and so on, a decimal may 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 7110 71 and 406 thousandths.

EXAMPLES FOR PRACTICE.

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

406

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

3. Write fifty-four, and fifty-four millionths.
4. Eighty-one, and 1 ten-thousandth.
5. One hundred, and 67 ten-thousandths.
6. Read the following numbers:

147. From the foregoing explanations and illustrations we derive the following important

PRINCIPLES OF DECIMAL NOTATION AND NUMERATION.

1. The value of any decimal figure depends upon its place from the decimal point: thus .3 is ten times .03.

2. Prefixing a cipher to a decimal decreases its value the same as dividing it by ten; thus, .03 is the value of .3.

3. Annexing a cipher to a decimal does not alter its value, since it does not change the place of the significant figures of the decimal; thus,, or .6, is the same as fo, or .60.

60

4. Decimals increase from right to left, and decrease from left to right, in a tenfold ratio; and therefore they may be added, subtracted, multiplied, and divided the same as whole numbers.

5. The denominator of a decimal, though never expressed, is always the unit, 1, with as many ciphers annexed as there are figures in the decimal.

6. To read decimals requires two numerations; first, from units, to find the name of the denominator, and second, towards units, to find the value of the numerator.

148. Having analyzed all the principles upon which the writing and reading of decimals depend, we will now present these principles in the form of rules.

RULE FOR DECIMAL NOTATION.

I. Write the decimal the same as a whole number, placing

What is the first principle of decimal notation? Second? Third? Fourth? Fifth? Sixth Rule for notation, first step?