21. Paid 12 dollars for some cloth, at the rate of of a dollar a yard; how many yards was purchased?

22. If 8 oranges cost of 1 dollars, what will 1 orange 30st? Ans. of a dollar.

23. A man bought of a farm and sold what part of the whole farm did he sell? eft?

of his share; what part had he Ans. Sold 14.

24. If a barrel of sugar is worth 22 dollars, what is of

it worth?

25. How many hours will it take miles, if he travel 32 miles an hour? 26. How many barrels of apples dollars, at 1 dollars a barrel ?

6

Ans. 15 dollars. a man to travel 136 Ans. 41 hours. can be bought for 18 Ans. 15 barrels.

27. If the smaller of two fractions be, and the difference, what is the greater?

Ans. .

28. If the sum of two fractions is 1g, and one of them is

Ans. %.

Ans. 11.

20, what is the other? 29. If the dividend be, and the quotient, what is the divisor? 30. If the divisor be, and the quotient 31, what is the dividend? Ans. 2. 31. How many bushels of oats worth of a dollar a bushel, will pay for of a barrel of flour worth 9 dollars a barrel? Ans. 15 bushels. 32. At of a dollar a rod, what will it cost to digof of 5 rods of ditch? Ans. dollars. 33. If a man has 247 bushels of clover seed, and he sells of it, how much has he left? Ans. 67 bushels. 34. A man had 6 lots of land, each containing 37 acres; how many acres did they all contain?

35. If of a ton of hay can be bought for 15 dollars, what part of a ton can be bought for 1 dollar?

DECIMAL FRACTIONS.

NOTATION AND NUMERATION.

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

Decimal fractions are commonly called decimals.

10

Since To T880, &c., the denominators of decimal fractions increase and decrease in a tenfold ratio, the same as simple numbers.

104. In the formation of Decimals a unit is divided into 10 equal parts, called tenths; each of these tenths is divided into 10 other equal parts called hundredths; each of these hundredths into 10 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.

105. The Decimal sign (.) is always placed before decimal figures to distinguish them from integers. It is commonly called the decimal point. Thus,

And universally, the value of a figure in any decimal place is the value of the same figure in the next left band place.

106. The relation of decimals and integers to each oth

er is clearly shown by the following

107. Since the denominator of tenths is 10, of hundredths 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 al ways 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.

6. Write fifty-eight hundredths.

7. Write two hundred thirty-six thousandths.

8. Write one thousand three hundred twenty ten-thou

sandths.

Ans. .1320.

9. Write seven hundred thirty-two ten-thousandths.

Read the following decimals:

108. 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.

406

EXAMPLES FOR PRACTICE.

1. Write twenty-four, and four tenths.

Ans. 24.4.

2. Write thirty-two, and five hundredths,

3. Write seventy-six, and forty-six thousandths.

4. Write one hundred twelve, and one hundred ninety thousandths. Ans. 112.190.

5. Write sixty-three, and forty-four ten-thousandths. 6. Write seventy-five, and one hundred forty ten-thou sandths.

7. Write five, and 5 hundred thousanths. 8. Write sixteen, and 21 ten-thousandths. 9. Write eight, and 234 hundred thousand 10. Write forty, and 75 hundred thousandths.

109. 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 altar its value, since it does not change the place of the significant figures of the decimal; thus,, or, .6, is the same as 60 or .60.

1009

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.

110. 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 ciphers where necessarg to give each significant figure its true local value.

II. Place the decimal point before the first figure.