Q. How may you obtain the circumference of a circle when you know its diameter? What is the length of the earth's diameter? What is its circumference ?

6. It is computed that a man travels on foot and on level ground about 314 miles per day. How many miles at this rate would a man travel in a year which contains 365 days? Ans.

Q. What is the computed rate of travel of a man on foot? How many days and fractions of a day in a year?

7. There are 544,743 white persons above 20 years of age, in the 26 States, who cannot read and write. What would be the cost of 3 years' schooling of the whole number, at 14 dollars for each person per year ? Ans.

Q. How many white persons in the 26 States who cannot read and write?

OF DECIMAL FRACTIONS.

88. When the unit 1 is divided into tenths, hundredths, thousandths, &c., the resulting fractions are called Decimal Fractions.

12

Thus, 10, 100, 100, 1000, 80, &c., are decimal fractions.

The denominator of a decimal fraction is not usually expressed, and to distinguish the numerator, which is alone written, from a whole number, a point, . called the decimal point, is placed on its left.

Thus, is written .2

105 1000

When the denominator of decimal fractions is expressed, it is always 1, with as many ciphers to the right as there are figures in the numerator.

Q. What are decimal fractions? Are they usually expressed like other fractions? Is the denominator usually expressed? How is the numerator distinguished from a whole number? What is the point called? Where is it placed? How would you express & decimally?

Is a decimal fraction? Why not? Into how many equal parts

is the unit divided in? Is a decimal fraction? Why? How is the unit divided in it? How is it expressed decimally? What is the denominator of a decimal fraction?

89. The place next to the decimal point is called the tenths place; the next to the right the hundredths place; the next the thousandths, &c.; so that the same number decreases in a tenfold proportion as we proceed from the decimal point to the right.

Thus, is written .4, the 4 being in the tenths place, and is 4 tenths.

To is written .04, the 4 being in the hundredths place, and is 4 hundredths.

Too is written .004, the 4 being in the thousandths place, and is 4 thousandths.

Todoo is written .0004, the four being in the ten thousandths place, and is 4 ten thousandths.

Q. What is the place next to the decimal point called? The next? The next? How does the value of the same number vary as we proceed to the right in decimals? How is 4 written decimally? What place does the 4 occupy? What is .4? How is written decimally? What place does the 4 take? How is written decimally? What place does the 4 occupy? Is .4 greater or less than .04? How many times greater? Is .04 greater or less than .004? How many times greater? How many times is .004 less than .4?

4

90. Decimals are numerated from the decimal point to the right. Thus:

The first three, four hundred and seventy-six thousandths. The first four, four thousand seven hundred and sixtynine ten thousandths.

The first five, forty-seven thousand six hundred and ninety-eight hundred thousandths.

The whole number is four hundred and seventy-six thousand nine hundred and eighty-five millionths.

A mixed number is a whole number and a decimal, and as whole numbers decrease from the left to the right (Art. 8), as has just been shown to be the case with decimal fractions, it follows that a mixed number may be written decimally by placing the decimal part to the right of the whole number, with the decimal point between them.

is written 4.7

Thus, four and seven tenths
four and seven hundreths

four and seven thousandths

66

"6

4.07
4.007

The

Q. How are decimals numerated? Read the decimal table? first figure is what? The two first? The whole number? Read the fraction .014? .1004? .50075? .170403? Do decimals decrease or increase from the left to the right? How is it with whole numbers? What is a mixed number? How may a mixed number be written decimally? Write five and four ten thousandths. Seventeen and three millionths.

5

50

91. Since 10, 100, 1000, 10000, &c., are fractions of equal value (Art. 69), their equivalent decimals, .5, .50, .500, .5000, &c., must also be of equal value. Hence, placing ciphers on the right of a decimal fraction, does not alter its value.

5

Again, is ten times greater than 5, one hundred times greater than 100, and one thousand times greater than 1000 (Art. 67); their equivalent decimals must vary in the same proportion. That is,

.5 is ten times greater than .05.

.5 is one hundred times greater than .005.

.5 is one thousand times greater than .0005, &c. Hence, placing a cipher on the left of a decimal frac tion diminishes its value tenfold.

.0005 is ten times less than .005, one hundred times less than .05, and one thousand times less than .5.

Q. What effect has placing ciphers on the right of a decimal? Why is not its value changed? Is of the same value as 50? Why? How do .5 and .50 compare in value? If a cipher be placed on the left

5

of a decimal, is the value of the fraction changed? Why? Is less or greater than 5? Why greater? How much greater? What are their equivalent decimals? How do .5 and .05 compare in value? How do and compare in value? What are their equivalent T decimals? How do .5 and .005 compare in value? If one cipher be placed on the left of a decimal, how many times is its value diminished? If two? If three? If four? How do the following decimals compare with each other: .1, .01, .001, .0001, .00001? How do the following decimals compare with each other in value: .1, .10, .100, .1000, .10000?

OF ADDITION OF DECIMAL FRACTIONS.

92. Addition of Decimal Fractions is as readily performed as addition of whole numbers. The only difficulty consists in placing the decimal point.

Since in whole numbers units are placed under units, tens under tens, &c. (Art. 17), so in addition of decimal fractions, tenths are placed under tenths, hundredths under hundredths, thousandths under thousandths, &c., and the sum is then found as in whole numbers.

OPERATION.

12.04

5.307

.0765

17.4235

Thus, add 12.04, 5.307 and .0765 together. In this example the tenths are placed under tenths, hundredths under hundredths, &c., which brings the decimal points directly under each other, and we then add as in whole numbers. The sum will evidently contain tenths only, if only tenths are to be added; it will contain tenths and hundredths, if tenths and hundredths are to be added, &c. The example taken containing tenths, hundredths, thousandths and tens of thousandths, the sum must contain them also. We have then the following

RULE.

I. Place the numbers one under the other so that tenths fall under tenths, hundredths under hundredths, &c.

II. Add as in addition of simple numbers, pointing off as many decimal places from the right hand as there are in the number which has the greatest number of decimal places among the given numbers.

Q. What is the only difficulty in addition of decimals? How are the decimals written down? How many decimal places in the sum ? What is the rule for adding decimals?

EXAMPLES.

1. Add 340.10, 1.004, .3, .0047, together.

Ans. 341.4087.

2. Add 7.10405, 30.04, .7632 and 104.300768 together. Ans. 142.208018.

3. Add 3 tenths, 3 hundredths, 33 thousandths, 333 ten thousandths together. Ans. .3963. 4. Add 4 tenths, 44 hundredths, 444 thousandths, 44 ten thousandths, and 4 millionths. Ans. 1.288404. 5. Add seventeen and six hundredths, thirty-seven thousand four hundred and ninety-one hundred thousandths, 129 and 3 hundredths. Ans. 146.46491.

SUBTRACTION OF DECIMAL FRACTIONS.

93. In subtraction of Decimal Fractions we have to find the difference between two decimal numbers. It differs but little from subtraction of whole numbers.

OPERATION.

25.250

12.379

Let it be required to take 12.379 from 25.25. Placing units under units, and tens under tens in the entire part of the two numbers, and tenths under tenths, hundredths under hundredths, &c., in the decimal part, we find there are no thousandths in the minuend under which to place the 9 thousandths of the subtrahend. But as placing a cipher on the right of a decimal does not change its value (Art. 91), we place a cipher in the thousandths place of the minuend, and then subtracting as in whole numbers, we have for the remainder 12.871. We have the following

RULE.

12.871

I. Set the less number under the greater, placing tenths under tenths, hundredths under hundredths, &c., and when the number of decimal places is unequal in the two num