78. How many feet will be required to make 36 boxes whose outer dimensions are the same as the inner dimensions given in Ex. 77, the boards being of the same thickness; and what is the difference in the capacity of the two sets of boxes in cubic inches. Ans. 1001ft.; 144144c. in.

DECIMAL FRACTIONS.

155. A DECIMAL FRACTION is a fraction whose denominator is 10, 100, 1000, or 1 with one or more ciphers annexed. NOTE 1. The word decimal is derived from the Latin decem, which signifies ten.

NOTE 2. By the word decimal we usually mean a decimal fraction.

156. The denominator of a Common Fraction may be any number whatever. Every principle and every operation in Common Fractions is equally applicable to Decimals.

157. The denominator of a decimal fraction is not usually expressed, since it can be easily determined, it being 1 with as many ciphers annexed as there are figures in the given decimal.

158. A decimal fraction is distinguished from a whole number by a period, called the decimal point or separatrix, placed before the decimal; the first figure at the right of the point is tenths; the second, hundredths; the third, thousandths; etc.; thus, .6, .06 180, .006 TO, etc., the figures in the decimal decreasing in value from left to right, as in whole numbers (Art. 15).

155. What is a Decimal Fraction?

=

Decimal, from what derived? What is usually meant by the word decimal? 156. A Common Fraction, what is its denominator? Are the principles of common fractions applicable to decimals? 157. Is the denominator of a decimal usually expressed? 158. How is a decimal fraction distinguished from a whole number? What is the first figure at the right of the point? Second? Third?

159. Since whole numbers and decimal fractions both decrease by the same law from left to right, they may be expressed together in the same example, and numerated as in the following

NUMERATION TABLE.

c. Thousands,

➤ Hundreds, Tens,

- Units,

Hundred-Thousandths,

Hundredths,

Thousandths,

∞ Ten-Thousandths,

Tenths,

∞ Ten-Millionths,

→ Hundred-Millionths,

Millionths,

Ten-Billionths,

Etc., etc. Ten-Trillionths,

-Hundred-Billionths,

∞ Billionths,

→Trillionths,

160. A whole number and decimal fraction written together, as in the above table, form a mixed number. The integral part is numerated from the decimal point toward the left, and the fraction from the same point toward the right, each figure, both in the whole number and decimal, taking its name and value by its distance from the decimal point. Hence,

161. Moving the decimal point one place toward the right, multiplies the number by 10; moving the point two places multiplies the number by 100, etc. Also moving the point one place to the left, divides the number by 10; moving the point two

places divides by 100, etc.

162. In reading a decimal, we may give the name to each figure separately, or we may read it as we read a whole number, and give the name of the right-hand figure only; thus, the expres sion .23 may be read and 13, or it may be read, for

and 18

159. Read the Numeration Table. 160. What is a mixed number? Which way is the integral part numerated? Which way the decimal? What determines the name and value of a figure? 161. How does moving the decimal point to the right affect the value of a number? How moving it to the left?

162. In what two ways may a decimal be read? Illustrate.

163. To read a decimal fraction as we read a whole number, requires two numerations; first, from the decimal point, to determine the denominator, and second, towards the point, to determine the numerator; thus, to read the following: .3578692, first, to determine the denominator or name of the right-hand figure, beginning at the 3, say, tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, ten-millionths; and then, to determine the value of the numerator, or name of the left-hand figure considered as an integer, beginning at the 2, say, units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, and then read, three million, five hundred and seventy-eight thousand, six hundred and ninetytwo ten-millionths.

164. Since multiplying both terms of a fraction by the same number does not alter its value (Art. 147, a, Note 1), annexing one or more ciphers to a decimal does not affect its value; thus, fo 100 =100, etc.; i. e. .2 = .20 —.200, etc.

165. Prefixing a cipher to a decimal, i. e. inserting a cipher between the separatrix and a decimal figure, diminishes the value of that figure to its previous value; for it removes the figure one place further from the decimal point (Art. 161); thus, .3, but .03 only 13, which is but off.

What is the effect of prefixing two, three, or more ciphers to a decimal?

166. A common fraction is sometimes annexed to a decimal;

thus, .21. This is equivalent to the complex fraction

common fraction is never to be counted as a decimal place, but it is always a fraction of a unit of that order represented by the preceding decimal figure; thus, in .2341, the is half of a thousandth.

163. To read a decimal requires how many numerations? First, which way? For what purpose? Second, which way? For what? Illustrate. 164. How is the value of a decimal affected by annexing a cipher? Why? 165. How by prefixing a cipher? Why? 166. A common fraction annexed to a decimal, what is it? Illustrate.

NOTATION AND NUMERATION OF DECIMAL FRACTIONS.

167. Let the pupil express in figures the following numbers:

1. Fifty-two hundredths.

2. Four hundred and sixteen thousandths.

3. Three hundred and forty-two ten-thousandths.

Ans. .52.

Ans. .416.

NOTE 1. An ambiguity often arises in enunciating a whole number and a decimal in the same example; thus, .203 is two hundred and three thousandths, and 200.003 is two hundred, and three thousandths. This ambiguity may, however, be avoided by placing the word decimal before the fraction; thus, 200.003 may be read two hundred and decimal three thousandths.

NOTE 2. In decimals, as in whole numbers (Art. 16), ciphers are used to fill places that would otherwise be vacant.

4. Write the decimal six hundred and forty-one thousandths. 5. Decimal five hundred and eighteen ten-thousandths.

6. Eight hundred and decimal eight thousandths.

7. Six thousand and decimal six millionths.

8. Nine hundred and thirty and eight tenths.

9. Decimal two hundred and forty-six ten-millionths.

10. One thousand and decimal two hundred-thousandths.

11. Eleven and eleven ten-billionths.

12. Six hundred and sixteen and sixteen trillionths.

13. Ten thousand and decimal four ten-thousandths.

14. Decimal three hundred twenty-five thousand, four hundred and eighty-seven hundred-millionths.

168. Write the following numbers in words, or read them

167. What uncertainty often exists in reading mixed numbers?

How can

this ambiguity be avoided? For what are ciphers used in the notation of decimals?

NOTE 1. Addition, subtraction, multiplication, and division of decimal fractions are performed precisely as the same operations in whole numbers, no further explanation being necessary, except to determine the place of the decimal point in the several results.

NOTE 2. The proofs are the same as in whole numbers.

PROBLEM 1.

169. To add decimal fractions:

RULE. Place tenths under tenths, hundredths under hundredths, etc.; then add as in whole numbers, and place the point in the sum directly under the points in the numbers added.

7. Add 3.546, 44.8693, 2.8769, and 734.68723. 8. 872.34, 6789.3274, 22.987, and 346.42.

9. Add 3582.47, 62.84693, .47249, and 7.458.

10. Add five hundred and decimal six thousandths; forty-five millionths; eighty-four million and decimal twelve millionths; seventy thousandths; and decimal three hundred and fifty-four hundred-thousandths. Ans. 84000500.079597.

11. What is the sum of one thousand two hundred twenty-six

168. How are addition, subtraction, multiplication, and division of decimals performed? Proofs? 169. Rule for addition? The point, where placed?