70. Writing Decimals. Although we write a decimal fraction without its denominator, the denominator is shown by the decimal point and is indicated in reading. " nine Thus 0.9 has no printed denominator, but when we say tenths we indicate the fact that the denominator is ten as well as state that the numerator is nine. Therefore, write only the numerator of the decimal, inserting such zeros as are necessary to indicate the denominator, and place the decimal point before the tenths. Thus, to write in figures two hundred one hundred-thousandths, we write 201, and prefix two zeros, preceded by a decimal point, so that the last figure is in hundred-thousandths' place. We then have 0.00201. Write in figures : EXERCISE 28 1. Seven tenths; one tenth; five tenths; nine tenths. 2. Twelve hundredths; fifty-five hundredths; nine hundredths; one hundredth; twenty hundredths. 3. One hundred twenty-five thousandths; forty-seven thousandths; ten thousandths; five thousandths. 4. One thousand two hundred fifty ten-thousandths; seven hundred sixty-five ten-thousandths. 5. 5 and 63 thousandths; 7 and 70 thousandths. 6. 1352 and 135 thousandths; 60 and 60 thousandths. 7. 1000 and 1 thousandth; 1000 and 10 thousandths. Express in figures as dollars and cents: 8. Two and seventy-five hundredths dollars. 10. One hundred and eight hundredths dollars. 11. One thousand three and three hundredths dollars. 12. Two hundred four and two hundredths dollars. 71. Reduction of Decimals. Since decimals are only a special form of fractions, we may reduce them as we reduce common fractions. For example, since f = 5% (§ 65), therefore 0.5 = 0.50. 72. Annexing Zeros. Because, as just shown, 0.5 0.50, therefore Annexing zeros to a decimal does not change its value. 73. Similar Decimals. Decimals that have the same number of decimal places are called similar decimals. Thus 0.75 and 0.25 are similar decimals, and so are 0.150 and 0.275; but 0.15 and 0.275 are dissimilar decimals. 74. Reduction of Dissimilar Decimals to Similar Decimals. Since we may give any decimals the same denominators (understood, not written) by annexing or cutting off zeros, Therefore, to reduce dissimilar decimals to similar decimals, give them the same number of decimal places by annexing or cutting off zeros. Thus 0.4, 0.72, and 0.125 may all be reduced to thousandths as follows: 0.400, 0.720, and 0.125. Similarly, 0.40 and 0.600 are equal to 0.4 and 0.6. 75. Reduction of Decimals to Common Fractions. We have 0.75%, which may be reduced to 3 (§ 65). In a similar way any decimal may be expressed as a common fraction. Therefore, to reduce a decimal to a common fraction, omit the decimal point, write the denominator of the decimal, and then reduce the common fraction to lowest terms. In a case like that of 0.331 we proceed as follows: Reduce to common fractions in lowest terms or to mixed 76. Reduction of Common Fractions to Decimals. Since a common fraction indicates division, we may divide the numerator by the denominator and thus reduce the fraction to the decimal form. For example, consider the fraction 3. We have found that we may annex zeros after the decimal point without changing the value; that is, 3 = 3.00. We may now divide exactly as in United States money, and the result is 0.75. That this is allowable may be seen from the following: 0.75 3.0 means 30 tenths, and 30 tenths ÷ 4 = 7 tenths with remainder 2 tenths. But 2 tenths 20 hundredths (1% = 10%), 4)3.00 and 20 hundredths ÷ 4 = 5 hundredths. Therefore = 0.75. In the same way, 0.13 = 0.175. Therefore, to reduce a common fraction to a decimal, place a decimal point after the numerator, and divide by the denominator as in United States money. A common fraction cannot always be reduced to an exact decimal. Thus = 0.31 = 0.33} = 0.333. It is customary to carry the division as far as necessary and then write a common fraction or the plus sign to indicate that the work is not complete; thus = 0.3331, or 0.333 +. Often this is written 0.333, using as many places as necessary. No common fraction equals an exact decimal if the denominator, when the fraction is reduced to lowest terms, contains other prime factors than 2 and 5. EXERCISE 31 Reduce to decimals, carrying the reduction only to thousandths in case of inexact decimals: 77. United States Money. The system of United States money is arranged on the decimal plan. Thus, This table, like the common operations with United States money, is already familiar to the pupil, and is inserted only for reference. The unit of United States money is the dollar, and the dimes and cents are written as decimal parts of the dollar. Thus $2.48 means 2 dollars and 48 hundredths of a dollar, or 2 dollars and 48 cents, the cent being one hundredth of a dollar. Similarly, $2.08 means 2 dollars and 8 hundredths of a dollar, or 2 dollars and 8 cents. 78. Coins. The coins of the United States, as issued at present, are as follows: 79. Paper Money.' Bank bills, United States treasury notes, and gold and silver certificates are largely used in place of coins. These are issued in denominations of $1, $2, $5, $10, $20, $50, $100, $500, $1000, $5000, and $10,000. Formerly paper money of smaller denominations was issued, as well as the gold dollar, the silver 20-cent, 5-cent, and 3-cent pieces, the nickel 3-cent piece, the bronze 2-cent piece, and the copper half cent. These are no longer issued. The mill was never coined. |