9. Forty-nine hundred ten thousandths. 10. Fifty-nine and sixty-seven ten thousandths. 11. Four hundred and sixty-nine ten thousandths. 12. Seventy-nine, and four hundred and fifteen millionths. 13. Sixty-seven, and two hundred and twenty-seven ten thousandths. 14. One hundred and five, and ninety-five ten millionths. UNITED STATES MONEY. 147. The denominations of United States Money correspond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar. EXAMPLES. 1. Express $37 and 26 cents and 5 mills, decimally. 3. Express $215 and 8 cents, decimally. 6. Express $15 6 cents 9 mills, decimally. ANNEXING AND PREFIXING CIPHERS. 148. Annexing a cipher is placing it on the right of a number. If a cipher is annexed to a decimal it makes one more decimal place, and, therefore, a cipher must also be added to the denominator (Art. 145). The numerator and denominator will therefore have been multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 118): hence, 147. If the denominations of Federal Money be expressed decimally, what is the unit? What part of a dollar is 1 dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill? Annexing ciphers to a decimal fraction does not alter its value. We may take as an example, .5 = ・ If we annex a cipher to the numerator, we must, at the same time, annex one to the denominator, which gives, Also, .4 40 100 .400 = 400 1000 = .40= .7.70.700.7000 = .70000. 149. Prefixing a cipher is placing it on the left of a number. If ciphers are prefixed to the numerator of a decimal fraction, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged while the denominator will be increased ten times for every cipher annexed; and hence, the value of the fraction will be diminished ten times for every cipher prefixed to the numerator (Art. 117). Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed. Take, for example, the fraction .3 100 = .03 . by prefixing one cipher. by prefixing two ciphers. .3 becomes 03 by prefixing three ciphers: in which the fraction is diminished ten times for every cipher prefixed. Does the annexing of 148. When is a cipher annexed to a number? ciphers to a decimal alter its value? Why not? become by annexing a cipher? What by annexing two ciphers? Three ciphers? What does 7 tenths become by annexing a cipher? By annexing two ciphers? By annexing three ciphers? 149. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the denominator? What effect, then, has it on the value of the fraction? ADDITION OF DECIMALS. 150. ADDITION of decimals is the operation of finding a single number which shall be equal in value to all the numbers added. It must be remembered, that only units of the same value can be added together. Therefore, in setting down decimal numbers for addition, figures expressing the same unit must be placed in the same column. The addition of decimals is then made in the same manner as that of whole numbers. 1. Find the sum of 87.06, 327.3 and .0567. Place the decimal points in the same column: this brings units of the same value in the same column then add as in whole numbers; hence, RULE.-I. Set down the numbers to be OPERATION. 87.06 327.3 .0567 414.4167 added so that figures of the same unit value shall stand in the same column. II. Add as in simple numbers, and point off in the sum, from the right hand, as many places for decimals as are equal to the greatest number of places in any of the numbers added. PROOF. The same as in simple numbers. EXAMPLES. 1. Add 6.035, 763.196, 445.3741, and 91.5754 together. 2. Add 465.103113, .78012, 1.34976, .3549, and 61.11. 3. Add 57.406 +97.004 + 4 + .6 + .06 + .3. 4. Add .0009 + 1.0436 + .4 + .05 + .047. 150. What is Addition? What parts of unity may be added together? How do you set down the numbers for addition? How will the decimal points fall? How do you then add? How many decimal places do you point off in the sum ? 5. Add .0049 + 49.0426 + 37.0410 + 360.0039. 7. Add 3.754, 47.5, .00857, 37.5 together. 12. Required the sum of 9 tens, 19 hundredths, 18 thousandths, 211 hundred-thousandths, and 19 millionths. 13. Find the sum of two, and twenty-five thousandths, five, and twenty-seven ten-thousandths, forty-seven, and one hundred twenty six-millionths, one hundred fifty, and seventeen ten-millionths. 14. Find the sum of three hundred twenty-seven thousandths, fifty-six ten-thousandths, four hundred, eighty-four millionths, and one thousand five hundred sixty hundred-millionths. 15. What is the sum of 5 hundredths, 27 thousandths, 476 hundred-thousandths, 190 ten-thousandths, and 1279 ten-mil lionths? 16. What is the sum of 25 dollars 12-cents 6 mills, 9 dollars 8 cents, 12 dollars 7 dimes 4 cents, 18 dollars 5 dimes 8 mills, and 20 dollars 9 mills? 17. What is the sum of 126 dollars 9 dimes, 420 dollars 75 cents 6 mills, 317 dollars 6 cents 1 mill, and 200 dollars 4 dimes 7 cents 3 mills? 18. A man bought 4 loads of hay, the first contained 1 ton 25 thousandths; the second, 997 thousandths of a ton; the third, 88 hundredths of a ton; and the fourth, 9876 ten-thousandths of a ton what was the entire weight of the four loads? 19. Paid for a span of horses, $225,50; for a carriage, $127,055, and for harness and robes, $75,28: what was the entire cost? 20. Bought a barrel of flour for $9,375; a cord of wood for $2,12; a barrel of apples for $1,62; and a quarter of beef for $6,09 what was the amount of my bill? 21. A farmer sold grain, as follows: wheat, for $296.75; corn, for $126,121; oats, for $97,371; rye, for $100,10; and barley, for $50,62: what was the amount of his sale? 22. A person made the following bill at a store; 5 yards of cloth, for $16,408; 2 hats, for $4,874; 4 pairs of shoes, for $6; 20 yards of calico, for $2, 378; and 12 skeins of silk, for $0,624: what was the amount of his bill? SUBTRACTION OF DECIMALS. 151. SUBTRACTION OF DECIMAL FRACTIONS is the operation of finding the difference between two decimal numbers. 1. From 6.304 to take .0563. 6.3040 .0563 6.2477 NOTE. In this example. a cipher is annexed to OPERATION. the minuend to make the number of decimal places equal to the number in the subtrahend. This does not alter the value of the minuend (Art. 148): hence, RULE. -1. Write the less number under the greater, so that figures of the same unit value shall fall in the same column. II. Subtract as in simple numbers, and point off the decimal places in the remainder, as in addition. PROOF.-Same as in simple numbers. EXAMPLES. 1. From 3278 take .0879. 2. From 291.10001 take 41.496. 3. From 10.00001 take .111111. 4. Required the difference between 57.49 and 5.768. 5. What is the difference between .3054 and 3.075? 6. Required the difference between 1745.3 and 173.45. 7. What is the difference between seven-tenths and 54 tenthousandths? 151. What is subtraction of decimal fractions? How do you set down the numbers for subtraction? How do you then subtract? How many decimal places do you point off in the remainder? |