multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 143): hence, Annexing ciphers to a decimal does not alter its value. Take as an example, .5 = If we annex a cipher to the decimal, we at the same time annex one to the denominator; thus, 196. PREFIXING a cipher is placing it on the left of a number. If ciphers are prefixed to a decimal, the same number of ciphers must be annexed to the denominator; for, the denominator must always contain as many ciphers as there are decimal places in the numerator. 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 decimal (Art. 142): hence, Prefixing ciphers to a decimal diminishes its value ten times for every cipher prefixed. Take, for example, the decimal .3 = 13%. .3 becomes .03 = 130 by prefixing one cipher; .3 becomes .003 = 100 .3 becomes .0003 103 = by prefixing two ciphers; in which the fraction is diminished ten times for every cipher prefixed. 196. 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 decimal? 197. Analysis of decimals. Analyze 62.25. It is composed of 6 tens, 2 units, 2 tenths, and 5 hundredths; or it is composed of 62 units and 25 hnndredths; or of 622 tenths and 5 hundredths; or 6225 hundredths. NOTE -Let it be remembered that a fractional unit of any one place is of the unit of the place next on the left, or of the unit which is 2 places to the left, or 100 of the fractional unit, which is three places to the left. ADDITION OF DECIMALS. 198. ADDITION OF DECIMALS is the operation of finding the sum of two or more decimal numbers. It must be remembered, that only units of the same value can be added together. Therefore, in setting down decimal numbers for addition, figures having the same unit value 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. ANALYSIS.-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, OPERATION. 87.06 327.3 .0567 414.4167 Rule. I. Set down the numbers to be added so that figures of the same unit value shall stand in the same column : 198. What is addition? What parts of a unit may be added to gether? 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? II. Add as in simple numbers, and point off in the sum, from the right hand, a number of places for decimals 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. 5. Add .0049 + 49.0426 + 37.0410 + 360.0039. 12. Required the sum of 9 tens, 19 hundredths, 18 thon sandths, 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 tenmillionths. 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 dol lars & cents, 12 dollars 7 dimes 4 cents, 18 dollars 5 dimes & 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 tenthousandths 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 $126.123; oats, for $97.371; rye, for $100.10; and barley, for $50.624: 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 $4.87; 4 pairs of shoes, for $6; 20 yards of calico, for $2.378; aud 12 skeins of silk, for $0.624 what was the amount of his bill? : 23. What is the sum of $99 87 cents 5 mills; $87 6 cents 18 mills; $59 42 cents 20 mills; $60 49 cents 16 mills; and $21 29 cents 13 mills? 24. What is the sum of $97 4 mills; $25 19 mills; $65 95 cents 6 mills; $4 874 cents 3 mills; and $55 14 cents 9 mills? 23. Mr. James bought of Mr. Squires, the grocer, the fol lowing articles: a bag of coffee, for $37.874; a chest of tea, for $50.009; a barrel of sugar, for $19 4 cents and 6 mills ; and 9 gallons of wine, for $27 69 cents and 15 mills: what was the amount of his bill? SUBTRACTION 199. SUBTRACTION OF DECIMALS is the operation of finding the difference between two decimal numbers. 1. From 6.304 take .0563 ANALYSIS. In this example a cipher is annexed to 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. 195): hence, Rule. OPERATION. 6.3040 6.2477 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 875.05 take .0467. 3. From 7141.604 take .09046. 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? 8. What is the difference between .105 and 1.00075 ? 199. What is subtraction of decimal fractions? How do you set down the numbers for subtraction? How do you then subtract 1 How many decimal places do you point off in the remainder? What is the proof? |