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Of the mixed number expressed in the table, the part on the left of the decimal point is the whole number, and that on the right the decimal. The decimal part is numerated from the left to the right, and the value represented is expressed in words thus : Two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety-two billionths. And the mixed number thus : Seven million six hundred fifty-four thousand three hundred twenty-one, and decimal two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety-two billionths.

267. From the table we deduce the following rules :

1. READ a decimal as though it were a whole number, adding the name of the right-hand order.

2. Write a decimal as though it were a whole number, supplying with ciphers such places as have no significant figures.

EXAMPLES.

Express orally, or write in words the following numbers :1. .056

6. 1.631 11. 1.000007 2. .1003 7. 48.07

12. 5.101016 3. .2786

8. 1.315 13. 1.000327 . .16302 9. 5.6001 14. 0.000001 5. .97500 10. 87.0006 15. 16.000000007

406 100000

13 TO 0

6 Ισσσ 19

333 10 oo

1031

Express in the decimal form by figures :-
16.
19.

22.

7106oo 17.

20.

Ισσσσσσ 23.
18.
'10000

21.
1000000

24. 1τσσσσσσ 25. Three hundred twenty-five, and seven tenths. 26. Four hundred sixty-five, and fourteen hundredths. 27. Ninety-three, and seven hundredths. 28. Twenty-four, and nine millionths. 29. Two hundred twenty-one, and nine hundred-thousandths. 30. Forty-nine thousand, and forty-nine thousandths.

31. Seventy-nine million two thousand, and one hundred five thousandths.

32. Sixty-nine thousand fifteen, and fifteen hundred-thousandths.

33. Eighty thousand, and eighty-three ten-thousandths.

34. Nine billion nineteen thousand nineteen, and nineteen hundredths.

35. Twenty-seven, and nine hundred twenty-seven thousandths.

36. Forty-nine trillion, and one trillionth.
37. Twenty-one, and one ten-thousandth.
38. Eighty-seven thousand, and eighty-seven millionths.

39. Ninety-nine thousand ninety-nine, and nine thousand nine billionths.

40. Seventeen, and one hundred seventeen ten-thousandths. 41. Thirty-three, and thirty-three hundredths. 42. Forty-seven thousand, and twenty-nine ten-millionths. 43. Fifteen, and four thousand seven hundred-thousandths. 44. Eleven thousand, and eleven hundredths. 45. Seventeen, and eighty-one quadrillionths. 46. Nine, and fifty-seven trillionths.

47. Sixty-nine thousand, and three hundred forty-nine thou. sandths.

268. Decimals, since they increase from right to left, and decrease from left to right, by the scale of ten, as do simple whole numbers, may be added, subtracted, multiplied, and divided in the same manner.

ADDITION OF DECIMALS.

269. 61.111.

Ex. 1. Add together 23.61, 161.5, 2.6789, and

Ans. 248.8999.

OPERATION.

2 3.6 1

We write the numbers so that figures of the 1 6 1.5

same decimal place shall stand in the same col

umn, and then, beginning at the right hand, add 2.6 7 8 9

them as whole numbers are added, and place 6 1.1 11 the decimal point in the result directly under

those above. 2 4 8.8 9 9 9 RULE. Write the numbers so that figures of the same decimal place shall stand in the same column.

Add as in whole numbers, and point off in the sum, from the right hand, as many places for decimals as equal the greatest number of decir mal places in any of the numbers added.

Proof. — The proof is the same as in addition of whole numbers.

EXAMPLES.

2. Add together the following numbers : 81.61356, 6716.31, 413.1678956, 35.14671, 3.1671, 314.6. Ans. 7564.0052656.

3. What is the sum of the following numbers: 1121.6116, 61.87, 46.67, 165.13, 676.167895 ? Ans. 2071.449495. 4. Add 7.61, 637.1, 6516.14, 67.1234, 6.1234 together.

Ans. 7234.0968. 5. Add 21.611, 6888.32, 3.6167 together.

Ans. 6913.5477. 6. Add together $ 15.06, $ 107.09, $ 1.625, and $ 93.765

7. I have bought a horse for $137.50, a wagon for $ 55.63, a whip for $1.375, and a halter for $ 0.871; what did they all cost?

Ans. $ 195.38. 8. What is the sum of twenty-three million ten; one thousand, and five hundred-thousandths; twenty-seven, and nineteen millionths ; seven, and five tenths? Ans. 23001044.500069.

9. Add the following numbers: fifty-nine, and fifty-nine thousandths; twenty-five thousand, and twenty-five ten-thousandths; five, and five millionths ; two hundred five, and five hundredths.

Ans. 25269.111505. 10. What is the sum of the following numbers : twenty-five, and seven millionths; one hundred forty-five, and six hundred forty-three thousandths; one hundred seventy-five, and eightynine hundredths ; seventeen, and three hundred forty-eight hundred-thousandths ?

Ans. 363.536487. 11. A farmer has sold at one time 3 tons and 75 hundredths of a ton of hay, at another time 11 tons and 7 tenths of a ton, and at a third time 16 tons and 125 thousandths of a ton. How much has he sold in all ?

Ans. 31.575. 12. Add together 73 and 29 hundredths, 87 and 47 thousandths, 3005 and 116 ten-thousandths, 28 and 3 hundredths, 29000 and 5 thousandths.

13. Add together two hundred nine thousand, and fortysix millionths ; ninety-eight thousand two hundred seven, and fifteen ten-thousandths ; fifteen, and eight hundredths; and forty-nine ten-thousandths.

Ans. 307222.086446.

SUBTRACTION OF DECIMALS.

OPERATION.

270. Ex. 1. From 61.9634 take 9.182. Ans. 52.7814.

Having written the less number under the great61.9 6 3 4 er, so that figures of the same decimal place stand 9.1 8 2 in the same column, we subtract as in whole num

bers, and place the decimal point in the result, as in 5 2.7 814

addition of decimals. RULE. Write the less number under the greater, so that figures of the same decimal place shall stand in the same column.

Subtract as in whole numbers, and point off the remainder as in addition of decimals.

Proof. - The proof is the same as in subtraction of whole numbers.

EXAMPLES.
2.
3.
4.

5. 3 9.3

5.
6.1

41.7 1.6 7 8 9 1.6 7 8 1.9 9 9 9 9 2 1.9 7 67 3 7.6 2 11 3.3 2 2 4.1 0 0 0 1 1 9.7 2 3 3 6. From 29.167 take 19.66711.

Ans. '9.49989. 7. From 91.61 take 2.6671.

Ans. 88.9429. 8. From 96.71 take 96.709.

9. Take twenty-seven, and twenty-eight thousandths from ninety-seven,and seven tenths.

Ans. 70.672. 10. Take one hundred fifteen, and seven hundredths from three hundred fifteen, and twenty-seven ten-thousandths.

Ans. 199.9327. 11. From twenty-nine million four thousand and five take twenty-nine thousand, and three hundred forty-nine thousand two hundred, and twenty-four hundred-thousandths.

Ans. 28625804.99976. 12. From one million take one millionth.

Ans. 999999.999999. 13. From $ 19 take $ 1.375.

Ans. $ 17.625. 14. A merchant bought flour to the amount of $ 316.87 and sold it for $ 400 ; how much did he gain by the sale ? 15. From 19 million take 19 billionths.

Ans. 18999999.999999981.

16. Charles Washburne has in one farm 93.45 acres, in another 124 acres, in a third 244.285 acres, and in wood-lots 216.136 acres ; how many acres more would he require to have exactly 1000 acres ?

MULTIPLICATION OF DECIMALS.

OPERATION.

100; = 245,792

10

1000

OPERATION.

271. Ex. 1. Multiply 76.81 by 3.2. Ans. 245.792.

We multiply as in whole numbers, and point off 7 6.8 1

on the right of the product as many figures for deci3.2 mals as there are decimal figures in the multiplicand 1 5 3 6 2

and multiplier counted together.' The reason for

pointing off the decimals in the product, as in the 2 3 0 43

operation, will be seen, if we convert the multipli2 4 5.7 92

cand and multiplier into common fractions, and mul

tiply them together. Thus, 76.81 = 76-80% 7681 and 3.2

is. Then 1080 x = 215732 245.792, Ans., the same as in the operation. 2. Multiply .1234 by .0046.

Ans. .000567 64. Since the number of figures in the product .1 2 3 4 is not equal to the number of decimals in the .00 46 multiplicand and multiplier, we supply the de

ficiency by placing ciphers on the left hand. 7 40 4

The reason of this process will appear, if we 493 6

perform the operation thus : .1234 =

and .0046 .000 5 6 7 64

Then 1.2.3.4. Х

.00056764, Ans., the same as in

the operation. RULE. Multiply as in whole numbers, and point off as many figures for decimals, in the product, as there are decimal figures in the multiplicand and multiplier.

If there be not so many figures in the product as there are decimal figures in the multiplicand and multiplier, supply the deficiency by prefixing ciphers.

Proof.- The proof is the same as in multiplication of whole numbers.

EXAMPLES. 3. Multiply 61.76 by .0071.

Ans. .438496. -4. Multiply .0716 by 1.326.

Ans. .0949416. 5. Multiply .61001 by .061. 6. Multiply 71.61 by 365.

Ans. 26137.65. 7. Multiply .1234 by 1234.

Ans. 152.2756.

1234 10000 10000

46 10000

5 67 64 100000000

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