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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 $0.87 J; 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. 307222.080446.

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

opeRATron. Having written the less number under the great6 1.9 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 numbers, and place the decimal point in the result, as in o l.l 8 14 addition ot'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.

SUBTRACTION OF DECIMALS.

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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.

271. Ex. 1. Multiply 70.81 by 3.2. Ans. 245.792.

OPERATION.

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 - and multiplier counted together. The reason for

pointing off the decimals in the product, as in the & o U 4 o operation, will be seen, if we convert the multipli2 4 5 7 9 2 cand an(l multiplier into common fractions, and multiply them together. Thus, 76.81 = 70iVff = 7l<WLi and 3.2 = 3ft - ff. Then *ftY X.?}= IP = ^iVft = 245.792, Ans , the same as in the operation.

2. Multiply .1234 by .0046. Ans. .00056764.

Operation. Since the number of figures in the product

.1 2 3 4 is not equal to the number of decimals in the .0046 multiplicand and multiplier, we supply the deficiency by placing ciphers on the left hand. 7 4 0 4 Tne reason 0f this process will appear, if we 4 9 3 6 perform the operation thus: .1234 = ftWo i

00056764 and-0046 = T^w The" Twat X TTJVW = .v u v o o o * ^hjij^ = .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.

8. Multiply 6.711 by 6543. Ans. 43910.073.

9. Multiply .0009 by .0009. Ans. .00000081.

10. What is the product of one thousand and twenty-five, multiplied by three hundred and twenty-seven ten-thousandths?

Ans. 33.5175.

11. What is the product of seventy-eight million two hundred five thousand and two, multiplied by fifty-three hundredths? Ans. 41448651.06.

12. Multiply one hundred and fifty-three thousandths by one hundred twenty-nine millionths. Ans. .000019737.

13. What will 26.7 yards of cloth cost, at $5.75 a yard?

Ans. $153,525.

14. What will 14.75 bushels of wheat cost, at $ 1.25 a bushel? Ans. $18.4375.

15. What will 375.6 pounds of sugar cost, at $0,125 per pound?

16. What will 26.58 cords of wood cost, at $5,625 a cord? Ans. $149.512^.

17. What will 28.75 tons of potash cost, at $ 125.78 per ton? Ans. $3616.175.

CONTRACTIONS IN MULTIPLICATION OF DECIMALS.

272. To multiply a decimal by 10, 100, 1000, &c. Remove the decimal point as many places to the right as there are ciphers in the mult iplier, annexing ciphers if required. Thus, 1.25 X 10 = 12.5; and 1.6 X 100 = 160.

Examples.

1. Multiply 131.634 by 1000. Ans. 131634.

2. Multiply 3478.9 by 100. Ans. 347890.

3. Multiply one thousandth by one thousand.

4. What is the profit on one million yards of cotton cloth, at $ 0.007 per yard. Ans. $ 7000.

273i° When it is not necessary that all the decimal places of the product should be retained, tedious multiplications may often be obviated, by contracting the work as follows : —

Write the units' place of the multiplier under that figure of the multiplicand whose place it is proposed to retain in the product, and dispose of all the other figures of the multiplier in an order contrary to the usual one. Then, in multiplying, begin, for each partial product, with that figure of the multiplicand which stands above the multiplying figure, observing to add to the product the number nearest to that which would have been carried if the places at the right had not been rejected. Write down the several partial products, so that the right-hand figure of each shall be in the same column, and their sum will be the product required.

Examples.

1. Multiply 3.141592 by 52.7438, retaining only four places for decimals in the product. Ans. 165.6995

FIRST OPERATION. SECOND OPERATION.

3.1 4 1 5 9 2 = Multiplicand 3.1 4 1 5 9 2

8 3 4 7.2 5 = Multiplier reversed. 5 2.7 4 3 8

1 5 7 07~9lT = Product by 5, + 1 25]l 3 2 73 6

'2 9424776

6 2 8 3 2 = Product by 2,

2 19 9 1 = Product by 7,
1 2 5 7 = Product by 4,

9 4= Product by 3, 4- 1 6283184
2 5 = Product by 8, -f 1 15707960

4 12566368
1 2 19 9 1144

1 6 5.6 9 9 5 = Product sought. 1 6 5.6 9 9 5(0 0 1 2 9 6

By comparison of the two methods of solution, it will be seen that the common one, as shown in the second operation, gives ten places of decimals, or six more than are required by the question, thus rendering unnecessary the several- figures on the right of the vertical line. By the contrasted way, the multiplier, for convenience, has its figures reversed, or placed contrary to the usual order, so that the product of each figure by the one of the multiplicand above it, must be of the order of ten-thousandths. The first figure, at the right, of each partial product, being of the order of ten-thousandths, is written in the same column. To the product by 5 we add 1, since, if the 2 in the multiplicand had not been rejected, there would have been 1 to carry to the product of the 9 by the 5; to the product by

2 we add 2, since the product of the rejected figures, 92, by 2, approximates to 2 hundred, which would require 2 to be carried; to the product by 7 we add 4. since the product of the two rejected figures, 59, by 7, would require 4 to be carried; to the pro

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