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7. What is the sum of thirty-six, and fifteen thousandths ; three hundred, and six hundred five ten-thousandths; five, and three millionths; sixty, and eighty-seven ten-millionths? Ans. 401.0755117.

8. What is the sum of fifty-four, and thirty-four hundredths; one, and nine ten-thousandths; three, and two hundred seven millionths; twenty-three thousandths; eight, and nine tenths; four, and one hundred thirty-five thousandths? Ans. 71.399107.

9. How many yards in three pieces of cloth, the first piece containing 18.375 yards, the second piece 41.625 yards, and the third piece 35.5 yards?

10. A's farm contains 61.843 acres, B's contains 143.75 acres, C's 218.4375 acres, and D's 21.9 acres; how many acres in the four farms?

11. My farm consists of 7 fields, containing 123 acres, 183 acres, 9 acres, 24 acres, 41% acres, 8 acres, and 1518 acres respectively; how many acres in my farm?

NOTE. Reduce the common fractions to decimals before adding. Ans. 93.6375.

12. A grocer has 2 barrels of A sugar, 5 barrels of B sugar, 3ğ barrels of C sugar, 3.0642 barrels of crushed sugar, and 8.925 barrels of pulverized sugar; how many barrels of sugar has he? Ans. 23.8642.

13. A tailor made 3 suits of clothes; for the first suit he used 2 yards of broadcloth, 3 yards of cassimere, and yards of satin; for the second suit 2.25 yards of broadcloth, 2.875 yards of cassimere, and 1 yard of satin; and for the third suit 5 yards of broadcloth, and 1 yards of satin. How many yards of each kind of goods did he use? How many yards of all? Ans. to last, 18.375.

SUBTRACTION.

153. 1. From 91.73 take 2.18. ANALYSIS. In each of these

OPERATION.

91.73
2.18

Ans. 89.55

2. From 2.9185 take 1.42.

OPERATION.

2.9185

1.42

Ans. 1.4985

3. From 124.65 take 95.58746.

OPERATION.

124.65
95.58746

Ans. 29.06254

three examples, we write the
subtrahend under the minu-
end, placing units under
units, tenths under tenths,
&c.
Commencing at the
right hand, we subtract as
in whole numbers, and in
the remainders we place the
decimal points directly under
those in the numbers above.
In the second example, the
number of decimal places in
the minuend is greater than
the number in the subtra-
hend, and in the third exam-
ple the number is less. In
both cases, we reduce both
minuend and subtrahend to
the same number of decimal
places, by annexing ciphers;
or we suppose the ciphers to

be annexed, before performing the subtraction. Hence the

RULE. I. Write the numbers so that the decimal points shall stand directly under each other.

II. Subtract as in whole numbers, and place the decimal point in the result directly under the points in the given numbers.

4. Find the difference between 714 and .916. Ans. 713.084,

5. How much greater is 2 than .298? 6. From 21.004 take 75 hundredths.

7. From 10.0302 take 2 ten-thousandths.

8. From 900 take .009.

Ans. 1.702.

Ans. 10.03.

Ans. 899.991.

Ans. .999999.

9. From two thousand take two thousandths.

10. From one take one millionth.

Explain subtraction of fractions. Give the rule, first step. Second.

11. From four hundred twenty-seven thousandths take four hundred twenty-seven millionths. Ans. .426573.

12. A man owned thirty-four hundredths of a township of land, and sold thirty-four thousandths of the township; how much did he still own? Ans. .306.

.35

MULTIPLICATION.

154. 1. What is the product of .35 multiplied by .5? OPERATION. ANALYSIS. We perform the multiplication the same as in whole numbers, and the only difficulty we meet with is in pointing off the decimal places in the product. To determine how many places to .175, Ans. point off, we may reduce the decimals to common fractions; thus, .35 = 85 and .5%. Perform

.5

100

175

ing the multiplication, and we have X, and this product, expressed decimally, is .175. Here we see that the product contains as many decimal places as are contained in both multiplicand and multiplier. Hence the following

RULE. Multiply as in whole numbers, and from the right hand of the product point off as many figures for decimals as there are decimal places in both factors.

NOTES. 1. If there be not as many figures in the product as there are decimals in both factors, supply the deficiency by prefixing ciphers, 2. To multiply a decimal by 10, 100, 1000, &c., remove the point as many places to the right as there are ciphers on the right of the multiplier.

EXAMPLES.

2. Multiply 1.245 by .27.

3. Multiply 79.347 by 23.15.

4. Multiply 350 by .7853.

5. Multiply one tenth by one tenth.

Ans. .33615. Ans. 1836.88305.

6. Multiply 25 by twenty-five hundredths.

Ans. 01. Ans. 6.25.

Explain multiplication of decimals. Give ruie. If the product have less decimal places than both factors, how proceed? How multiply by 10, 100, 1000, &c.?

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9. Multiply .006 by 1000.

10. Multiply 23 by .009.

11- Multiply sixty-four thousandths by thirteen millionths Ans. .000000832.

12. Multiply eighty-seven ten-thousandths by three hun dred fifty-two hundred-thousandths.

Ans. 1.

13. Multiply one million by one millionth. 14. Multiply sixteen thousand by sixteen ten-thousandths.

Ans. 25.6.

15. If a cord of wood be worth 2.37 bushels of wheat, how many bushels of wheat must be given for 9.58 cords of wood? Ans. 22.7046 bushels.

DIVISION.

155. 1. What is the quotient of .175 divided by .5?

OPERATION.

.5) .175

Ans. .35

ANALYSIS. We perform the division the same as in whole numbers, and the only difficulty we meet with is in pointing off the decimal places in the quotient. To determine how many places to point off, decimals to common fractions; thus, .175= Performing the division, and we have 5 175 10 X 1000 $

we may reduce the 15%, and .5=15.

10

175

1000

10

=

and this quotient, expressed decimally, is .35. dividend contains as many decimal places as divisor and quotient. Hence the following

35

100

Here we see that the are contained in both

RULE. Divide as in whole numbers, and from the right hand of the quotient point off as many places for decimals as the decimal places in the dividend exceed those in the

divisor.

Explain division of decimals. Give rule.

NOTES. 1. If the number of figures in the quotient be less than the excess of the decimal places in the dividend over those in the divisor, the deficiency must be supplied by prefixing ciphers.

2. If there be a remainder after dividing the dividend, annex ciphers, and continue the division: the ciphers annexed are decimals of the dividend.

3. The dividend must always contain at least as many decimal places as the divisor, before commencing the division.

4. In most business transactions, the division is considered sufficiently exact when the quotient is carried to 4 decimal places, unless great accuracy is required.

5. To divide by 10, 100, 1000, &c., remove the decimal point as many places to the left as there are ciphers on the right hand of the divisor.

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9. Divide 3 by 3; divide 3 by .3; 3 by .03; 30 by .03.

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16. Divide fifteen, and eight hundred seventy-five thou

sandths, by twenty-five ten-thousandths.

Ans. 6350.

17. Divide 365 by 100.

18. Divide 785.4 by 1000.

Ans. .7854.

19. Divide one thousand by one thousandth.

Ans. 1000000.

When are ciphers prefixed to the quotient? If there be a remainder, how proceed? If the dividend have less decimal places than the divisor, how proceed? How divide by 10, 100, 1000, &c.?

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