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RULE. Make the repetends similar and conterminous, and find their sum, as in common Addition. Divide this sum by as many 9's as there are places in the repetend, and the remainder is the repetend of the sum, which must be set under the figures added, with ciphers on the left when it has not so many places as the repetends. Carry the quotient of this division to the next column, and proceed with the rest as with finite decimals.

2. Add 27.56+5.632 + 6.7 + 16.356+.71 and 6.1234 together. Ans. 63.1690670868888. 3. Add 2.765 +7.16674 +3.671 +.7 and .1728 together.

4. Add 5.16345 +8.6381 +3.75 together.

Ans. 14.55436.

Ans. 17.55919120847374090302.

5. Reduce the following numbers to decimals, and find their

sum: †,†, and f.

Ans. .587301.

SECTION XXXII.

SUBTRACTION OF CIRCULATING DECIMALS

EXAMPLE.

1. From 87.1645 take 19.479167.

OPERATION.

87.1645 87.164545 19.479167 19.479167

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67.685377

Having made the numbers similar and conterminous, we subtract as in whole numbers, and find the remainder of the circulate to be 5378, from which we subtract 1, and write the remainder in its place, and proceed with the other part of the question as in whole numbers. The reason why 1 should be added to the repetend may be shown as follows. The minuend may be considered 164545, and the subtrahend 7819; we then proceed with these numbers as in Case II. of Subtraction of Vulgar Fractions; and the numerator 5377 will be the repeating decimal. Q. E. D.

OPERATION.

164848 7916 83333

999

RULE.

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Make the repetends similar and conterminous, and subtract as usual; observing, that if the repetend of the subtrahend be greater than the repetend of the minuend, then the remainder on the right must be less by unity than it would be if the expressions were finite.

2. From 7.1 take 5.02.

Ans. 2.08.

3. From 315.87 take 78.0378. Ans. 237.838072095497.

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10. From

take 17.

Ans. 5.0462. Ans. 14.8951. Ans. 2.405951.

Ans. .246753.

Ans. .158730.

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11. From 5.12345 take 2.3523456.

Ans. 2.7711055821666927777988888599994.

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= 2776

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units to the product; for Thus we see the repeating

RULE. Turn both the terms into their equivalent vulgar fractions, and find the product of those fractions as usual. Then change the vulgar fraction expressing the product into an equivalent decimal, and it will be the product required. But, if the multiplicand ONLY has a repetend, multiply as in whole numbers, and add to the right-hand place of the product as many units as there are tens in the product of the lefthand place of the repetend. The product will then contain a repetend whose places are equal to those in the multiplicand.

3. Multiply 87.32586 by 4.37. 4. Multiply 3.145 by 4.297.

Ans. 381.6140338.
Ans. 13.5169533.
Ans. 8s.

5. What is the value of .285714 of a guinea ?
6. What is the value of .461607142857 of a ton ?

Ans. 9cwt. Oqr. 26lb.

7. What is the value of .284931506 of a year?

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RULE. Change both the divisor and the dividend into their equivalent vulgar fractions, and find their quotient as usual.

Change the vulgar fraction expressing the quotient into its equivalent decimal, and it

will be the quotient required.

2. Divide 345.8 by .6.

3. Divide 234.6 by .7.

Ans. 518.83.

Ans. 301.714285.

4. Divide .36 by .25. Ans. 1.4229249011857707509881.

SECTION XXXV.

MENTAL OPERATIONS IN FRACTIONS, &c.

If any number be divided into two equal parts, and into two unequal parts, the product of the two unequal parts together with the square, of half the difference of the two unequal parts is equal to the square of one of the equal parts. Also,

The product of any two numbers is equal to the square of

half their sum, less the square of half their difference. See Euclid's Elements, Book Second, Proposition Fifth.

NOTE.

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- A number is said to be squared when it is multiplied by itself; thus, the square of 5 is 5 x 5 = 25.

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From the above proposition we deduce the following rules.

To multiply any number containing a half by itself.

RULE 1. Multiply the whole number given in the question by the next larger whole number, and to the product add the square of the half ·1.

=

1. Multiply 5 by 51.

OPERATION.

5×6=30; × 1 = 1; 30 += 30 Ans.

NOTE. The whole number given is 5, and the next larger whole number is 6.

2. Multiply 7 by 71.
3. Multiply 3 by 31.
4. Multiply 9 by 91.
5. Multiply 11 by 11.
6. Multiply 201 by 201.
7. Multiply 30 by 301.

NOTE.

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The same principle will hold good if we multiply any number by itself whose unit is a 5.

RULE 2. Take the next least number that ends in a cipher, and multiply it by the next larger number ending in a cipher, and add to the product the square of 5=25, and the result will be the product.

8. Multiply 25 by 25. Ans. 625. The next less number ending in a cipher is 20, and the next larger is 30; 30 × 20 = 600; 5 × 5=25; 600 +- 25 = 625

Ans.

9. Multiply 35 by 35.

10. Multiply 85 by 85. 11. Multiply 95 by 95.

Ans. 1225.

Ans. 7225.

Ans. 9025.

To find the product of two mixed numbers, whose fractional part is a half, and whose difference is a unit.

RULE 3. Multiply the larger number without the fraction by itself, and from the product subtract the fractional part multiplied by itself, and the result will be the product.

12. Multiply 6 by 74.

Ans. 48.

OPERATION.

=

7×7=49; 1×1; 49-483 Ans.

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Ans. 802. Ans. 143.

16. Multiply 89 by 90.

Ans. 3992.

Ans. 8099.

NOTE. If the fractional parts of the numbers approach within a }, 4, or

, &c., of the larger number, the principle is the same.

17. What is the product of 43 multiplied by 53. Ans. 248.

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To find the product of two numbers, one of which is as much less than either 20, 30, 40, &c., as the other is more than either of these numbers.

RULE 4.- Multiply the 20, 30, or 40, as the case may be, by itself, and subtract from the product the square of half the difference of the two numbers to be multiplied, and the result will be the product.

22. Multiply 28 by 32.

Ans. 896.

We find that 28 is as much less than 30 as 32 is more than 30; we therefore multiply 30 by 30=900, and from this prod

uct we subtract the square of 24; 900

4896 Ans.

23. What is the product of 75 by 85?

Ans. 6375.

Ans. 6391.

Ans. 9991.

Ans. 391.

24. What is the product of 83 by 77 ?
25. What is the product of 97 by 103 ?
26. What is the product of 17 by 23?
27. What cost 18cwt. of steel, at $22 per cwt. ?

Ans. $396.

28. What cost 27 tons of hay, at $33 per ton?

Ans. $891.

29. What cost 64 gallons of oil, at $0.56 per gallon?

Ans. $35.84.

30. What cost 28 tons of hay, at $32 per ton?

Ans. $896.

31. What cost 49 tons of iron, at $ 51 per ton?

Ans. $2499.

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