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211. To reduce decimals to a common denominator. 1. Reduce .5, .24, .7836 and .375 to a common denominator.

ANALYSIS. A common denominator must contain OPERATION. .5000

as many decimal places as are equal to the greatest .2400

number of decimal figures in any of the given deci.7836

mals. We find that the third number contains four .3750 decimal places, and hence 10000 must be a com

denominator. As annexing ciphers to decimals does not alter their value, we give to each number four decimal places, by annexing ciphers, and thus reduce the given decimals to a common denominator. Hence,

RULE. Give to each number the same number of decimal places, by annexing ciphers.

Notes.-1. If the numbers be reduced to the denominator of that one of the given numbers having the greatest number of decimal places, they will have their least common decimal denominator.

2. An integer may readily be reduced to decimals by placing the decimal point after units, and annexing ciphers; one cipher reducing it to tenths, two ciphers to hundredths, three ciphers to thousandths, and so on.

EXAMPLES FOR PRACTICE. 1. Reduce .18, .456, .0075, .000001, .05, .3789, .5943786, and .001 to their least common denominator.

2. Reduce 12 thousandths, 185 millionths, 936 billionths, and 7 trillionths to their least common denominator.

3. Reduce 57.3, 900, 4.7555, and 100.000001 to their least common denominator.

CASE II.

212. To reduce a decimal to a common fraction. 1. Reduce .375 to an equivalent common fraction.

Analysis. Writing the decimal

figures, .375, over the common de.375

nominator, 1000, we have 10=. Hence,

OPERATION.

37 5 1000

RULE.

Omit the decimal point, and supply the proper denomi

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Ans..

NOTE.—The decimal.13} may properly be called a complex decimal.
10. Reduce .57} to a common fraction.
11. Reduce .66; to a common fraction.

Ans.
12. Reduce .4444 to a common fraction.
13. Reduce .024; to a common fraction.

Ans. 1377 14. Reduce .9843 to a common fraction. 15. Express 7.4 by an integer and a common fraction.

Ans. 73. 16. Express 24.74 by an integer and a common fraction. 17. Reduce 2.1875 to an improper fraction.

Ans. 36 18. Reduce 1.64 to an improper fraction. 19. Reduce 7.496 to an improper fraction.

Ans. 933

CASE III.

213. To reduce a common fraction to a decimal. 1. Reduce to its equivalent decimal.

ANALYSIS. We first annex .625

the same number of ciphers to both terms of the fraction; this does not alter its value, (174,

FIRST OPERATION.

5 000 8000

625 1000

r

OPERATION.

SECOND OPERATION.

III); we then divide both re8) 5.000

sulting terms by 8, the signifi

cant figure of the denominator, .625

and obtain the decimal denominator, 1000. Omitting the denominator, and prefixing the sign, we have the equivalent decimal, .625.

In the second operation, we omit the intermediate steps, and obtain the result, practically, by annexing the three ciphers to the numerator, 5, and dividing the result by the denominator, 8. 2. Reduce iz to a decimal.

ANALYSIS. Dividing as in the former ex125 ) 3.000 ample, we obtain a quotient of 2 figures, 24. .024

But since 3 ciphers have been annexed to the

numerator, 3, there must be three places in the required decimal; hence we prefix 1 cipher to the quotient figures, 24. The reason of this is shown also in the following operation.

iis = 738880 = ido = .024 214. From these illustrations we derive the following

RULE. I. Annex ciphers to the numerator, and divide by the denominator.

II. Point off as many decimal places in the result as are equal to the number of ciphers annexed.

NOTE.-- If the division is not exact when a sufficient number of decimal figures bave been obtained, the sign, +, may be annexed to the decimal to indicate that there is still a remainder. When this remainder is such that the next decimal figure would be 5 or greater than 5, the last figure of the terminated decimal may be increased by 1, and the sign, -, annexed. And in general, + denotes that the written decimal is too small, and - denotes that the written decimal is too large; the error always being less than one half of a unit in the last place of the decimal.

Ans. .75. Ans. .3125.

EXAMPLES FOR PRACTICE. 1. Reduce a to a decimal. 2. Reduce to a decimal. 3. Reduce s to a decimal. 4. Reduce il to a decimal. 5. Reduce ji to a decimal. 6. Reduce z's to a decimal. 7. Reduce to a decimal.

Ans. .04. Ans. .068.

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2 3

.

1 67

8. Reduce li to a decimal.

Ans. .59375 9. Reduce

ī 2800

to a decimal. 10. Reduce to a decimal.

Ans. .2916711. Reduce 97 to a decimal. 12. Reduce 3 to a decimal.

Ans. .767857+. 13. Reduce 73 to the decimal form.

Ans. 7.125. 14. Reduce 56 to the decimal form. Ans. 56.078125. 15. Reduce 325 to the decimal form. 16. Reduce 244 to a simple decimal. 17. Reduce 5.7819 to a simple decimal. 18. Reduce .3111, to a simple decimal. Ans. .30088. 19. Reduce 4.0.3 to a simple decimal.

Ans. 4.008. 20. Reduce .30,198 1o to a simple decimal.

ADDITION.

OPERATION.

215. Since the same law of local value extends both to the right and left of units' place; that is, since decimals and simple integers increase and decrease uniformly by the scale of ten, it is evident that decimals may be added, subtracted, multiplied and divided in the same manner as integers. 216. 1. What is the sum of 4.75, 246, 37.56 and 12.248 ?

ANALYSIS. We write the numbers so that units of 4.75 like orders, whether integral or decimal, shall stand

.246 in the same columns; that is, units under units, tenths 37.56 under tenths, etc. This brings the decimal points 12.248

directly under each other. Commencing at the right 51.804

hand, we add each column separately, carrying 1 for

every ten, according to the decimal scale; and in the result we place the decimal point between units and tenths, or directly under the decimal points in the numbers added. Hence the following

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

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

EXAMPLES FOR PRACTICE.

1. Add .375, .24, .536, -78567, .4637, and .57439.

Ans. 2.97476. 2. Add 5.3756, 85.473, 9.2, 46.37859, and 45.248377.

Ans. 191.675567. 3. Add .5, .37, .489, .6372, .47856, and .02524. 4. Add .463, .325}, .1674, and .275,16.

Ans. 1.2296625. 5. Add 4.64, 7.3233, 5.37843, and 2.648781. 6. Add 4.3785, 23, 54, and 12.4872. Ans. 24.9609+

7. What is the sum of 137 thousandths, 435 thousandths, 836 thousandths, 937 thousandths, and 496 thousandths ?

Ans. 2.841. 8. What is the sum of one hundred two ten-thousandths, thirteen thousand four hundred twenty-six hundred-thousandths, five hundred sixty-seven millionths, three millionths, and twenty-four thousand seven hundred-thousandths ?

9. A farm has five corners; from the first to the second is 31.72 rods; from the second to the third, 48.41 rods; from the third to the fourth, 152.17 rods; from the fourth to the fifth, 95.36 rods; and from the fifth to the first, 56.18 rods. What is the whole distance around the farm ?

10. Find the sum of $4, 36, 156, and 172in decimals, correct tò the fourth place.

Ans. .6669+. Note.—In the reduction of each fraction, carry the decimal to at least the fifth place, in order to insure accuracy in the fourth place.

11. A man owns 4 city lots, containing 16, rods, 15 %, rods, 18] rods, and 1445 rods of land, respectively; how many rods in all ?

12. What is the sum of 4; decimal units of the first order, 2; of the second order, 9 of the third order, and 3 of the fourth order?

Ans. .486929. 13. What is the approximate sum of 1 decimal unit of the first

, a , 1 order, i of a unit of the fourth order, } of a unit of the fifth order, & of a unit of the sixth order, and 4 of a unit of the seventh order ?

Ans. .1053605143_,

37

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