RULE FOR DECIMAL NUMERATION.

I. Numerate from the decimal point, to determine the de

nominator.

II. Numerate towards the decimal point, to determine the numerator.

III. Read the decimal as a whole number, giving it the nume of its lowest decimal unit, or right hand figure.

6. Write twelve hundred ten-thousandths.

7. Write 9 hundred-thousandths.

8. Read the following decimals.

Ans. .00009.

.32760

040721

Ans. 400.9.

9. Write four hundred, and nine tenths.

10. Write twenty-seven, and fifty-six hundredths.

11. Write eighty-five, and one hundred fifty thousandths. 12. Write one thousand, and twelve millionths.

13. Write three hundred sixty-five, and one thousand eight hundred seven hundred-thousandths.

Ans. 365.01807.

14. Write nine hundred ninety, and three thousand two hundred fourteen millionths. Ans. 990.003214.

15. Read the following numbers:

71.03

126.326

11.0003

240.01376

34.800000

9.1263476

REDUCTION.

CASE I.

111. To reduce decimals to a common denominator.

1. Reduce .3, .09, .0426, .214 to a common denominator.

OPERATION.

.3000

.0900

.0426
.2140

ANALYSIS. A common denominator must contain as many decimal places as is equal to the greatest number of decimal figures in any of the given decimals. We find that the third number contains four decimal places, and hence 10000 must be a common 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, annexing ciphers.

EXAMPLES FOR PRACTICE.

2. Reduce .7, .073, .42, .0020 and .007 to a common denominator.

3. Reduce .004, .00032, .6, .37 and .0314 to a common denominator.

4. Reduce 1 tenth, 46 hundredths, 15 thousandths, 462 ten-thousandths, and 28 hundred-thousandths, to a common denominator.

5. Reduce 9 thousandths, 9 ten-thousandths, 9 hundredthousandths and 9 millionths to a common denominator. 6. Reduce 42.07, 102.006, 7.80, 400.01234 to a com mon denominator.

7. Reduce 300.3, 8.1003, 14.12614, 210.000009, and 1000.02 to a common denominator.

CASE II.

112. To reduce a decimal to a common fraction. 1. Reduce .125 to an equivalent common fraction.

OPERATION.

ANALYSIS. Writing the decimal figures, .125=125.125, over the common denominator, 1000,

RULE. Omit the decimal point, supply the proper denominator, and then reduce the fraction to its lowest terms.

EXAMPLES FOR PRACTICE.

1. Reduce .08 to a common fraction.
2. Reduce .625 to a common fraction.
3. Reduce .375 to a common fraction.
4. Reduce .008 to a common fraction.
5. Reduce .4 to a common fraction.
6. Reduce .024 to a common fraction,

CASE III.

Ans..

Ans. t.

Ans. t.

Ans.

Ans. .

Ans. T

113. To reduce a common fraction to a decimal

2. Reduce to its equivalent decimal.

OPERATION.

4)3.0(7 tenths.

2.8

420(5 hundredths.
20

or 4)3.00

Ans. .75.

.75 Ans.

ANALYSIS. Since we can not divide the numerator 3, by 4, wè reduce it to tenths by annexing a cipher, and then dividing we obtain 7 tenths, and a remainder of 2 tenths. Reducing this remainder to hun dredths by annexing a cipher, and dividing by 4, we obtain 5 hundredths. The sum of the quotients gives .75, the required answer.

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.

10. What decimal is equivalent to 17?
11. What decimal is equivalent to?
12. What decimal is equivalent to?

ADDITION.

Ans. .5.

Ans. .25.

Ans. .4.

Ans. .8.

Ans. .125.

Ans. .9.

Ans. .625.

Ans. .04.

Ans. .3125.

Ans. .85.

Ans. .1875.

Ans. .016.

114. 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. 1. What is the sum of 4.314, 36.42, 120.0042, and .4276? .

OPERATION.

4.314

36.42

120.0042

.4276 161.1658

ANALYSIS. We write the numbers so that the figures of like orders of units shall stand in the same columns; that is, units under units, tenths under tenths, hundredths under hundredths, &c. This brings the decimal points directly under each other. Commencing at the right hand, we add each column separately, and carry as in whole numbers, and in the result we place a decimal point between units and tenths, or directly under the decimal point in the numbers added From this example we derive 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.

2. What is the sum of 2.7, 30.84, 75.1, 126.414 and 3.06? Ans. 238.114. 3. What is the sum of 1.7, 4.45, 6.75, 1.705, .50 and .05? Ans. 15.155. 4. Add 105.7, 19.4, 1119.05, 648.006 and 19.041.

5. Add 48.1, .0481, 4.81, .00481, 481.

Ans. 1911.197.

Ans. 533.96291.

6. Add 1.151, 13.29, 116.283, 9.0275 and .61.

Ans. 140.3615.

7. Add .8, .087, .626, .8885 and .49628. 8. What is the sum of 91.003, 16.4691, 160.00471, 700.05, 900.0006, .035? Ans. 1867.55891.

9. 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.

10. How many acres of land in four farms, containing respectively, 61.843 acres, 120.75 acres, 142.4056 acres, and 180.750 acres? Ans. 505.7486.

How many yards of cloth in 3 pieces, the first containing 211⁄2 yards, the second 36 yards, and the third 40.15 yards?

Ans. 98.40.

12. A man owns 4 city lots, containing 321, 36%, 40%, 42.73 rods of land respectively; how many rods in all?

Ans. 152.205 rods.