8. Five hundred and 25 thousandths.

Ans. .525.

Ans. .3007.

9. Three tenths and 7 ten-thousandths.

10. Seven hundredths and 9 ten-thousandths.

11. Three hundred and 78 ten-thousandths.

12. Five tenths, 6 hundredths, and 7 hundred-thousandths. 13. Four hundredths, seven ten-thousandths, and 6 hundredthousandths.

14. Nine hundred and sixty-nine hundred-thousandths. 15. Two tenths and three millions.

16. Five hundredths, six thousandths, and eight millionths. 17. Thirty-five thousand and eight millionths.

18. Ninety-three hundred and seven ten-millionths. 19. Eighteen thousand and one hundred-millionths.

20. Two million, 6 thousand, and 9 hundred-millionths. Express the following common fractions and mixed numbers in the decimal form :

146, Let the pupil show the truth of the following principles of the decimal notation, and illustrate them with examples :

I. Changing the decimal point one place toward the right multiplies by ten; two places multiplies by one hundred, etc. II. Changing the decimal point one place towards the left divides by ten; two places, divides by one hundred, etc.

III. Placing a cipher between the decimal point and a decimal divides the decimal by ten.

IV. Annexing ciphers to the right of a decimal does not change the value of the decimal.

REDUCTION OF DECIMALS.

147, There are two cases of the reduction of decimals :

1st. To reduce decimals to common fractions.

2d. To reduce common fractions to decimals.

CASE 1.

148. To reduce a decimal to a common fraction.

1. Reduce .45 to a common fraction.

SOLUTION.-.45 expressed in the form of a common fraction is, which reduced to its lowest terms equals 20

Hence the following

OPERATION.

.45==, Ans.

RULE. Write the denominator under the decimal, omitting the decimal point, and reduce the common fraction to its lowest

Ans. 17.

Ans. 34. 11. 17.0125.

12. Reduce the mixed decimal .23 to a common fraction.

SOLUTION. This is 23 tenths, and writing the

23
10'

denominator, we have which equals

OPERATION.

ૐ

23

or .24

10'

10

%, which reduced to its lowest terms equals.

Therefore, etc.

Reduce the following to common fractions or mixed numbers:

13. .81.

14. .163.

15. .25%.

16. .45.

149. To reduce a common fraction to a decimal.

1. Reduce § to a decimal.

SOLUTION. equals of 5. 5 equals 50 tenths; of 50 tenths is 6 tenths and 2 tenths remaining; 2 tenths equals 20 hundredths; of 20 hundredths equals 2 hundredths and 4 hundredths remaining; 4 hundredths equals 40 thousandths; of 40 thousandths is 5 thousandths. Therefore, & equals .625. Hence the following

OPERATION.

= of 5 =8)5 000

.625

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

II. Point off as many decimal places in the quotient as there are ciphers annexed.

NOTE. In many cases the division will not terminate, and the com mon fraction cannot then be exactly expressed by a simple decimal. Reduce the following common fractions to decimals :—

150. Addition of Decimals is the process of finding the sum of two or more decimals.

1. What is the sum of 11.96, 25.075, 84.306, and 90.728 ?

OPERATION,

11.96

25.075

84.300

90.728

SOLUTION. We write the numbers so that figures of the same order shall stand in the same vertical column, and commence at the right to add. 8 thousandths, plus 6 thousandths, plus 5 thousandths, equal 19 thousandths, which equals 1 hundredth and 9 thousandths; we write the 9 thousandths, and add the 1 hundredth to the next sum. 2 hundredths, plus 7 hundredths, plus 6 hundredths, equal 15 hundredths, and the 1 hundredth added is 16 hundredths, which equals 1 tenth and 6 hundredths, etc.

212.069

RULE.-I. Write the numbers so that units of the same order shall stand in the same vertical column.

II. Add as in whole numbers, placing the decimal point in its proper place in the sum.

2. Find the sum of 79.76, 85.08, 36.125, 140.309.

Ans. 341.274.

3. Find the sum of 87.09, 58.37, 95.42, 237.675.

Ans. 478,555.

4. Add 18.79, 147.072, 856.709, 185.8761, 397.05784.

Ans. 1605.50494.

5. Add 59.874, 435.095, 672.328, 976.309, 8467.500843. Ans. 10611.106843. 6. Add together 9 and 7 tenths, 41 and 8 hundredths, 75 and 54 hundredths, 128 and 187 thousandths. Ans. 254.507. 7. Add together 187 and 5 thousandths, 49 and 9 hundredthousandths, 1876 and 245 millionths, 187 ten-thousandths, and 999 ten-millionths. Ans. 2112.0241349.

8. Add 798 and 9 ten-thousandths, 17 millionths, 18 thousandths, and 98 ten-millionths, 67 hundred-thousandths, and 95 ten-millionths. Ans. 798.0196063. 9. Find the sum of the answers of the problems in addition of decimals, beginning with the second problem.

Ans. 16200.9915242.

SUBTRACTION OF DECIMALS.

151. Subtraction of Decimals is the process of finding the difference between two decimals.

1. From 972.163 take 856.235.

OPERATION.

972.163 856.235

115.928

SOLUTION. We write the numbers so that figures of the same order stand in the same column, and commence at the right to subtract. We cannot subtract 5 thousandths from 3 thousandths, hence we take 1 hundredth from 6 hundredths. 1 hundredth equals ten thousandths, which added to 3 thousandths equals 13 thousandths; 5 thousandths from 13 thousandths leaves 8 thousandths, etc.

RULE.-I. Write the smaller number under the greater, so that gures of the same order stand in the same column.

II. Subtract as in simple numbers, and place the decimal

point so as to give the proper name to the different figures.

2. From 707.325 take 623.452.
3. From 826.438 take 734.936.
4. From 1230.207 take 384.1231.

5. From 78.3057 take 29.084.
6. From 2.07 take 1.432765.

Ans. 83.873.

Ans. 91.502. Ans. 846.0839.

Ans. 49.2217.

Ans. .637235.

7. From 1 and 1 hundredth take 1 tenth and 1 millionth.

Ans. .909999.

8. From 3 tenths take 3 hundred-millionths.

Ans. .29999997.

9. From 21 take 2 thousandths and 2 billionths.

Ans. 2.4974999975.

10. Find the sum of the answers in subtraction of decimals,

from the second problem.

Ans. 1074.9253339675.

MULTIPLICATION OF DECIMALS.

152. Multiplication of Decimals is the process of multiplying when one or both factors are decimals.

1. Multiply 7.23 by .46.

SOLUTION.-7.23 multiplied by 46 equals 332.58, and multiplied by 46 hundredths the product will be 1 hundredth as great, which by removing the decimal point two places to the left, becomes 3.3258. Hence, 7.23 multiplied by .46 equals 3.3258.

SOLUTION 2D.-7.23.46783 X

723X46

OPERATION.

7.23

.46

4338

2892

3.3258

1000 = Todoo × 33258

σ

3.3258. From either of these solutions we derive the following

RULE.-Multiply as in whole numbers, and point off as many decimal places in the product as there are in both multiplicand and multiplier, prefixing ciphers if necessary.

2. Multiply 27.13 by .67. 3. Multiply 43.08 by 2.36. 4. Multiply 79.52 by .019. 5. Multiply 8.534 by 20.074. 6. Multiply 123.107 by 1.52. 7. Multiply 512.073 by 35.08. 8. Multiply 54.0079 by 7.072. 9. Multiply 1.08096 by 3.5702. 10. Multiply .03507 by .005873. 11. Multiply 2.0709 by .000246.

Ans. 18.1771.

Ans. 101.6688.

Ans. 1.51088. Ans. 171.311516.

Ans. 187.12264. Ans. 17963.52084. Ans. 381.9438688.

Ans. 3.859243392. Ans. .00020596611. Ans. .0005094414.

12. Multiply 1 hundredth by 1 thousandth, and add 1 tenth Ans. .10001.

to the product.

13. What is the product of one-tenth by one-tenth? one hundred by one-hundreth? one million by one-millionth?

14. What cost 672.325 pounds of coffee, at $0.1075 a pound? Ans. 72.2749375.