Rule II.— Write the numerator, and then place the decimal point so that the right-hand term shall express the denomination of the decimal, using ciphers when necessary.

NOTE.-To avoid ambiguity, where integers and decimals occur in the same written number, a comma should be placed between them; thus, three hundred and seven ten-thousandths should be written .0307, but three hundred, and seven ten-thousandths, 300.0007.

Express the following in decimal form:

2. Twenty-five hundredths. 3. 2 tenths and 8 hundredths. 4. 7 tenths and 9 hundredths. 5. Twenty-five thousandths. 6. 4 tenths and 7 thousandths. 7. Seven tenths and 8 thousandths.

8. Five hundred, and 25 thousandths.

9. Three tenths and 7 ten-thousandths.

10. Seven hundredths and 9 tenthousandths.

11. Three hundred, and 78 tenthousandths.

12. Five tenths, 6 hundredths, and 7 hundred thousandths.

13. Four hundredths, seven tenthousandths, and 6 hundred-thousandths.

14. Nine hundred and sixtynine hundred-thousandths.

15. Two tenths and three millionths.

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, and 6 thousand and 9 hundred-millionths.

1. Moving the decimal point one place to the right, multi plies the decimal by 10; two places, multiplies by 100; etc

For, if the point be moved one place to the right, each figure will express ten times as much as before, hence the whole decimal will be ten times as great; etc.

2. Moving the decimal point one place to the left, divides the decimal by 10; two places, divides by 100; etc.

For, if the point be moved one place to the left, each figure will express 1 tenth of its previous value, hence the whole decimal will be only 1 tenth as great; etc.

3. Placing a cipher between the decimal point and the decimal, divides the decimal by 10.

For, this moves each figure one place to the right in the scale, in which case they express 1 tenth as much as before, and hence the decimal is only 1 tenth as great.

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

For, each figure retains the same place as before, and hence expresses the same value as before, and consequently the value of the decimal is unchanged.

MENTAL EXERCISES.

1. How many tenths in 1? in ? in ? in ?

2. How many hundredths in ? in ? in ? in? in
3. How many hundredths in ? in
4. How many halves in .5? in .50? 5ths in .2? in .4?

? in? in ? in

?

?

4

5. How many 4ths in .25? in .75? eighths in .125? in .375? 6. Express as a common fraction and reduce to lowest terms, .5; .4; .8; 25; .50; .75; .80; .125; .625.

REDUCTION OF DECIMALS.

204. There are two cases of the reduction of decimals: 1st. To reduce decimals to common fractions.

2d. To reduce common fractions to decimals.

CASE I.

205. 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 45 which reduced to its lowest terms equals. 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 terms.

12. Reduce the complex decimal .2 to a common fraction. SOLUTION.-.2 is 23 tenths, which, by writing OPERATION.

2} 10'

the denominator, becomes which equals

10' .2}=

23_ }
10 10

=

, which reduced to its lowest terms equals 1. Therefore, etc.

30 = Ans.

206. 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 equal, 20 hundredths; of 20 hundredths equals 2 hundredths and 4 hundredths remaining; 4 hundredths equal 40 thousandths; of 40 thousandths is 5 thousandths. Therefore, equals .625.

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 used.

NOTES.-1. In many cases the division will not terminate; the common fraction cannot then be exactly expressed by a decimal. Such decimals are called interminate or infinite decimals.

2. The symbol + annexed to a decimal indicates that it contains other decimal terms. The symbol annexed to a decimal indicates that the last decimal term is increased by 1. term is greater than 5.

WRITTEN

This is often done when the next

EXERCISES.

ADDITION OF DECIMALS.

207. 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, 90.728?

OPERATION.

11.96

25.075

84.306

90.728

SOLUTION.-We write the numbers so that terms of the same order shall stand in the same column, and begin at the right to add. 8 thousandths, plus 6 thousandths, plus 5 thousandths, are 19 thousandths, which equals 1 hundredth and 9 thousandths; we write the 9 thousandths, and add the 1 hundredth to the next column: 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 nuu dredths; we write the 6 hundredths, etc.

212.069

Rule.-I. Write the numbers so that terms of the same order stand in the same column.

II. Add as in whole numbers, and place the decimal point between the units and tenths of 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. What is the sum of $25, $371, $28.371, $50.064, $15 37, $573, $153, and $23.87? Ans. $253.88.

7. Add 9 and 7 tenths, 41 and 8 hundredths, 75 and 54 hundredths, 128 and 187 thousandths. Ans. 254.507.

8. Add 187 and 5 thousandths, 49 and 9 hundred-thonsandths, 1876 and 245 millionths, 187 ten-thousandths, and 999 ten-millionths. Ans. 2112.0241349.

9. Add 798 and 9 ten-thousandths, 17 millionths, 18 thousandths and 98 ten-millionths, 67 hundred-thousandths and 95 ten-millionths. Ans. 798.0196063.

10. Find the sum of 3 tenths, 6 hundredths, 4 thousandths, 3 ten-thousandths, and 6 hundred-thousandths. Ans. .3696245.

SUBTRACTION OF DECIMALS.

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

OPERATION.

972.163

856.235

115.928

1. From 972.163 take 856.235. SOLUTION.-We write the numbers so that terms of the same order stand in the same column, and begin at the right to subtract. We cannot subtract 5 thousandths from 3 thousandths, hence we add ten thousandths to 3 thousandths, which equals 13 thousandths; 5 thousandths from 13 thousandths leaves 8 thousandths, which we write in the order of thousandths: since we have added 10 thousandths or 1 hundredth to the minuend, we must add 1 hundredth to the subtrahend; 1 hundredth and 3 hundredths are 4 hundredths; 4 hundredths from 6 hundredths leaves 2 hundredths, etc. Rule.-I. Write the subtrahend under the minuend, so that terms of the same order stand in the same column.

II. Subtract as in whole numbers, and place the decimal point between the units and tenths of the remainder.

2. From 707.325 take 623.452. 3. From 826.438 take 734.936.

4. From 78.3057 take 29.084.

5. From 1230.207 take 384.1231.

6. From 2.07 take 1.432765.

7. From .3 take 3 hundred-millionths.

Ans. 83.873.

Ans. 91.502. Ans. 49.2217.

Ans. 846.0839.

Ans. .637235. Ans. .29999997.

8. From 1 and .001 take .01 and .000001. Ans. .990999.

9. From 2 take 2 thousandths and 2 billionths.

Ans. 2.4974999975