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ADDITION OF DECIMALS. 207. Addition of Decimals is the process of finding the sum of two or more decimals.

MENTAL EXERCISES.

Add
1 x and %.
2, 3 and .5.
&. .04 and .03.
4. .6 and .04.

15. 25 and .5.

6. .07 and .08.
7. .05 and .136
8. 2.5 and 4.25

WRITTEN EXERCISES.

OPERATION.

1. What is the sum of 11.96, 25.075, 84.306, 90.728? SOLUTION.-We write the numbers 80 that terms of the same order shall stand in the same column, 11.96 and begin at the right to add. 8 thousandths, plus 6

25.075 thousandths, plus 5 thousandths, are 19 thousandths, 84.306 which equals 1 hundredth and 9 thousandths; we

90.728 write the 9 thousandths, and add the 1 hundredth to the next column: 2 hundredths, plus 7 hundredths,

212.069 plus 6 hundredths, equal 15 hundredths, and the i hundredth added is 16 hundredths, which equals 1 tenth and 6 nundredths; we write the 6 bundredths, etc.

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 or 87.09, 5&37, 95.42, 237.675.

Ans. 478.555 4. Add 18.79, 147.072, 856.709, 185.8761, 397.05784.

Ans. 160,5.50494 5. Add 59.874, 435.095, 672.328, 976.309, 8467.500843.

Ans. 10611.106843. 6. What is the sum of $25%, $37}, $28.37], $50.064, 815 371, $57}, $15], and $23.871?

Ans. $253.885 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-thon. sandths, 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, 61 hundredths, 41 thousandths, 31 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.

MENTAL EXERCISES. 1. Subtract ty from yo.

5. Subtract % from .9. 2. Subtract .4 from .9.

6. Subtract .05 from .5. 3. Subtract .06 from 08.

7. Subtract .25 frum .63. 4. Subtract .05 from .55.

8. Subtract .75 from 1.08.

WRITTEN EXERCISES. 1. From 972.163 take 856.235. SOLUTION.-We write the numbers 80 that terms of OPERATION. the same order stand in the same column, and begin 972.163 at the right to subtract. We cannot subtract 5 thou

856.235 sandths from 3 thousandths, hence we add ten thou

115.928 sandths 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.-1. 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.

Ans. 83.873. 3. From 826.438 take 734.936.

Ans. 91.502. 4. From 78.3057 take 29.084.

Ans. 49.2217. . 6. From 1230.207 take 384.1231

Ans. 846.083-9. 6. From 2.07 take 1.432765.

Ans. .637235. 7. From .3 take 3 hundred-millionths. Ans. .29999997. & From 1 and .001 take .01 and .000001, Ans. .990999. 9. From 24 take 2 thousandths and 2 billionths.

Ans. 2.4974999976

MULTIPLICATION OF DECIMALS. 209. Multiplication of Decimals is the process of finding the product when one or both factors are decimals.

MENTAL EXERCISES. What is the value of 1. 2 times to? 3X.4?

5. XX.6? .6X.7? 2. 2 times .6? 5X.6 ?

6. 5X Ito? 6X.087 3. Å times Zo? .5x.7? 7. %x%%? .8X.18? 4. A times it? .4*.89 8. .6x,.159 .12X.129

9. How many places in the product of units multiplied by tenthel Tenths by tenths ? Units by hundredths! Tenths by hundredths! Hundredths by hundredths ?

10. How many decimal places in the product if there are two in the multiplicand and one in the multiplier? If there are two in each factor? If three in multiplicand and two in multiplier ?

WRITTEN EXERCISES. 1. Multiply 7.23 by .46.

OPERA ION. SOLUTION.—7.23 multiplied by 46 equals 332.58, and

7.23 multiplied by 46 hundredths the product is 1 hundredth

.46 as great, which, by removing the decimal point two

4338 places to the left, becomes 3.3258. Hence, 7.23 mul

2892 tiplied by .46 equals 3.3258.

3.3258 SOLUTION 2D.—7.23X.46=78X4=8888=rotooX 33258–3.3258. From either of these solutions we derive the following

Rule.-Multiply as in whole numbers, and from the right of the product point off as many decimal places as there are in both factors, prefixing ciphers when necessary. 2. Multiply 27.13 by .67.

Ans. 18.1771. 3. Multiply 43.08 by 2.36.

Ans. 101.6688. 4. Multiply 79.52 by .019.

Ans. 1.51088. 5. Multiply 8.534 by 20.074. Ans. 171.311516. 6 Multiply 123.107 by 1.52.

Ans. 187.12264. 7. Multiply 512.073 by 35.08. Ans. 17963.52084. 8. Multiply 54.0079 by 7.072. Ans. 381.9438688. 9. Multiply 1.08096 by 3.5702. Ans. 3.859243392. 10. Multiply .03507 by .005873. Ans. .00020596611. u. Multiply 2.0709 by .000246. Ans. .0005094414. 12. Value of 94 x.084 x 10?

Ans. 8.3125.

18. Value of .5 of 1 *.054?

Ans. .0175. 14. Value of .07 of $x 300 x.094?

Ans. 1.65. 16. Value of $25.67 x .051+$15 x 6.037?

Ans. $91.8575. 16. 200 +.02 - **.84 +0.81–.1270 xf of 7})?

Ans. 164.7432. 17. Multiply 1 hundredth by 1 thousandth, and add 1 tenth to the product.

Ans. .10001. 18. What is the product of one-tenth by one-tenth? ono hundred by one-hundredth? one million by one-millionth?

DIVISION OF DECIMALS.

210. Division of Decimals is the process of finding the quotient when une or both terms are decimals.

MENTAL EXERCISES. What is the value vi 1. A -27.4;-2?.

6. 6-8? 6;.3? ;-.03? 2. Po ;-69.24;-6?

7. .8:48 8;-.47 8.04? 3. .50:-2? .5:27

8. 7.2-4.6? .72;.67 .72;.06? 4. .150:-2? .15--2?

9. 8.4+.7? .84;-.7? .844.07? 5. 6+-4? 0:4?.06+-4? 10, 96;.89 9.8;-.8? .96;.8?

11. What is the product of tenths by tenths! What then is the quotient of hundredths by tenths ? What is the product of tenths by hundredths ? What is the quotient of thousandths divided by tenths !

12. If there are three decimal places in the multiplicand and two in the multiplier, how many in the product? If there are ive in the dividend and three in the divisor, how many in the quotient?

WRITTEN EXERCISES.

OPERATION.

1. Divide 12.3084 by 2.34. SOLUTION.—Dividing as in whole numbers, we obtain the quotient 626; now,

since the

2.34)12.3084(5.26 dividend is the product of the divisor and 1170 quotient, the number of decimal places in the

608 dividend should equal the number in the divi

468 sor, plus the number in the quotient; hence the number of decimal places in the quotient

1404 equals the number of decimal places in the

1404 dividend, minus the number of places in the divisor; hence there should be four minus two, or two decimal places in the quotient, therefore the quotient is 5.26.

BOLUTION 22.–12.3084+2.34=Yödteto x }!!= Sidox 190 sito of 526=5.26. From either of these solutions we derive the following

Rule.-I. Annex ciphers to the dividend, if necessary to make the number of decimals equal to the number of deci mal places in the divisor.

II. Divide as in whole numbers, annexing ciphers to the dividend when needed to continue the division.

III. Point off as many decimals in the quotient as the number of decimal places in the dividend exceeds the number in the divisor.

NOTES.-1. When there are ciphers at the right of the divisor, cut them off, diride by the significant part, and then point off as many decimal places as before, plus the number of ciphers cut off.

2. Make complex decimals pure or divide them like common mixed numbers, or multiply both by the L. C. M. of the denominators, and then divide. 2. Divide 272.636 by 6.37.

Ans. 42.8. 3. Divide 281.8585 by 3.85.

Ans. 73.21. 4. Divide 57.85728 by 8.32.

Ans. 6.954. 5. Divide 50.38218 by 67.8.

Ans. .7431 6. Divide 31.421154 by 85.43.

Ans. .3678. 7. Divide 268.32615 by 4.705.

Ans. 57.03. 8. Divide 2.3697792 by 34.56.

Ans. .06857. 9. Divide 3411.3424 bv 5.678.

Ans. 600.8. 10. Divide 195.388698 by 32.185.

Ans. 6.0708. 11. Divide 1678.543 by 185.79321. Ans. 9.03446+.

What is the value of 12. .065+,026 ? Ans. 2.5.17. 1+.125.

Ans. 8. 13. .375+.025 ? Ans. 15. 18. .517.015, Ans. 34. 14. .008.04? Ans. .2. 19. 1554-.0625. Ans. 2480, 15. .012-.06? Ans. .2. 20. .00625; 25. Ans. .00025. 16. .004 : 2.5 ? Ans. .0016. / 21. 25-:.00625. Ans. 4000 29. (8.04:17+47-200 x 5000)-7? Ans. 1174.245. 28. (789-789)+(.75-.075.75 of 8)? Ans. 194.62. 24. 45.181+3.6 x.31=(3.5–3:48)? Ans. 8.5436411 26. ($347.84:$10.87) *.0025+.014 x 50? Ans. 32. 26. ($1080 x 3.27):$10.90—($790 --$3.95) ? Ans. 124.

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