read, four hundred units and thirty-five thousandths; while .435 is read, four hundred and thirty-five thousandths. When the mixed number is concrete, (yards,) instead of saying 400 units, &c., say, 400 yards and 35 thousandths.

2. Another method: Read the whole number, add the word deci mal, then read the fraction. Thus, read, 25.025, twenty-five, decimal, twenty-five thousandths; or, twenty-five, decimal, nought-two-five.

3. Before reading the decimal, let the pupil ascertain the name of the right hand order. Begin at the separatrix, point to each order successively, and pronounce its name; thus, tenths, hundredths, thousandths, ten-thousandths, &c.

ART. 176. To WRITE DECIMALS; First, write the number of parts expressed by the decimal, as a whole number; then prefix ciphers till the right hand figure stands in the required order.

Thus, in writing twenty-five ten-thousandths, first write 25, then prefix two ciphers, so that, beginning at tenths to name the orders, 5 may stand in the order of ten-thousandths; .0025

REVIEW.-178. How are whole numbers and decimals written in the same line? What are they called? What the decimal point called? 174. If the dollar be taken as the unit, what are dimes? Cents? Mills? How express any sum in U. S. Money? Give examples.

175. What the rule for reading decimals? REM. 1. How read abstract mixed numbers? Why? 2. What other method? 176. What the rule for writing decimals

25 10000'

NOTE. To change a Com. Fraction whose denominator is 1, with ciphers annexed, as, to a decimal form, prefix so many ciphers to the numerator, that the number of places in the decimal shall equal the ciphers in the denominator, Art. 169.

Express these Fractions and Mixed numbers in Decimals.

8. 105 2. 13,7, 24,80, 301250, 610000, 1000000•

EXERCISES IN WRITING DECIMALS.

3. Four tenths.

19. Eighty-thousand and six

4. Two tenths and six hun- ten-millionths. dredths.

5. Thirty-five hundredths.

6. Eight hundredths.

7. Five thousandths.

8. Three hundred and four thousandths.

9. Four thousand one hundred and twenty-five ten-thousandths.

20. Two hundred-millionths. 21. Nine hundred and seven hundred-millionths.

22. 20 million 20 thousand and three hundred-millionths.

23. One million ten thousand and one hundred-millionths.

24. Twenty units and twentyfive hundredths.

10. Two hundred and five ten- 25. One hundred and six units thousandths. and thirty-seven thousandths.

11. Eight ten-thousandths.

26. One thousand units and

12. 20 thousand three hundred one thousandth. and four hundred-thousandths. 13. Six hundred and five hundred-thousandths.

14. Ninehundred-thousandths. 15. Three hundred thousand and four millionths.

16. 203 millionths.

17. Seven millionths.

18. Twenty-four ten-millionths.

27. Two hundred units and twenty-five thousandths.

28. 29 units and 29 millionths. 29. One million and five bil lionths.

30. Two hundred units and two ten-billionths.

31. Sixty-five units and six thousand and five millionths.

ART. 177. Mixed decimal numbers may be read as though they were entirely decimals.

Thus, 6.56%

it may be read, 65 tenths.

Hence, when a mixed decimal number is expressed in decimals,

it may be written as a whole number, and the point so placed, that the right hand figure shall stand in the required order.

32. One hundred and forty-three tenths.

33. Three thousand two hundred and four hundredths.

34. Ten thousand five hundred and two ten-ihcusandths.

ADDITION OF DECIMALS.

ART. 178. 1. Add 4 and 4 ten-thousandths; 28 and 35 thousandths; 8 and 7 hundredths; and 9404 hundredthousandths.

SOLUTION. Since only numbers of the same unit value can be added, write figures of the same order under each other.

OPERATION.

4.0004

28.035

8.07 ..09404

Since ten units of any order (Art. 173) make one unit of the order next higher, add the figures in each order, and carry one for every ten, as in Addition of Simple Numbers, placing the separatrix under the points above.

Ans. 40.19944

2. Find the sum of 3 units and 25 hundredths; 6 units and 4 tenths; and 35 hundredths.

Ans. 10.

Rule for Addition.— Write numbers of the same order under each other, tenths under tenths, hundredths under hundredths, &c.

Then add, as in Addition of Simple Numbers, and point off in the amount as many places for decimals as are equal to the greatest number of decimal places in either of the given numbers.

PROOF.-The same as in Addition of Simple Numbers.

3. Add 21.611; 6888.32; 3.4167 Ans. 6913.3477 4. Add 6.61; 636.1; 6516.14; 67.1234; and 5.1233 Ans. 7231.0967

REVIEW.-177. How may mixed decimals be read? Give an example. When a mixed decimal is expressed in words, how written?

178. In writing decimals to be added, why place figures of the same order under each other? Why carry one for every ten? Where place the separating point? How prove Addition of decimals?

5. Add 4 and 8 tenths; 43 and 31 hundredths; 74 and 19 thousandths; 11 and 204 thousandths. Ans. 133.333 6. Add 45 and 19 thousandths; 7 and 71 hundredthousandths; 93 and 4327 ten-thousandths; 6 and 401 ten-thousandths. Ans. 151.49251

7. Add 432 and 432 thousandths; 61 and 793 tenthousandths; 100 and 7794 hundred-thousandths; 6.009; 1000 and 1001 ten-thousandths. Ans. 1599.69834

8. Add 16 and 41 thousandths; 9 and 94 millionths; 33 and 27 hundredths; 8 and 969 thousandths; 32 and 719906 millionths. Ans. 100.

9. Add 204 and 9 ten-thousandths; 103 and 9 hundred-millionths; 42 and 9099 millionths; 430 and 99 hundredths; 220.0000009 Ans. 999.99999999 10. Add 35 ten-thousandths; .00035; 35 millionths and 35 ten-millionths. Ans. .0038885

For additional problems, see Ray's Test Examples.

SUBTRACTION OF DECIMALS.

ART. 179. 1. From 20 and 14 thousandths, subtract 7 and 21 ten-thousandths.

OPERATION.

20.014

7.0021

SOLUTION.-Write numbers of the same order under each other, because the difference only between numbers of the same unit value can be found. Since annexing ciphers to a decimal does not alter its value, Art. 172, Ans. 13.0119 regard the place of ten-thousandths in the minuend, as occupied by a cipher. Then Subtract as in Simple Numbers.

*2. From 5 and .03 take 2 and .115

Ans. 2.915

Rule for Subtraction.- Write the less number under the greater, tenths under tenths, hundredths under hundredths, &c. Subtract, as in Simple Numbers, and point off the decimal places as in Addition of Decimals.

PROOF. The same as in Subtraction of Simple Numbers.

REVIEW.-179. In Subtraction of decimals, why are tenths written under tenths, hundredths under hundredths, &e.?

13. 25 thousandths take 25 millionths. Ans. .024975

MULTIPLICATION OF DECIMALS.

ART. 180. Multiplication of decimals embraces two cases. 1. To multiply together a decimal and a whole number. 2. To multiply together two decimals.

ART. 181. 1. Multiply 125 thousandths by 9.

.125

SOLUTION.-Regard 125 as the numerator of a OPERATION. fraction, the denominator being 1000. Then, since a fraction is multiplied by multiplying its numerator, 125 × 9=1125 thousandths = 1.125, by proper pointing (Art. 177).

9

Ans. 1.125

In this case, the number of decimal places in the product must equal the number in the multiplicand. If 9 be multiplied by .125, the product will also be 1. 125 (Art 30).

*2. Multiply 35 hundredths by 7. *3. Multiply 2 tenths by 8 tenths.

OPERATION.

Ans. 2.45

.2X.8X8=1.16 Ans.

4. Multiply 2 hundredths by 4 tenths.

OPERATION.

.02X.480X10=1000.008 Ans.

Here, the decimals are converted into common fractions, for

REVIEW.-179. How is the subtraction performed? Why? Give the Rule. 180. What two cases are embraced in Multiplication of decimals? 181. How multiply a decimal fraction by a whole number?