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17. When both factors contain decimals, there must be as many places of decimals in the product as there are decimals in both the factors. According to article 16. Con. III. a unit in the product, before it is pointed, is of the same value of a unit in the factor when the comma is removed. Thus the product of 42,6 by 3, is-1278 before it is pointed, but it does not represent so many whole numbers, but tenths, because when the comma is removed from 42,6, it is in tenths.

If multiplying a number, containing decimals, by a whole number, gives a product, before it is pointed, of the same denomination with the decimal expression in the factor, then by multiplying two numbers together, both having decimals, the product will be of the same denomination, which the decimals of both the face tors placed one after the other, would express. The product of 36,48 by 8,16 is 2976768, and the decimals of the factors, written one after the other, would become ten thousandths, consequently the product is in ten thousandths, and must be divided by 10,000, which cuts off the four right hand figures, and renders it 297,6768.

Pup. From what you have said, the reason of pointing appears very plain, and I think I shall be able to answer any questions in multiplication of decimals.

Tut. To multiply decimals, or whole numbers accompanied by decimals ;

Place the factors and multiply as in whole numbers ; then point off from the product so many places for decimals as there are decimals in both the factors.

Multiply 223,86 by 2,500.
Multiply 8,84960 by 2,618.
Multiply 520,3 by 0,417.

Ans. 559,65000. Ans. 22,2425280.

Ans. 110,424.

18. It is sometimes necessary to put one or more ciphers on the left of the product, in order to give the number of decimals required. If you multiply 0,24 by 0,0004, the product of 24 by 4 will be 96, consisting of but two figures, whereas the product ought to consist of 6 figures, and ciphers must be prefixed, thus, 0,000096. Multiply 0,217 by 0,0431.

Ans. 0,0093527.

When decimals are to be multiplied by 10, 100, 1000, &c. it is done by removing the point 1, 2, 3, &c. places towards the right. Multiply 0,468 by 10.

Ans. 4,68. Multiply 0,62 4053 by 1000.

Ans. 624,053.

19. In division of numbers the magnitude of the quotient does not depend on the magnitude of the divisor and dividend, but on their relative magnitudes. If, then, it be required to divide 48,24 by 12, we should observe that 48,24 amount to 4824 hundredths. (17. Con. III.) Now as the divisor and dividend must be of the same denomination, 12 must be reduced to hundredths by annexing ciphers. Thus to divide 48,24 by 12, the numbers will stand as follows,

1 2 0 0 ) 4 8 2 4 ( 4

4 8 0 0

24

Hence to divide any number accompanied by decimals, by another number having decimals ;

Equalize the decimal places of the two numbers, viz. if the decimal places in the dividend exceed those in the dirisor, annex ciphers to the divisor, till its decimal places are equal to those of the dividend ; and if those of the divisor exceed those of the divi. dend, annex ciphers to the dividend till its decimals are equal to those of the divisor. Then proceed exactly as with whole numbers.

Divide 5673,21 by 23,0.
Divide 27845,96 by 9,8732.
Divide 200,5 by 231.

Ans. 246,660 &c. Ans. 2820,3581 &c.

Ans. 0,0867 &c.

Vulgar fractions may be reduced to decimal fractions by annexing ciphers to the numerator, and dividing it by the denominator.

20. This rule is founded on the following reasons ;The numerator being converted into decimal parts by annexing ciphers, can be divided by the denominator, and by this means becomes a decimal; because the decimal obtained, must bear the same proportion to its denominator that the numerator of the vulgar fraction bears to that denominator. If we wish to reduce į to a decimal, we place'a cipher at the right of the numerator, and divide it by the denominator, which gives 5 for a quotient, which is 0,5 and has the same proportion to 10, that I has to 2. Again, if we reduce pe to a decimal, we obtain 0,5; and 5 has the same proportion to 10 that 6 has to 12, and the same of any fraction whatever.

Reduce to a decimal.
Reduce to a decimal.
Reduce to a decimal.

Ans. 0,75.

Ans. 0,8. Ans. 0,875.

All operations with federal money are according to the rules of decimals. In writing any sum in federal money first write the dollars, then write the comma, and at the right of it place the cents and mills. The cents and mills are decimals of a dollar; and in all operations with Federal Money, the dollars are considered as whole numbers, and the cents and mills as decimals.

Questions. What are vulgar fractions ?

How does it appear that writing the remainder after division, over the divisor, expresses the true value of the remainder?

What are the two terms of the fraction called ?
Why are they called numerator and denominator?
How do you reduce fractions to their least terms ?

Give the reasons for proceeding as you do for obtaining a common divisor?

How do you multiply fractions together ?

Why does multiplying the numerator increase the fraction, and multiplying the denominator decrease it?

How do you divide one fraction by another?
How are whole numbers expressed fractionwise !
How do you find a common denominator ?

How does it appear that this will give a common deDominator ?

What do you do when there are mixed numbers ?
What do you do when there are compound fractions ?
How are fractions added ?
How are they subtracted ?
What are decimal fractions ?
How are they written?
How do they differ from vulgar fractions ?

How do you know what the denominator of a decimal fraction is, and why is it not expressed ?

How do ciphers, placed at the right of decimals, affect them?

How do you distinguish decimals from whole numbers ?

How do you point the figures in addition and subtraction of decimals ?

How do you point them in multiplication and division of decimals?

How does it appear that this will give a true expresgion ?

How do you reduce a vulgar to a decimal fraction ?

How does it appear that this will give the true value of the vulgar fraction?

In what respect is Federal Money like decimals ?

Examples for Practice.

1. What is the greatest common divisor of 720 and

Ans. 8. 1736 ? 2. Multiply of 7 by a

Ans. 13 3. Multiply f of 5 by 1 of 4.

Aps. 4. Multiply of by of 1 of 114.

Ans.

. 4 5. Divide 4 by .

Ans. 6. Divide of by of 1.

Ans. 7. Divide 3 by 9.

Ans. .

8. Divide 4204} by of 112.

Ans. 4218 9. Add }, 91 and of į together.

Ans. 910 1. 10. Add of 41, of and 94 together. Ans. 1256 11. From 961 take 144.

Ans. 8111 12. From 141 take of 19.

Ans. 112 13. From 13} take of 15.

Ans. 2. Add 126,042—0,032–6,568 and 16 together.

Ans. 148,642. 15. Add 0,00004–2,--14,6 and 12,6666 together.

Ans. 29,26664.

16. Add 4,--0,6-140 and 600 together. Ans. 744,6. 17. From 16 take 0,4689.

Ans. 15,5311. 18. From 2 take 0,16289.

Ans. 1,83711. 19. Multiply 146,003 by 62,16. Ang. 9075,54648. 20. Multiply 2,046 by 0,0004. Ans. 0,0008 184. 21. Divide 336,04 by 24,6.

Ans. 13,66 &c. 22. Divide 689,6 by 42,368.

Ans. 16,274 &c. 23. Reduce li to a decimal. Ans. 0,51612 &c. 24. Reduce 1216 to a decimal. Ans. 0,14228 &c.

CONVERSATION IV.

COMPOUND ADDITION, COMPOUND SUBTRACTION, AND REDUCTION.

Tut. What is Compound Addition?

Pup. I cannot tell. I should think by the name, that it was the addition of compound numbers; but what kind I cannot tell.

Tut. It is the addition of compound numbers, viz. numbers which increase in a different manner from the simple numbers, by its taking a certain number of one dedomination to make one of the next greater, and a differént number of this greater to make one of the next greater.

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