He gave them of a bushel apiece, expressed in the form of common fractions; but it is proposed to express it in decimals.

I first suppose each bushel to be divided into 10 equal parts or tenths. The five bushels make. I perceive that I cannot divide into exactly 8 parts, therefore I suppose each of these parts to be divided into 10 equal parts; these parts will be hundredths. 5= {9}• 500 But 500 cannot be divided by 8 exactly, therefore I suppose these parts to be divided again into 10 parts each. These parts will be thou5000 may be divided by 8 exactly, f Ans. .625 of a bushel each.

sandths. 5= 5000

of 5000 is 625

1000

1000

1000

or .625. Instead of trying until I find a number that may be exactly divided, I can perform the work as I make the trials. For instance, I say 5 bushels are equal to 50 of a bushel. of 50 is, and there are left to be divided into 8 parts. I then suppose these 2 tenths to be divided into ten equal parts each. They will make 20 parts, and the parts are hundredths. of 20 are, and there are left to be divided into 8 parts. I suppose these 4 hundredths to be divided into 10 parts each. They will make 40 parts, aud the parts will be thousandths. of 18 is To Bringing the parts, To, and together, they make

100

of a bushel each, as before.

The operation may be performed as follows:

50 (8 48

1000

or .625

Then

I write the 5 as a dividend and the 8 as a divisor. I multiply 5 by 10, (that is, I annex a zero) in order to reduce the 5 to tenths. Then of 50 is 6, which I write in the quotient and place a point before it, because it is tenths. There is 2 remainder. I multiply the 2 by 10, in order to reduce it to hundredths. of 20 is 2, and there is 4 remainder. I multiply the 4 by 10, in order to reduce it to

thousandths.

each, as before.

of 40 is 5. The answer is .625 bushels

In Art. X. it was shown, that when there is a remainder after division, in order to complete the quotient, it must be written over the divisor, and annexed to the quotient. This fraction may be reduced to a decimal, by annexing zeros, and continuing the division.

Divide 57 barrels of flour equally among 16 men.

57 (16
48

In this example the answer, according to Art. X., is 3 bushels. But instead of expressing it so, I annex a zero to the remainder 9, which reduces it to tenths, then dividing, I obtain 5 tenths to put into the quotient, and I separate it from the 3 by a point. There is now a remainder 10, which I reduce to hundredths, by annexing a zero. And then I divide again, and so on, until there is no remainder.

The first remainder is 9, this is 9 bushels, which is yet to be divided among the 16 persons; when I annex a zero I reduce it to tenths. The second remainder 10 is so many tenths of a bushel, which is yet to be divided among the 16 persons. When I annex a zero to this I reduce it to hundredths. The next remainder is 4 hundredths, which is yet to be divided. By annexing a zero to this it is reduced to thousandths, and so on.

The division in this example stops at ten-thousandths; the reason is, because 10000 is exactly divisible by 16. If I take of 1999 I obtain 5625 or .5625, as above.

16

10000

100009

There are many common fractions which require so many

figures to express their value exactly in decimals, as to render them very inconvenient. There are many also, the value of which cannot be exactly expressed in decimals. In most calculations, however, it will be sufficient to use an approximate value. The degree of approximation necessary, must always be determined by the nature of the case. For example, in making out a single sum of money, it is considered sufficiently exact if it is right within something less than 1 cent, that is, within less than of a dollar. But if several sums are to be put together, or if a sum is to be multiplied, mills or thousandths of a dollar must be taken into the account, and sometimes tenths of mills or ten-thousandths. In general, in questions of business, three or four decimal places will be sufficiently exact. And even where very great exactness is required, it is not very often necessary to use more than six or seven decimal places.

A merchant bought 4 pieces of cloth; the first contained 28 yards; the second 344; the third 30; and the fourth 42 yards. How many yards in the whole?

In reducing these fractions to decimals, they will be sufficiently exact if we stop at hundredths, since To of a yard is only about of an inch.

is exactly .6. If we were to continue the division of 3, it would be .28571, &c.; in fact it would never terminate; but .28 is within about one of of a yard, therefore sufficiently exact. is not so much as, therefore the first figure is in the hundredths' place. The true value is .0666, &c., but because is more than of τόσο I call it .07 instead of .06. 3 is equal to .7777, &c. This would never terminate. Its value is nearer .78 than .77, therefore I use .78.

6

1000

When the decimal used is smaller than the true one, it is well to make the mark+after it, to show that something more should be added, as 2 = 28. When the fraction is too large the mark should be made to show that something should be subtracted, as.07.

The numbers to be added will now stand thus :

From the above observations we obtain the following general rule for changing a common fraction to a decimal: An ner a zero to the numerator, and divide it by the denominator, and then if there be a remainder, annex another zero, and divide again, and so on, until there is no remainder, or until a fraction is obtained, which is sufficiently exact for the purpose required.

Note. When one zero is annexed, the quotient will be tenths, when two zeros are annexed, the quotient will be hundredths, and so on. Therefore, if when one zero is annexed, the dividend is not so large as the divisor, a zero must be put in the quotient with a point before ít, and in the same manner after two or more zeros are annexed, if it is not yet divisible, as many zeros must be placed in the quotient.

14783

Two men talking of their ages, one said he was 37, 3847 years old, and the other said he was 6427 years old. What was the difference of their ages?

If it is required to find an answer within 1 minute, it will be necessary to continue the decimals to seven places, for 1 minute is of a year. If the answer is required only within hours, five places are sufficient; if only within days, four places are sufficient.

6424364.8520000 37-444737.2602313+

Ans. 27.5917687 years.

It is evident that units must be subtracted from units, tenths from tenths, &c. If the decimal places in the two numbers are not alike, they may be made alike by annexing zeros. After the numbers are prepared, subtraction is per formed precisely as in whole numbers.

Multiplication of Decimals.

XXVII. How many yards of cloth are there in seven pieces, each piece containing 19 yards?

19궁

= 19.875
7

Ans. 139.125 = 139,12% 139 yards.

N. B. All the operations on decimals are performed in precisely the same manner as whole numbers. All the difficulty consists in finding where the separatrix, or decimal point, is to be placed. This is of the utmost importance, since if an error of a single place be made in this, their value is rendered ten times too large or ten times too small. The purpose of this article and the next is to show where the point must be placed in multiplying and dividing.

In the above example there are decimals in the multiplicand, but none in the multiplier. It is evident from what we have seen in adding and subtracting decimals, that in this case there must be as many decimal places in the product, as there are in the multiplicand. It may perhaps be more satisfactory if we analyze it.

7 times 5 thousandths are 35 thousandths, that is, 3 hundredths and 5 thousandths. Reserving the hundredths, I write the 5 thousandths. Then 7 times 7 hundredths are 49 hundredths, and 3 (which I reserved) are 52 hundredths, that is, 5 tenths and 2 hundredths. I write the two hundredths, reserving the 5 tenths. Then 7 times 8 tenths are 56 tenths, and 5 (which I reserved) are 61 tenths, that is, 6 whole ones and 1 tenth. I write the 1 tenth, reserving the 6 units. Then 7 times 9 are 63, and 6 are 69, &c. It is evident then, that there must be thousandths in the product, as there are in the multiplicand. The point must be made between the third and fourth figure from the right, as in the multiplicand, and the answer will stand thus, 139.125 yards.

Rule. When there are decimal figures in the multiplicand only, cut off as many places from the right of the product for decimals, as there are in the multiplicand.

If a ship is worth 24683 dollars, what is a man's share worth, who owns of her.

}= .375

375

. The question then is, to find of