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5 2 5
DECIMAL FRACTIONS, I LII. Q. When such fractions as these occur, viz. iv, 1802 18s, how is a unit supposed to be divided ?
A. Into 10 equal parts, called tenths; and each tenth into 10 other equal parts, called hundredths, and each hundredth" into 10 more equal parts, called thousandths, &c.
Q. How is it customary to write such expressions ?
A. By taking away the denominator, and placing a comma before the numerator.
Let me see you write down, in this manner, or to the
A. From the Latin word decem, signifying ten; because they increase and decrease in a tenfold proportion, like whole numbers. Q. What are all other fractions called ?
4. Vulgar, or Common Fractions.
Q. In whole numbers, we are accustomed to call the right-hand figure, units, from which we begin to reckon, or numerale; hence it was found convenient to make the same place a starting point in decimals; and to do this, we make use of a comma; what, then, is the use of this comma ?
A. It merely shows where the units' place is.
A. Because it separates the decimals from the whole numbers.
Q. What is the first figure at the right of the separatrix called ?
sandths, the fourth ten thousandths, and so on, aš in the numeration of whole numbers.
Let me see you write down again to in the form of a decimal.
Q. As the first figure at the right of the separatrix is tenths, in writing down To, then, where must a cipher be placed ?
A. In the tenths' place.
A. By placing 2 ciphers at the right of the separatrix, that is, before the 7.
Let me see you write it down.
A. Because in Toro, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals.
Q. What does ,5 signify?'
Q. What does ,05 signify?
Q. Now, as is =, and as multiplying Táo by 10 produces
, which is also equal to 1, how much less in value is ,05 than ,5? A. Ten times. Q. Why?
A. Because the parts in tá are ten times smaller than in B; and, as the numerator is the same in both expressions, consequently, the value is lessened 10 times.
Q. How, then, do decimal figures decrease in value from the left towards the right ?
A. In a tenfold proportion.
R. We have seen that ,5 is 10 times as much in value as ,05, or Top; what effect, then, does a cipher have placed at the left of decimals?
A. It decreases their value in a tenfold proportion.
Q. Since decimals decrease from the left to the right in a tenfold proporcion, how, then, must they increase from the right to the left ?
A. In the same proportion.
Q. Since it was shown, that ,5 = ; 25 = ; what, then, will always be the denominator of any decimal expression ?
A. The figure 1, with as many ciphers placed at the right of it as there are decimal places.
Let me see you write down the following decimals on your slate, and change them into a common or vulgar fraction, by placing their proper denominators under each, viz. ,5 ,05,005 ,62 ,0225 ,37.
Q. ,25 is or=#, and ,5 is = }; which, then, is the most in value, ,25 or ,5 ?
Q. By what, then, is the value of any decimal figures determined ?
A. By their distance from the units' place, or separatrix.
Q. When a whole number and decimal are joined together, thus, 2,5, what is the expression called ?
A. A mixed number.
Q. As any whole number may be reduced to tenths, hundredths, thousandths, &c. by annexing ciphers, (for multiplying by 10, 100, &c. thus, 5 is 50 tenths, 500 hundredths, &c.; how, ihen, may any mixed number be read, as 25,4 ?
A. 254 tenths, giving the name of the decimal to all the figures.
Q. How is 25,36 read?
Q. What would 5125 thousandths be, written in the form of a vul. gar or common fraction ?
This is evident from the fact, that is an improper fraction), reduced 10 a mixed number again, is equal to 5,125.
The pupil may learn the names of any decimal expression, as far as ten-millionths, also how to read or write decimals, from the follownig Table :
read 5 Tenths, 18=. , 06.
read 6 Hundredths. co= ,025. read 25 Thousandths. TO007
read 1328 Ten-Thousandths. 78=..7,8.
read 7, and 8 Tenths. 61008000=..6,000009. read 6, and 9 Millionths.
26 =.26,25.... read 26, and 25 Hundredths. 3T0oooooo=..3,000000 8 read 3, and 8 Ten-Millionths. 365=3 65,0.0000 0 0 read 365.
Exercises for the Slate. Write in decimal form 7 tenths, 42 hundredths, 62 and 25 hundredths, 7 and 426 thousandths, 24 thousandths, 3 ten-thousandths, 4 hundredths, 2 ten-thousandths, 3 millionths.
Write the fractional part of the following numbers in the form of decimals, viz. 676, 1, 6278, 27, 3180, 2621060, 32 δόσ, οτσσσσσ, 45Ισσσσσσ, τσσσσσσσ, 5τσόσσ.
Write the following decimal numbers in the form of vulgar
8. 6,005. 3. 23,75.
9. 3,00025. A. 3Toto. 4. 11,8.
Q. What money is adapted to decimal rules ?
Ă. As 10 dimes make a dollar, and 10 cents a dime, &c., dimes are 10ths of a dollar, cents are 100ths, and mills are 1000ths of a dollar.
Q. How are 3 dollars 2 dimes 4 cents and 5 mills written?
ADDITION OF DECIMALS.
LIII. Q. As we have seen that decimals increase from right to left in the same proportion as units, tens, hundreds, &c., how, then, may all the operations of decimals be performed ?
A. As in whole numbers. Note.—The only difficulty which ever arises, consists in determining where the decimal point ought to be placed. This will be noticed in its proper place.
1. A merchant bought 510 barrels of rice at one time for $27166, at another 107 of a barrel for $4,255, at another 17% of a barrel for $76, and at another 100 of a barrel for $26020; how many barrels did he buy in all ? and what did they cost him? OPERATION.
As we have seen Barrels.
Dollars. that decimals cor5,2
27,825 respond with the de,62
4,255 nominations of Fed278
eral Money, hence
we may write the de89
2,627 cimals down,placing
dimes under dimes, Ans. 6,988 barrels, for $ 35,427 cents under cents,
&c., that is, tenths under tenths, hundredths under hundredths, &c., and add them up as in Addition of Federal Money. From these illustrations we derive the following
A. Tenths under tenths, hundredths under hundredths, and so on.
Q. How do you proceed to add ?