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altered, (§ XLI.) I may therefore multiply both the 7 and the 16 in the above Fraction, by 10, 100, 1,000 or any other number, without altering the value. Multiply both, then, by 10,000 and the Fraction becomes 70000. Now (§ XLI.) I may divide both numerator and denominator by the same number without altering the value. Divide them both then, by 16, and the Fraction becomes 1375=( LVIII.) 4375 Ans.

10000

Either mode of explanation gives us the following rule.

NATOR.

ANNEX CYPHERS TO THE NUMERATOR AND DIVIDE BY THE DENOMITHE DECIMAL WILL CONSIST OF AS MANY PLACES AS THERE ARE CYPHERS ANNEXED.

NOTE. If there are not as many quotient figures as the rule requires, prefix cyphers enough to make out the number. An improper Fraction must, of course, be first reduced to a mixed number.

14. Reduce to a decimal.

25

A. .0016

15. Reduce to a decimal. A. .028

16. Reduce, to a decimal. A. .05625

17. Reduce to a decimal. A. .3333333+

NOTE. We see here, that we may go on forever, and the decimal will contin ue to repeat 33, &c.

18. Reduce to a decimal. A. .18181818181818+ NOTE. This decimal goes on, repeating like the other; but it repeats two figures, 18, instead of one, as before.

Decimals, which continually repeat the same figures, are called REPEATING DECIMALS, or REPETENDS.

When one figure is repeated, the decimal is called a SINGLE REPETEND; when two or more, a COMPOUND REPETEND.

Repeating decimals are also called CIRCULATES, or CIRCULATING DECI MALS. Properly, the term circulate, belongs to compound repetends.

When other decimal figures precede the repetend, it is called a MIXED REPETEND; when otherwise, a PURE, or SIMPLE REPETEND.

Repetends are also called INFINITE DECIMALS; decimals which terminate, or come to an end, are called FINITE.

In single repetends, the repeating figure is commonly written only once, with a point over it, thus, .3. In compound repetends, all the repeating figures are once, written, and a point placed over both the first and last, thus, is. This notation shows us all the figures that repeat, and if it is necessary to extend the decimal lower, for the purposes of Addition, Multiplication, &c. we can write it down immediately, without the trouble of calculation. For further particulars on this subject see § LXIX.

19. Change to a decimal. A. .142857

20. Change to a decimal.

A. .6

21. Reduce 23 to a decimal. A. .008 22. Reduce to a decimal. A. .6875

20

23. Reduce 17 to a decimal. A. .85 24. Reduce

to a decimal.

A. .03125

25. Reduce to a decimal. A. .037

26. Reduce to a decimal. A. .0384615

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27. Reduce, da, 44, 1750, xŝu, ata, o8o, J, and to decimals.

§ LX. 1. Reduce 1 pt. to the decimal of a gallon.

Ans. 1 pt.

Ans. 9h.

gal.=.125

day=.375

2. Reduce 9 hours to the decimal of a day.

3. Reduce 2 ft. 6 in. to the decimal of a yard. Ans. 2ft. 6in.=30in.=38-4yd.=.83 4. Reduce 5 fur. 16 rds. to the decimal of a mile.

Ans. .675

5. Reduce 12s. 6d. 3 qrs. to the decimal of a pound. Ans. .628125

NOTE. When decimals, whether finite or infinite run on to a great number of places, it will usually be sufficient to take them to three or four, and neglect the rest. For the lower decimals become so small that they are of little importance. If on carrying the decimal as many places as you wish, the remainder left is less than half the divisor, the decimal, as it stands will be sufficiently ac curate; if it be more, increase the last figure of the decimal by 1. It will be seen that neither of these decimals will be perfectly accurate, the former, being too small, and the latter too large. When a decimal is thus taken too small, the signis usually annexed to it, signifying that the true decimal is more. When it is taken too large, the sign is annexed, signifying that the true decimal is less. Decimals written with these signs are called APPROXIMATES, because they are only an approximation to the truth.

6. Reduce 47£. 16s. 73 d. to a decimal expression.

A. £47.8322916

NOTE. It would be sufficient to say £47.832+. We have carried the decimals several places, thronghout these examples, for the purpose of testing the accuracy of the pupil.

7. Reduce 83£; 19; 51 to a decimal.

Ans. £83.972916 or £83.973—.

8. Reduce 2 gals. to the decimal of a hhd.

Ans. 031746 or .032-.

9. Reduce 15s. 9d. 3 qrs. to the decimal of a pound. Ans. .790625 or .79+ or .791-. For practical purposes, there is a shorter mode of reducing shillings, pence and farthings to the decimal of a pound. It is not perfectly accurate, but sufficiently so, in most cases. Take the last example.

As 1 shilling is of a pound, so 2 shillings are

of a pound. For every 2 shillings, then, we have 1 tenth

10

1

in the decimal, and, as 15s. contains 2s. 7 times, we shall have .7. There is a shilling over. This is a half of i or a half of %==.05. For 15s. then, we have the decimal .75. Thus far, the method is perfectly accurate. 1 farthing is of a £., because I£. contains 20×12 X4 960 farthings. Now 960+ of itself=1000.— Therefore, any number of farthings, or 960ths. of £1., increased by of itself, will express the same value in 1,000ths of £1. If the farthings be 12, a 24th. part of them will be if more than 12, more than . In this case, then, if we add 1 to the farthings, they will be 1,000ths., with an error, less than a 1,000th. If they are 36, a 24th. part will be 1; if more, more than 1. In this case, then, if we add 2 to the farthings, they will be 1,000ths. with an error, less than a thousandth. If they are 24 or 48, a 24th. is 1, or 2, which added, gives 1,000ths exactly.

39

Now in the example above, we have 93d.=39 qrs.= 3 of a pound, 4 of a pound =.041, the decimal for the pence and farthings. Hence the whole decimal, for shillings, pence and farthings, is .75+.041.791, corresponding to one of the answers above.

Hence, to express shillings, pence and farthings in the decimal of a pound,

CALL EVERY 2 SHILLINGS, 1 tenth, eveRY ODD SHILLING 5 HUNDREDTHS, AND THE FARTHINGS IN THe pence and farthings, SO MANY THOUSANDTHS, adding 1 IF THE NUMBER IS BETWEEN 12 AND 36, AND 2, IF ABOVE 36.

NOTE. AS 48 farthings make a shilling, we never have occasion to go above this number. The above mode of finding the decimal of a pound, is called finding the decimal BY INSPECTION; that is, by sight, or without the trouble of a written calculation. Take the following examples.

10. Reduce 17s. 6d. 3qrs. to the decimal of a pound?

Ans. £0.878.

11. Reduce £19; 8; 7; 2, to a decimal expression.

Ans. £19.431.

12. Reduce £18; 16; 91, to a decimal expression.

Ans. £18.840.

13. Reduce 13s. 3d. 1qr. to the decimal of a pound. Ans. £0.664.

14. Reduce 14s. 8d. 3qrs.; 19s. 10d.; 16s. 73d. and 18s. 91d. to decimals.

15. Reduce £16; 17; 3; and £15; 8; 2, to decimals.

£11; 19; 5; £13; 9; 1,

ADDITION.

§ LXI. 1. A man paid debts as follows; to A, $37.3, to B, $94. 05, to C, $127.003, and to D, $1,843.375. How much did he pay

in all?

We have given this first example, in Federal Money, because the pupil will see at once that dimes should be added to dimes, cents to cents, &c. Of course he will place the numbers so that these denominations shall stand under each other.

37.3

94.05

127.003 $1,843.375

Ans. $2,101.728

2. Add 37.3; 94.05; 127.003, and 1,843.375. These numbers are the same as those in the last example, except that they are not Federal Money. They must therefore evidently be written down and added as before; for as dimes must be added to dimes, cents to cents, &c., so tenthe must be added to tenths, hundredths to hundredths, &c. This, it will be observed, brings all the decimal points in the given numbers under each other and the point in the result, of course, immediately under them. Hence the rule,

WRITE THE NUMBERS SO THAT THE DECIMAL POINTS MAY BE UNDER EACH OTHER; ADD AS IN WHOLE NUMBERS, AND PLACE THE POINT IN THE ANSWER DIRECTLY UNDER THE OTHER POINTS.

NOTE. Repetends should be extended, at least as far, and usually one place farther, than the longest finite decimal, before adding. When numbers are given in vulgar Fractions, they must be reduced to decimals, before addition takes place; and when the numbers are compound, the lower denominations should be made decimals of the higher.

EXAMPLES FOR PRACTICE.

3. A man purchased at one time, 7 cwt. of sugar; at another, 17% cwt.; at another, 15 cwt.; at another, 204 cwt. How many cwt. did he purchase in all? A. 60.744+

qr.

1 na.

4. In 3 pieces of cloth, are contained as follows: in the first, 25 yds. 3 qrs. 2 na.; in the second, 19 yds. 1 qr. 3 na.; and in the third, 17 yds. 1 What is the whole amount in yards and decimals. A. 62.625. 5. In one cask are contained 28 gals. 2 qts. 1 pt.; in another, 36 gals. 3 qts.; in another, 27 gals. 3 qts.; and in another, 16 gals. 1 pt. How much in all ? A. 109.25 gals.

6. One cask of sugar weighs 1 cwt. 2 qrs. 14 lb. ; another, 3 cwt. 14 lb.; another, 2 cwt. 1 qr.; and another, 1 cwt. 14 lb. How much do all weigh? A. 8.125 cwt.

7. If I travel 35 mls. 3 fur. 20 rds., one day; 41 mls. 2 fur. 10 rds., another; 17 mls. 4 fur., another; and 26 mls. 3 fur. 20 rds., another; how far do I travel in all ? A. 109.65625 mls.

8. If I pay $21 for one hhd. of molasses; $19 for another; $17 for another; 23 for another; and 18 for another; how much do I give for all? A. $101.00 9. Add $12.34565;-$7.891; $2.34; $14 and $0.0011. A. $36.57775.

10. Add .014; .9816; 1.32; 2.15914; .72913 and 3.0047.

A. 8.20857.

11. Add 27.148; 918.73; 14,016; 294,304.001; .7,138 and 221.701. A. 309,488.2938.

12. Add 312.984; 21.3918; 2,700.42; 3.173; 27.2 and 581.07. A. 3,646.2388.

MULTIPLICATION.

§ LXII. 1. At $1.25 per yd. what cost 3 yds. of cloth? This may be performed, either by Addition or by Multiplication. BY MULTIPLICATION.

BY ADDITION.
$1.25

1.25

1.25

$1.25

3

$3.75 Ans.

$3.75 Ans.

We have exhibited both modes, to show where the point ought to be placed, as this is the only difficulty in decimal calculations. When the multiplier is a whole number, then, there must be as many decimals in the product, as in the multiplicand.

$125
.3

2. At $125 a pipe, what cost .3 of a pipe of wine? $125 On the left we have found what 3 whole 3 pipes cost, by multiplying by 3. Now .3 is the tenth part as much as 3 whole ones. $375 Of course, we must take the tenth part of $37.5 Ans. the price of 3 whole pipes, for the price of .3 of a pipe. But (§ xxx.) cutting off one figure divides by 10. Of course, we have one decimal place in the product by .3 This is shown on the right. When the multiplicand is a whole number, then, there must be as many decimals in the product, as in the multiplier.

3. At $1.25 per lb. what cost .3 pound of tea?

If it were 3 whole pounds, we should obtain $3.75, as in Ex. 1. But it is the tenth part of 3 lb, being .3 lb., and therefore, the price will be a tenth part as much=$0.375 Ans.

When neither factor is a whole number, then, there must be as many decimals in the product as in both factors. Observation of the following, may, perhaps, render the principle more clear. We take the same numbers, making them at first whole, and moving the decimal point one place at each step.

125

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12.5

1.25

.125

.125

.125

.0125

[blocks in formation]

Hence the rule,

MULTIPLY AS IN WHOLE NUMBERs, and point ofF AS MANY DECIMALS

IN THE PRODUCT, AS THERE ARE IN BOTH FACTORS.

NOTE. From observing the three latter illustrations, the pupil will see that when there are not as many figures in the product as the rule requires, cyphers must be prefixed to make out the number.

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