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EDUCE the Fractional part into Decimals of the highest name mentioned, then state the question and proceed as in the Rule of Three Direct, observing to point off the Decimal places as has been taught in Multiplication and Division of Decimals.

1. Suppose I give 6s. 3d. for 43 yards of cloth; what will 48 yards of the same come to at that rate?

Answ. 31..1907 or 31. 3s. 93d.

4.75: 3125:: 48.56 one Lo

2. If 25. of Tea cost. 5s. what will 1431b. come to at the same rate? Answ. 71...375 or 71. 7s. 6d.

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3. If 1. of sugar cost 11d. what will 4hhds. each weighing neat 4 Cut. 2qrs. 14. cost at the same rate? Answ. 1011. .4417 or 1011. 8s. 10d.

00892855:489583:

18.5

4. A grocer buys 4 chests of tea, each weighing neat 2Cwt. 3grs. 14. for 906l. 10s. what rate did he give tb.? Answ. .7038 or 14s. 02d.

11.5: 906.
6.5:00 89 2858

5. An oilman bought 4 tuns, 201 gallons of Florence oil for 240/. 16s. 6d. but by misfortune it chanced to leak out 24 gallons: I desire to know at what rate he may sell the remainder gallon to be no loser?

or 4s. 03d. //85: 240-825:: k

CHAP. VII.

Answ..20322

OF CIRCULATING DECIMALS.

HE following method of managing Circulating Deci

metic that I have seen, I chose to deliver it by itself, detached from the common doctrine of Decimals, before laid down.

And first it will be proper to shew how to multiply and divide by 9, 99, 999, &c. in a contracted and very easy way.

I. To multiply by 9, 99, 999, &c.

Write as many cyphers as there are nines in the multiplier to the right hand of the multiplicand, and from the result subtract the multiplicand and the remainder will be the product.

1. Let it be required to multiply 456 by 9, &c.

One cypher added to 456 makes 4560-456 × 10
From which subtract the multiplic. 456=456 × 1

The remainder is the product 4104 456 × 10-1

2. Two cyphers added to 456 make 45600456×100 From which subtract 456 456 X1

The remainder is the product 44144-456 × 100-1

468 by 94212, 3726 × 99—368874,7568 × 999-7560432

II. To divide by 9, 99, 999, &c.

Divide the given dividend into periods of as many places as there are 9's in the divisor, beginning from the left hand, and annex as many cyphers to the right hand of the number as may be wanted to complete a period.

Then write the figures of the left hand period, under the next to the right hand, add these together, and place their sum under the third period, (if the sum amount to more figures than are in a period, the highest will of course fall under the lowest place of the second period.) In like manner add this sum to the period, and place the result under the fourth and so on: Lastly, under the last figures place that figure, which would have been placed there (if any) suppose the work had been to proceed a period farther.

Add them all together; and cancel as many figures as there were cyphers annexed to the dividend; then from the figures that remain, cut off with a comma, from the right hand towards the left, as many figures as the divisor contains nines; so shall the figures to the left of the comma be the quotient, and those on the right the remainder, which if it be all nines, add 1 to the quotient.

Application.

Let it be required to divide 87325 by 99?

The dividend with a cypher to make 3 periods, 87.32.50 The first period written under the second,

The sum of 87 and 32 is

87.

1.19

1

882 070

Under

Under the last place I set 1, because if 50 and 119 were added, the one would be so placed, the whole sum is 882.07, the last O being cancelled for 0 added, i. c. 882 the quotient and 7 remainder.

Those decimals which are produced from vulgar fractions whose denominators measure their numerators with cyphers annexed are called finite or terminate decimals, because they consist of a determinate number of places.

Decimals, (produced from vulgar fractions, whose denominators do not measure their numerators) in which a figure is repeated continually, or in which the same figures circulate continually, are called Circulating Decimals, and the circulating figures are called Repetends, and if one figure only repeat, it is called a Single Repetend, as

.1111..3333.

To avoid the trouble of writing down unnecessary figures, a single repetend is denoted by a point (') over the repeating figure, viz. the decimal 11111 is expressed by .1, .3333 by .3'.

If other figures rise before the repeating figure, as 16.0833, or .083'; .06; such decimals are called mixed single Repetends,

Such as have figures circulating alternately, or every third, fourth, &c. the same, are called compound Repetends, such as .410101, .128123123123.

And if other figures arise before the figures which circulate, then the decimal is called a mixed compound Repetend. Note. Mixt Repetends, single or compound, may be

called mixt Circulates.

Compound Repetends are distinguished by a point over the first and last repeating figure: Thus, .010101 may be written .O'1', and .123123123.1′23′. .15656, .1′56'.

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As in multiplying and dividing by these in perfect deci. mals, it requires frequently that the decimal must be extended to a pretty large number of places to prevent a very considerable error resulting from their imperfection; to remedy this, and to make the result perfect with less trouble, it will be useful to consider their generation.

Now

1

Now as 9 in 10 is contained once and one remains, unity with cyphers annexed being divided by 9 ad in finitum, the quotient figures will still be 1, i. c. 4, which being reduced to a decimal, will produce the circulating decimal .I': and since .I' is the decimal equivalent to.2' will be equal to ; .3' to (3) ; .4' to ; .5' to ; .6' to (); 7' to ; .8 to; and .9' to (==) 1.

Therefore every single repetend is equal to a vulgar fraction whose numerator is the repeating figure and de

nominator 9.

99) 1.0000 (.0101

99

100

99

1, &c.

999) 1.000000 (.001001 999

1000

Again, being reduced to a decimal makes .010101, &c. and makes.001001001,&c.

99

or.0'1.0'01'; now
every compound repetend of
two figures will be some mul-
tiple of .O'I' and the same mul-
tiple of the vulgar fraction
equal thereto, that is the vul-
gar fraction whose numerator
is the two repeating figures,
and the denominator 99.
That is
.0'2'

.01';;= (37).0'3';= 9

(TT =).0'9', 37=(=).2′7′′.

In like manner every compound repetend of three figures is shewn to be produced from a vulgar fraction, whose numerator is the three repeating figures, and denominator 999.

And so universally, we may conceive that a decimal fraction, consisting only of a repetend, is equal to a vulgar fraction whose numerator is that repetend, and the denominator a number consisting of as many nines, as there are places in the repetend.

Next to find a vulgar fraction equal to a mixt circulate, consider the next circulate as divisible into its finite and circulating

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