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2. Required the continued product of 3, 3, 5, and of 3.

3 13 x 3

Here Z X X X X =

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13 B
4 1

3. Required the product of

4. Required the product of

4 $

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8 4 X 2

of 9.
of 5.

5. Required the product of,, and . 6. Required the product of, 3, and 3. 7. Required the product of 3, 3, and 4. 8. Required the product of §, and 9. Required the product of 6, and 10. Required the product of 3 of 3, and § of 34. 11. Required the product of 3 and 4. 12. Required the product of 5, 3,

of %, and 4.

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DIVISION OF VULGAR FRACTIONS.

* PREPARE the fractions as before in multiplication: then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide: but if not, invert the terms of the divisor, and multiply the dividend by it, as in multiplication.

EXAMPLES.

= 13, by the first method.

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1. Divide 25 by §. Here 35 ÷ 3 = {
2. Divide by Here=× = × = = 4.
13.

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RULE OF THREE IN VULGAR FRACTIONS.

MAKE the necessary preparations as before directed (p. 36, 37); then multiply continually together the second and third terms, and the first with its parts inverted as in division, for the answer †.

* Division being the reverse of multiplication, the reason of the rule is evident.

Note. A fraction is best divided by an integer, by dividing the numerator by it; but if it will not exactly divide, then multiply the denominator by it.

This is only multiplying the 2d and 3d terms together, and dividing the product by the first, as in the rule of three in whole numbers.

EXAMPLES.

1. If of a yard of velvet cost of a pound sterling; what will of a yard

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Ans. 17 18 4 d.

2. What will 33 oz of silver cost, at 6s 4d an ounce ?
3. If of a ship be worth 2731 2s 6d; what are of her worth?

Ans. 2271 12s 1d.

4. What is the purchase of 1230l bank-stock, at 1083 per cent ?

5. What is the interest of 2731 15s for a year, at 3 per cent?

Ans. 13361 1s 9d.

Ans. 87 17s 11 d.

6. If of a ship be worth 731 1s 3d; what part of her is worth 250/ 10s?

Ans..

7. What length must be cut off a board that is 7 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad?

Ans. 18 inches.

8. What quantity of shalloon that is of a yard wide, will line 9 yards of cloth, that is 2 yards wide?

Ans. 31 yds. is 5s the bushel; Ans. 4 oz. poles broad, will Ans. 13 poles.

9. If the penny loaf weigh 6 oz. when the price of wheat what ought it to weigh when the wheat is 8s 6d the bushel? 10. How much in length, of a piece of land that is 11 make an acre of land? 11. If a courier perform a certain journey in 35 days, travelling 139 hours a day; how long would he be in performing the same, travelling only 11 hours a day?

Ans. 4065 days.

12. A regiment of soldiers, consisting of 976 men, are to be new clothed; each coat to contain 2 yards of cloth that is 13 yard wide, and lined with shalloon 7 yard wide; how many yards of shalloon will line them?

Ans. 4351 yds 1 qr 29 nails. Scholium. A rule for operations of this nature, where the first term is unity, was long in great use under the name of PRACTICE, and was broken down into a variety of separate cases adapted to the peculiar circumstances of each question. It was doubtless owing to the apparent complexity produced by the number of cases that it was generally considered very difficult of acquisition, and has now fallen into very general disuse. It is, however, an exceedingly useful process for daily purposes, and is, in fact, of very easy acquirement; but though the present Editor intended to give a page or two on the subject here, he is compelled to omit it for want of sufficient disposable space. A very systematic view of it is given by Mr. Rutherford in his edition of Gray's Arithmetic, and to that the student is referred *.

* The Editor takes this opportunity also to remark, that the very best work with which he is acquainted on Arithmetic for commercial purposes, and apart from ulterior views, is one bearing the title of "The Quadrantal System of Arithmetic, by Daniel Harrison." The scientific reader, too, will find some articles worthy of his attention in the work; though from its object it does not take a strictly scientific form.

DECIMAL FRACTIONS.

A DECIMAL FRACTION is that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus is 4, and is 24, and 17 is 074, and is 00124; where

74 1000

100000

ciphers are prefixed to make up as many places as there are ciphers in the denominator, when there is a deficiency in the figures. Thus, the understood denominator of a decimal is always either ten, or some power of ten; whence its name.

A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point. Thus, 3-25 is the same as 3100, or foo.

25

325

Ciphers on the right-hand of decimals make no alteration in their value; for ·4, or ·40, or ·400 are decimals having all the same value, each being = , or 3. But when they are placed on the left-hand, they decrease the value in a tenfold proportion: Thus, 4 is, or 4 tenths; but 04 is only 1, or 4 hundredths, and 004 is only Too, or four thousandths.

In decimals as well as in whole numbers, the values of the places increase towards the left-hand, and decrease towards the right, both in the same tenfold proportion; as in the following scale or table of notation.

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ADDITION OF DECIMALS.

SET the numbers under each other according to the value of their places, as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right-hand, add up all the columns of numbers as in integers; and point off as many places for decimals, as are in the greatest number of decimal places as any of the lines that are added; or, place the point directly below all the other points.

EXAMPLES.

1. Add together 29.0146, and 3146·5, and 2109, and 62417, and 14:16.

29.0146 3146.5

2109.

62417

14.16

5299-29877 the sum.

2. What is the sum of 276, 39 213, 72014 9, 417, and 5032 ?

Ans. 77779-113.

3. What is the sum of 7530, 16.201, 30142, 957.13, 672119, and 03014? Ans. 8513 09653.

4. What is the sum of 312-09, 3.5711, 7195-6, 71-498, 9739-215, 179, and ⚫0027 ? Ans. 17500 9768.

SUBTRACTION OF DECIMALS.

PLACE the numbers under each other according to the value of their places, as in the last rule. Then, beginning at the right-hand, subtract as in whole numbers, and point off the decimals as in addition.

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MULTIPLICATION OF DECIMALS.

* PLACE the factors, and multiply them together the same as if they were whole numbers. Then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers.

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1000

* The rule will be evident from this example :-Let it be required to multiply ·12 by 361; these numbers are equivalent to 1 and 6; the product of which is 361 1332 = 04332, by the nature of Notation, which consists of as many places as there are ciphers, that is, of as many places as there are in both numbers. And in like manner we reason for any other numbers. As a general investigation, however, let the one factor have m decimal places and the other n; and let all the figures of the first number, taken as integers, be expressed by M, and all those M N of the other by N. Then the actual numbers are and

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10m 10. Whence, their product is

10m+n that is, there are m + n decimals in the quotient.

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To multiply decimals by 1 with any number of ciphers, as by 10, or 100,

or 1000.

THIS is done by only removing the decimal point so many places farther to the right-hand, as there are ciphers in the multiplier; and subjoining ciphers if need be.

1. The product of 51.3 and 1000 is 51300.

2. The product of 2.714 and 100 is

3. The product of 916 and 1000 is

4. The product of 21:31 and 10000 is

CONTRACTION II.

To contract the operation so as to retain only as many decimals in the product as may be thought necessary, when the product would naturally contain several more places.

REMOVE the decimal point of the multiplier (if necessary) until the left-hand figure is an integer in the unit's place; and so many places as you have moved the decimal point in the multiplier to the left or to the right, remove, on the contrary, the decimal point in the multiplicand to the right or to the left. Then, place the multiplier under the multiplicand in the usual way; and begin to multiply by the left-hand figure of the multiplier, retaining in the product only so many decimals as you wish to have at last. Then, multiply by the remaining figures in the multiplier one by one, from the left towards the right; as you proceed, set each product one figure more to the left-hand; and, of course, leave out one figure more to the right-hand in each successive multiplication. The sum of these successive lines of products will give the general product required. It will always be better to calculate one place of decimals more than are required by the question. See the subsequent example and remarks.

In multiplying be very careful to increase the first right-hand figure retained in each line by what would be carried on from the figures omitted, in this manner: viz. add 1 if the preceding number fall between 5 and 14, 2 from 15 to 24, 3 from 25 to 34, 4 from 35 to 44, and so on. This process will usually make the general product true to the last place of decimals.

EXAMPLES.

1. Multiply 2.714986 by 924 1035, so as to retain only 4 places of decimals in the product.

This is evidently the same as to multiply 271-4986 by 9-241035; where the decimal point in the multiplicand is moved 2 places to the right-hand, and that in the multiplier 2 to the left.

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