then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required.

EXAMPLES.

1. Required the product of and 3. Here, X = 3% =, the answer.

Or, & × = × 1 =

2. Required the continued product of, 34, 5, and 2 of 3.

RULE.* 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, then invert the terms of the divisor, and multiply the dividend by it, as in Multiplication.

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

• 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.

RULE OF THREE IN VULGAR FRACTIONS.

RULE.* Make the necessary preparations as before directed; then multiply continually together, the second and third terms, and the first with its terms inverted as in Division, for the answer.

EXAMPLES.

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

2. What will 33 oz. of silver cost, at 6s. 4d. an ounce? Ans. 17. 1s. 41d. 3. If of a ship be worth 2731. 2s. 6d., what is of her worth?

Ans. 2277. 12s. 1d.

4. What is the purchase of 12301. bank-stock, at 108§ per cent?

Ans. 13361. 1s. 9d.

5. What is the interest of 2737. 15s. for a year, at 34 per cent?

Ans. 87. 17s. 114d.

6. If of a ship be worth 737. Is. 3d., what part of her is worth 250l. 10s.?

Ans..

7. What length must be cut off a board, that is 73 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. 9. If the penny-loaf weigh 6 oz. when the price of wheat is 5s. the bushel; what ought it to weigh when the wheat is at 8s. 6d. the bushel? Ans. 4 oz.

10. How much in length, of a piece of land that is 11 poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth ?

11. If a courier perform a certain journey in 351⁄2 days, a-day; how long would he be in performing the same, hours a-day?

12. A regiment of soldiers, consisting of 976 men, are to be new clothed, each coat to contain 2 yards of cloth that is 1ğ yard wide, and lined with shalloon yard wide; how many yards of shalloon will line them?

Ans. 4531 yds. 1 qr. 2 nails.

DECIMAL FRACTIONS.

A DECIMAL FRACTION is that which has for its denominator, a 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.

This is only multiplying the second and third terms together, and dividing the product by the first, as in the Rule of Three in whole numbers.

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Thus, is 5, and is 25, and 75% is 075, and is 00124; where ciphers are prefixed to make up as many places as are in the numerator, when there is a deficiency of figures.

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 3.25, or 188

328

Ciphers on the right hand of decimals make no alteration in their value; for *5 or 50 or 500, are decimals having all the same value, being each = % or 4. But if they are placed on the left hand, they decrease the value in a tenfold proportion. Thus 5 is or 5 tenths, but 05 is only or 5 hundredths, and 005 is but 10 or 5 thousandths.

The first place of decimals, counted from the left hand towards the right, is called the place of primes, or 10ths; the second is the place of seconds, or 100ths; the third is the place of thirds, 1000ths; and so on. For, 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.

RULE.-Set the numbers under each other according to the value of their places, like 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 in any of the lines that are added; or place the point directly below all the other points.

EXAMPLES.

1. To 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.

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2. To find the sum of 376.25 +86·125+ 637-4725 + 6·5+ 41·02 + 358.865.

Ans. 1506 2325.

3. Required the sum of 3.5+ 47.25 +2.0073 +92701 + 1·5.

Ans. 981-2673.

4. Required the sum of 27654321 + 112 +0·65 + 12·5 + ·0463.

Ans. 455 5173.

SUBTRACTION OF DECIMALS.

RULE.-Place the numbers under each other according to the places, as in the last rule. Then, beginning at the right hand, whole numbers, and point off the decimals as in Addition.

value of their subtract as in

RULE.*_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.

EXAMPLES.

1. Multiply 321096

by •2465

1605480

1926576

1284384

642192

Ans. 0791501640 the product.

361 1000

* The rule will be evident from this example: Let it be required to multiply 12 by 361; these numbers are equivalent to and the product of which is 4333。 =*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 for any other numbers.

To multiply decimals by 1 with any number of ciphers, as 10, or 100, or 1000, &c. 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.

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

SET the units' place of the multiplier under that figure of the multiplicand whose place is the same as is to be retained for the last in the product; and dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in.-Then, in multiplying, reject all the figures that are more to the right than each multiplying figure; and set down the products, so that their right hand figures may fall in a column straight below each other; but observing to increase the first figure of every line with what would arise from the figures omitted, in this manner, namely 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c.; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure.

EXAMPLES.

1. To multiply 27-14986 by 92-41035, so as to retain only four places of decimals in the product.

2. Multiply 480-14936 by 2.72416, retaining only four decimals in the product.

3. Multiply 2490 3048 by 573286, retaining only five decimals in the product. 4. Multiply 325-701428 by 7218393, retaining only three decimals in the product.