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THE RULES OF PROPORTION IN DECIMALS.`.

Note. The examples following are promiscuously arranged to exercise the scholar in the Rule of Three Direct, Inverse, Compound Proportion, &c. Decimals have the same properties as whole numbers, the only difficulty being in pointing off the decimals, a repetition of the rules already given would, therefore, be superfluous.

(1.) If 3.75 yards of cloth cost 8s. 9d. what will 2571⁄2 yards cost?

First 8s. 98.4375 and 257 yds.=2575

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£. s. d.

3.75)112.65625(30.04166, &c.—30..0..10 answer.

(2.) If cwt. of tobacco cost 41. 18s. how much may I buy at the same rate for 71.?

(3.) Bought 3.5 yards of cloth for 27. 14s. 3d. what must I give for 27.75 yards?

(4.) Sold 75 chaldrons of lime, at 11s. 6d. per chaldron, what is the amount?

(5.) A goldsmith sold a tankard for 10 67. at the rate of 5s. 6d. per oz.; what did it weigh?

(6.) If 12 men can perform a piece of work in 100 days, in how many days would 20 men perform the same? (7.) In 75411 ducats, at 4s. 4d. each, how many dollars, at 4s. 53d. each?

(8.) If 5400 bricks be required to pave a yard, when the bricks are 5 foot long, and 25 broad, how man will be required of 75 foot long and foot broad?

(9.) If I buy 14 yards of cloth for 10 guineas, how many ells Flemish can I buy for 283·8757.?

(10.) If 12 oz. of plate cost 10s. 11d. what will a service, weighing 327 61875 oz. cost?

(11.) How many yards of flannel that is one English ell in width, will be sufficient to line a cloak, containing 18 yards of cloth yard wide?

(12.) If 248 men in 60 hours dig a trench, containing 13924 solid yards of earth, how many men in 1188

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PART 1.] THE RULES OF PROPORTION IN DECIMALS. 103:

hours, will dig a similar trench, containing 26460 solid yards of earth; the earth being cast at the same distance from these men as the former?

(13.) If 2 men can do 125 rods of ditching in 65 days, in how many days can 18 men do 2424 rods? (14.) If of of a ship be worth 1477. 118. 3d. what is the whole worth?

(15.) If a piece of Arras hanging be 6 yards long, and 4 yards broad, how many square ells Flemish are contained therein?

(16.) If a wedge of gold, weighing 17 lb. troy, be worth 6794. what is the value of 1 grain of that gold?

(17.) What will be the expence of tiling an out-house that is 273:5 feet long, and 51.75 feet broad, with tiles at 11s. 10 d. per thousand, supposing every square of tiling to take up 1000 tiles?

(18) A man, with his family consisting of 4 persons, usually drink 7.8 gallons of beer in a week, how much would they drink in 22.5 weeks, if the family were to be increased by 3 persons?

(19.) I agreed for the carriage of 2:5 tons of goods 2:9 miles, for 075 guinea, what is that per cwt. for a mile?

(20.) If a traveller perform a journey in 35.5 days,, when the days are 13 625 hours long; in how many days of 11-9 hours long would he perform the samejourney?

(21.) The earth turns round on its axis from west to east in 23 hours 56 minutes, and the circumference of every circle on it surface is supposed to be divided into 360 degrees. At the equator a degree is 69.07 English miles; at Madras, Barbadoes, &c. 67-21 English miles; at Madrid, Philadelphia, &c. 52.85 English miles; and at Petersburg 34-53 English miles. How many miles per hour are the inhabitants in each of these places carried from west to east by the revolution of the earth on its axis?

(22.) Goliath, the Philistine, is said to have been 6 cubits high, each cubit 1 foot 7·168 inches English, what was his height in English feet?

CIRCULATING, OR REPEATING DECIMALS: Definition 1. When the denominator of a vulgar fraction, in its lowest terms, is not compounded of 2 or 5, or both, the decimal produced from such a vulgar fraction will be infinite; it is called a repetend, or circulating de cimal, from a continual repetition of the same figures.

2. A single repetend is a decimal, where only one figure repeats, as 222, &c. or 3333, &c. and these may be expressed by putting a mark over the first figure. Thus 222, &c. may be denoted by 2′, and ·3333, &c. by '3'.

3. A compound repetend has the same figures circulating alternately, as 575757, &c. or 57235723, &c. and these may be distinguished by marking the first and last repeating figure. Thus 5757, &c. may be written •5'7', and '57235723, &c. 5'723'.

4. Pure repetends are such as have no figures in them but what belong to the repetend, as '3', '5', 4'73', &c.

5. Mixed repetends are such as have ciphers or signifi cant figures, between the repetend and the decimal point, or such as have whole numbers to the left hand of the de cimal point, as '04', '07′53', •473′', •357′3′, 6·5′, 4·3′75′, &c. 6. Dissimilar repetends are such as begin at different places from the decimal points, as 2'53', '475'2', &c..

7. Similar repetends are such as begin at an equal distance from the decimal points, as 35'4', 2.75'34', &c. 8. Conterminous repetends are such as end at the same distance from the decimal points, as '125', '3'54', &c.

9. Similar and conterminous repetends are such as begin and end at the same place after the decimal points, as 53.2753', 4.63'25', and 46'32', &c.

REDUCTION OF CIRCULATING DECIMALS. Proposition 1. To reduce a pure repetend to its equiva ·lent vulgar fraction.

RULE.

Make the given decimal the numerator, and let the denominator be a number consisting of so many nines as there are figures in the repetend. The terms of this fraction, divided by their greatest common measure, will give the least equivalent vulgar fraction required.

Prop. 2. To reduce a mixed repetend to its equivalent vulgar fraction.

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