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. 2. Borrowed 10 English guineas at 28s. each, and 24 English crowns at Os. and 8d. each; how many pistoles át! 22s. each, will pay the debt ?

Ans. 20. 3 3. Four men brought each 171. 10s. sterling value in gold into the mint, how many guineas at 21s. each must tliey receive in return ?

Ans. 66 guin. 14s. 4. A silversmith received three ingots of silver, each weighing 27 ounces, with directions to make them into spoons of 2 oz., cups of 5 oz., salts of 1 02., and snuff-boxes of 2 07.., and deliver au equal number of each ; what was the number?

Ans. 8 of each, and 1 oz. over. 5. Admit a ship's cargo from Bordeaux to be 250 pipes, 1:30 hhds. and 150 quarter casks, si hhds.] how many gal. lons in all; allowing every pint to be a pound, what burden was the ship of ? Ans. 44415 gals. and the ship's burden

was 158 tons, 12 cut. 2 qrs. 6. In 15 pieces of cloth, each piece 20 yards, how many French Ells?

Ans. 200. 7. In 10 bales of cloth, each bale 12 pieces, and each piece 25 Flemish Ells, how many yards ? Ans. 2250.

8. The forward wheels of a wagon are 14. feet in ciscumference, and the hind wheels 15 foc: and 9 inches; how many more times will the forward wheels turn round than che hind wheels, in running from Boston to New-York, in being 248 miles ?

Ans. 7167. 9. How many times will a ship 97 feet 6 inches long, sail her length in the distance of 12800 leagues and ten yards?

Ans. 2079508. 10. The sun is 95,000,000 of miles from the earth, and, a cannon ball at its first discharge flies about a mile in 7 seconds; how long would a cannon ball be, at that rate in flying from here to the sun ? Ans. 22 yr. 216 d. 12 h. 40 m.*.***

11. The sun travels through o signs of the zodiac in half a year; how many degrees, minutes, and seconds ?'

Ans. 180 deg. 10800 min. 648000 sec. 12. How many strokes does a regular clock strike in 36% days, or a year?

Ans. 56940. 13. How long will it take to count a million, at the rate of 50 a minute ? Ans. 333 h. 20 m. or 13 d. 21 h. 20 m.

$14. The national debt of England amounts to about 279 wpillions of pounds sterling; how long would it take to count this debt in dollars (4s. 6d. sterling) reckoning without intermission twelve hours a day at the rate of 50 dols. a minute, and 365 days to the year?

Ans. 94 years, 134 days, 5 hours, 20 min.

FRACTIONS. "FRACTIONS, or broken numbers, are expressions for any assignable part of a unit or whole number, and (in general) are of two kinds, viz.

VULGAR AND DECIMAL. A Vulgar Fraction, is represented by two numbers placed me above another, with a line drawn between them, thus,

&c. signifies three fourths, five eighths, &c. "Lue figure above the line, is called the numerator, and urat below it, the denominator ; ..

5 Numerator. Thus, Denominator. The denominator (which is the divisor in division) shows how many parts the integer is divided into; and the nume rator (which is the remainder after division) shows how ma ny of those parts are meant by the fraction.

A fraction is said to be in its least or lowest terms, wlien it is expressed by the least numbers possible, as when reduced to its lowest terms will be ), and is equal to 1, &c.

PROBLEM 1. "To abbreviate or reduce fractions to their lowest terms.

Rule.-Divide the terms of the given fraction by any number which will divide them without a remainder, and the quotients again in the vamo manner; and so on, till it appears that there is no number treater than 1, which will divide them, and the fraction will be in its bilst terms.

EXAMPLES. 1. Reduce if to its lowest terins.

(3) (2) 8)=j=in=} the Answer. 2. Reduce to its lowest terms.

Ans. } 3. Reduced to its lowest terms.

Ans. * 1. Reduce to its lowest terms.

Ans.

102

Ans.

5. Abbreviate as much ùs possible. 6. Reduce 25 to its lowest terms.

Ans. : 7. Reduce 44 to its lowest terms.

Ans. 8. Reduce 32 to its lowest terms.

Ans. 9. Reduce 177 to its lowest terms.

Ans. 10. Reduce to its lowest terms.

Ans. PROBLEM II. To find the value of a fraction in the known parts of the integer, as to coin, weight, measure, &c.

Rule.-Multiply the numerator by the common parts of the integer und divide by the denominator, &c.

EXAMPLES. 1. What is the value of of a pound sterling ?

Numer. 2

20 shillings in a pound.

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3)1244 2. What is the value of is of a pound sterling I

Ans. 18s. 5d. 2 qrs. 3. Reduce of a shilling to its proper quantity. Ans. 4d 4. What is the value of of a shilling? . Ans. 4 d. 5. What is the value of 12 of a pound troy? Ans. 902. * 6. How much is ir of a hundred weight ?

M{

Ans. 3 qrs. 7 lb. 104 oz. 7. What is the value of of a mile?

Ans. 6 fur. 26 po. 11f1 8. How much is 7 of a cwt. ? Ans. 3 qrs. 3 lb. 1 oz. 12 do 9. Reduce of an Ell English to its proper quantity..

Ans. 2 yrs. 34 na 10. How much is 4 of a hhd. of wine ? Ans. 54 god.

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11. What is the value of of a day?

Ans, 16 h. 36 min. 551. SCC.

PROBLEM III. Tu reduce any given quantity to the fraction of any greatcr denomination of the same kind.

Rule.-Reduce the given quantity to the lowest term inentioned for a numerator; then reduce the integral part to the same term, for a denominator ; which will be the fraction required.

EXAMPLES. 1. Reduce 13s. 6d. 2qrs. to the fraction of a pound. in 20 integral part - 13 6 2 given sum,

12

12

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960 Denominator. 050 Num. Ans. * 2. What part of a hundred weight is 3 qrs. 14 lb. ?

3 qrs. 14 lb.=98 lb. Ans. = 3. What part of a yard is 3 qrs. 3 na.? Ans. 4. What part of a pound sterling is 13s. 4d.? Ans. 7. What part of a civil year is 3 weeks, 4 days?

Ans. = l. What part of a mile is 6 fur. 26 po. 3 yds. 2 ft. ? Jur. po. yds. ft. fcet. 6 20 3 2=4400 Num.

a mile =5280 Denom. Ans, 44:= 7. Reduce 7 oz. 4 pwt. to the fraction of a pound troy.

Ans. 10. What part of an acre is 2 roods, 20 poles ? Ans.

9. Reduce 54 gallons to the fraction of a hogshead of vine.

Ans. 10. What part of a hogshead is 9 gallons ? Ans. Í 11. What part of a pound troy is 10 oz. 10 pwt. 10 grs.

Ans. DECIMAL FRACTIONS. A Décimal Fraction is that whose denominator is a unit, with a cipher, or ciphers annexed to it, Thus, tortor niso &c. dor

The integer is always divided eitlier into 10, 100, 1000 &c. equal parts; consequently the denominator of the frac tion will always be either 10, 100, 1000, or 10000, &c. which being understood, need not be expressed; for the true value of the fraction may be expressed by writing the numératoi only with a point before it on the left hand thus, is writ. ten ,5; M. ,45; 120,725, &c.

But if the numerator has not so inany places as the de nominator has ciphers, put so many ciphers before it, viz. at the left hand, as will make up the defect; so write iko thus, ,05; and 10 mo thus, ,006, &c.

Nore. The point prefixed is called the separatrix.

Decimals are counted from the left towards the right hand, and each figure takes its value by its distance from the unit's place; if it be in the first place after units, (or se. parating point) it signifies tenths ; if in the second, hun dredths, &c. decreasing in each place in a tenfold propor 110.7, as in the following :

NUMERATION TABLE.

Millions.
. C. Thousands.
« X. Thousands.
A Thousands.
co Hundreds.
do Tens.
- Units.

u Tenth parts.
co Hundredth parts.
A Thousandth parts.
er X. Thousandth parts.

C. Thousandth parts.
Millionth parts.

EOS
7 6 5 4 3 2 1
Whole numbers.

Decimals. Ciphers placed at the right hand of a decimal fraction do not alter its value, sirce every significant figure continues to possess the same place : so ,5 ,50 and ,500 are all the same value, and equal to or .

But ciphers placed at the left hand of decimals, decrease their value in a tenfold proportion, by removing them further from the decimal point. Thus, ,5 ,05 ,005, &c. &re fixe tenth parts, five hundredth parts, five thousandth parts, de respectively. It is therefore evident that the magnitude

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