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14. The national debt of England amounts to about 279 millions of pounds sterling; how long would it take to count this debt in doilars (49. Cd. sterling) reckoning without intermission twelve hours a day at the rate of 50 dollars a minute, and 365 days to the year?

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

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

VULGAR AND DECIMAL. A Vulgar Fruction, is represented by two numbers placed one above another, with a line drawn between them, thus, ,, &c. sigoifies three-fourths, 've-eights, &c.

The figure above the line, is called the numerator, anul that below it, the denominator, Thus,

S 5 Numerator.

9 Denominator. The denominator (which is the divisor in division) shews how many parts the integer is divided into; and the numerator (which is the remainder after division) shews how many of those parts are meant by the fraction.

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

PROBLEM I.
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, anil the quo-
tients again in the same manner; and so on, till · appears
that there is no number greater than 1, which w: il divide
them, and the fraction will be in its least terms

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EXAMPLES.

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5 5 67

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1. Roduce 141 to its lowest terms.

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14*=*== the Answer. 2. Reduce 6 to its lowest terms.

Ans, 3. Reduce g to its lowest terms.

A is. A 4. Reduce 45 to its lowest terms.

Ans. 5. Abbreviate as much as possible.

Bis. 6. Reduce 825 to its lowest terms.

Ans. 7. Reduce 144 to its lowest terins.

Ans. 8. Redace 3% to its lowest terms.

Ans. 9. Reduce 171 to its lowest terms.

Ans. 19 10. Reduce 589 to its lowest terms,

Aus. PROBLE-1 1. To find the value of a fraction in the known parts of the integer, as to coin, weighit, measure, &c.

RÜLE. Multiply the numerator by the common parts of the integer, and divide by the denominator, &c. What is the value of of a pound sterling?

Numer: 2

20 shillings in a pound.

EXAMPLES,

Denom. 3)40(13s. 4d. Ans.

S

10
9

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1 12

5)1204

12 2. What is the value of t of a pound sterling?

jus. 18s. 5d. 2fqrs. 3. Reduces of a shilling to its proper quantity.

Ans. 41d. 4. What is the value of of a shilling? Ans. 4.d. 5. What is value of ik of a pound troy? Ans. 90%.

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6. How much is i of an hundred weight !

Ans. 3qrs, 7lb. 10_10z. 7. What is the value of á of a mile :

Ins. 6fur. 26po. 11ft. 8. How much is į of an cwt. ?

Ans. 3qrs. 3lb. 1oz. 12,dr. 9. Reduce of an Ell English to its proper quantity,

Ans. 2qrs. 3}na. 10. IIow much is of a hhd. of wine ? Ans. 54gal. 11. What is the value of is of a day?

Ins. 16h. 36min. 551 sec.

PROBLEM III. To reduce any given quantity to the fraction of any greater denomination of the same kind.

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

EXAMPLES.
1. Reduce 139. 6d. 2qrs. to the fraction of a pound.

20 Integral part 13 6 2 given sum.
12

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Ans. 25

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960 Denominator, 650 Num. Ans. f*= 2. What part of an hundred weight is Sqrs. 146. sqrs. 14ib.=9815.

Ans. = 3. What part of a yard is 3qrs. Ina. Ans. 4. What part of a pound sterling is 13s. 4d. Ans. 5. What part of a civil year is 3 weeks, 4 days ?

y's 6. What part of a mile is 6fur. 26po. Syds. 21t. ? fur. po. yd. ft. feet. 6 32=4400 Num. a mile z52 0 Den m.

Ans. 440.0 7. Reduce 70%. 4pwt. to the fraction of a pound iroy,

ins. } 8. What part of an acre is 2 roods, 20 poies ? ins.

73

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

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

Ans. 396

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DECIMAL FRACTIONS.
A Decimal Fruction is that whose denominator is an
unit, with a cypher, or cyphers annexed to it, Thus, ,
Tom To, &c. &c.

The integer is alivays divided either into 10, 100, 1000, &c. equal parts; consequently the denominator of the fraction 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 numerator only with a point before it on the left hand thus, jo, is written ,5; 10,45; 106,725, &c.

But if the numerator has not so many places as the denominator has cyphers, put so many cyphers before it, viz. at the left hand, as will make up the defect; so write To thus, ,05; and 1060 thus, ,006, &c. NOTE. 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 separating point) it signifies tenths; if in the second. hundredt!is, &c. decreasing in each place in a tenfold proportion, as in the following

NUMERATION TABLE.

O Tent! parts.
w Hundredth parts.
A Thousandth parts.
eX.Thousanuth parts.
C. Thousandth parts.
Millionth parts.

en Millions.

C. Thousands.
X. Thousands.
A Thousands.
co Hundreds.

Tens.
Units.

Whole Numbers.

Decimals.

pare

Cyphers placed at the rig!:t hand of a decimal fraction do not alter its valne, since every significant figure continues to possess the same place : so ,5 ,50

and

,500 are all tie sa!ne value, and egnal to to ord.

Bat cyphers placed at the left hand of decimals, deorease their value in a tenfold proportion, by removing tien further from the decimal point. Thus, ,5 ,05 ,005, &c. are five tenth parts, five hundredth parts, five thousandth parts, &c. respectively. It is therefore evident that the magnitude of a decimal fraction, compared with another, does not depend upon the number of its figures, but upon the value of its first left hand figure : for instance, a fraction beginning with any figure less than 99 such as ,899229, &c. if extended to an infinite number of figures, will not equal ,9.

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RULE. 1. Place the numbers, wnęther mixed or pure decimals, under each other, according to the value of their places.

2. Find their sum as in whole numbers, and point of so many places for the decimals, as are equal to the greatest number of decimal parts in any of the given numbers.

EXAMPLES.

1. Find the sum of 41,653+36,05 +24,009+1,6

141,653

36,05 Thus,

1,6

24,009

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13.

Sum, 103,512 which is 103 integers, and 30% parts of an unit. Or, it is 103 units, and 3 tenth parts, 1 hun. dredth part, and 2 thousandth parts of an unit, or 1.

Hence we may observe, that decimals, and FEDERAL Money, are subject to one, and the same law of notation, and consequently of operation.

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