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14. The national debt of England amounts to about 279 millious 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 ruinbers, 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 one above another, with a line drawn between them, thus, 1,, &c. signifies three fourths, five eighths, &c.

The figure above the line, is called the numerator, and that 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 numerator (which is the remainder after division) shows how muny 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 , and ia 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, and the quotients again in the same manner; and so on, till it appears that there is no number greater than 1, which will divide them, and the fraction will be in its least terms.

EXAMPLES. 1. Reduce its to its lowest terms.

(3) (2) 8)144=j=6=} the Answer. 2. Reduce or to its lowest terms.

Ans. 3. Reduce is to its lowest terms.

Ans. 4. Reduce to its lowest terms.

Ans,

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5. Abbreviate as much as possible.

Ans, 6. Reduce a to its lowest terms.

Ans. 7. Reduce to its lowest terms.

Ans. 2 를 8. Reduce 32 to its lowest terms.

Ans. 9. Reduce ili to its lowest terms.

Ans. 10. Reduce bil to its lowest terms.

Ans.

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

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

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

Numer.

20 shillings iy a pound. Denom. 3)40(13s. 4d. Ans.

3

10
9

112

3)124

12 2. What is the value of ij of a pound sterling

Ans. 18s. 5d. 21. qro. 3. Reduce of a shilling to its proper quantity. Ans. 4 d. 4. What is the value of of a shilling? Ans. 41d. 5. What is the value of le of a pound troy? Ans. Ouz. 6. How much is of a hundred weight ?

Ans. 3 qrs. 7 lb. 10,21 oz. 7. What is tlie value of of a mile?

Ans. 6 fur. 26 po. 11 ft. 8. How much is of a cyt. ? Ans. 3 qrs. 3 lb. 1 oz. 124 dr. 9. Reduce of an Ell English to its proper quantity,

Ans. 2 yrs. 3, na. 10. How much is of a hhd. of wine ? Ans. 54 gal,

11. What is the value of is of a day?

Ans. 16 h. 36 min. 551 sec.

PROBLEM III.

To reduce any given quantity to the fraction of any greater denomination of the saine kind.

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

EXAMPLES.
I. Reduce 13s. 63. 2qrs. to the fractiou of a porid.
20 integral part

13 6 2 given sum.
12

12

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Ans. lá Ans.

960 Denominator. 650 Num. Ans. =8£. 2. What part of a hundred weight is 3 qrs.

14 lb. ?
3 qrs. 14 lb. 98 10.

Ans. 11
What part of a yard is qrs.
4. What part of a pound sterling is 13s. 4d. ?
5. What part of a civil year is 3 weeks, 4 days?

Ans. =*
6. What part of a mile is 6 fur. 26 po. 3 yds. 2.ft. ?
fur. po. yds. ft. feet.
6 26 3 2=4400 Num.:
a mile =5280 Denom.

Ans.
7. Reduce 7 oz. $pwt. to the fraction of a pound troy.

Ans.

8. What part of an acre is 2 roods, 20 poles: ? Ans. 9. Reduce 54 gallons to the fraction of a hogshead of wine. 10. What part of a hogshead is 9 gallons ? 11. What part of a pound troy is 10 oz. 10 pwt. 10 grs.

Ans. Ans. }

Ans.

5.0 510

DECIMAL FRACTIONS. A Decimal Fraction is that whose denominator is a unit, with a cipher, or ciphers annexed to it, Thus, it, Tón, tish, &c. &c.

The integer is always 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, so is written ,5; 1,45; 4,725, &c.

But if the numerator has not so many places as the denominator has ciphers, put so many ciphers before it, viz. at the left hand, as will make up the detect; so write tão thus, ,05; and your 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 uvits, (or separating point) it signifies tenths; it in the second, hundredths, &c. decreasing in each place in a tenfold proportion, as in the following

NUMERATION TABLE.

Millions.

C. Thousands.
07 X. Thousands.

'Thousands.
co Hundreds.

o Tenth parts.
co Hundredth parts.
A Thousandth parts.
SX. Thousandth parts.
C. Tbousandth parts.

Millionth parts.

Tens.

Units.

7 6 5 4 3 2 1

2 3 4 5 6 7 Whole numbers.

Decimals, Ciphers placed at the right hand of a decimal fraction do not alter its value, since every significant figure continues to possess the same place: 60 ,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 ,45 ,005, &c. &re five tenth parts, five hundredth parts, five thousandth parts, &c. respectively. It is therefore evident that the magnitude

of a decimal fraction, conpared 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 ,9 such as ,899229, &c. if extended to an infinite number of figures, will not equal ,9.

ADDITION OF DECIMALS. RULE.-1. Place the numbers, whether mixed or pure decimals, únder each other, according to the value of their places.

2. Find their sum as in whole numbers, and point off 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

41,653

36,05 * Thus,

24,009
1,6

Sum, 103,312, which is 103 integers, and due parts of a unit. Or, it is 103 units, and 3 tenth parts, 1 hundredth part, and 3 thousandth parts of a 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.

For since dollar is the money unit; and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, or unit, it is evident that any number of dollars, dinies, cents and mills, is simply the expression of dollars, and decimal parts of a dollar: Thus, 11 dollars, 6 dimes, 5 cents,=11,65 or 110 dol. &c. 2. Add the following mixed numbers together.

(3)

(4) Yards. Ounces. Dollars. 46,23456 12,3456 48,9108 24,90400 7,891

1,8191 17,00111 2,34

3,1030 3,01111 5,6

,7012

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