11. What is the value of of a day?

Ans. 16 h. 36 min. 55 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 mentioned 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. 20 integral part

13 6 2. given sum.

12

240

4

960 Denominator.

12

2. What part of a hundred weight is 3 qrs. 14 lb. ?

3 qrs. 14 lb. 98 lb.

3. What part of a yard is 3 qrs. 3 na.?

Ans.

Ans.

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

Ans. 33 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. 100

7. Reduce 7 oz. 4 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 vine.

10. What part of a hogshead is 9 gallons? 11. What part of a pound troy is 10 oz. 10 p

A Decimal Fraction is that whose denominator is a unit, with a cipher, or ciphers annexed to it, Thus, To, Táo, Tôðõ, &c. &c.

The integer is always divided either 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 numerator only with a point before it on the left hand thus, is written,5; 45,45; 725,725, &c.

100

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 defect; so write thus,,05; and 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 se parating point) it signifies tenths; if in the second, hun dredths, &c. decreasing in each place in a tenfold propor tion, as in the following

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: 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. 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 ,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, unaer 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 n any of the given numbers.

EXAMPLES.

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

Sum, 103,312, which is 103 integers, and

a unit.

parts of

Or, it is 103 units, and 3-tenth parts, 1 hundredth

part, and 2 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, dimes, 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 115 dol. &c.

2. Add the following mixed numbers together.

(2)

5. Add the following sums of Dollars together, viz. $12,34565+7,891+2,34+14,+,0011

6.

Ans. $36,57775, or $36, 5di. 7cts. 775-mills. Add the following parts of an acre together, viz. ,7569+,25+,654+,199.

Ans. 1,8599 acres.

7. Add 72,5+32,071+2,1574+371,4+2,75.

Ans. 480,8784 8. Add 30,07+200,71+59,4+3207,1. Ans. 3497,28 9. Add 71,467+27,94+16,084+98,009+86,5. Ans. 300 10. Add ,7509+,0074+,69+,8408+,6109. Ans. 2,9 11. Add ,6+,099+,37+,905+,026.

Ans. 2

12. To 9,999999 add one millionth part of a unit, and the sum will be 10.

13. Find the sum of

Twenty-five hundredths,

Three hundred and sixty-five thousandthis,

Six tenths, and nine millionths,

Ans. 1,215009

SUBTRACTION OF DECIMALS.

RULE. Place the numbers according to their value; then subtrat sin whole numbers, and point off the decimals as in Addition.

13. From a unit, or 1, subtract the millionth part of itself. Ans. ,999999

MULTIPLICATION OF DECIMALS.

RULE.-1. Whether they be mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers.

2. Point off so many figures from the product as there are decimal places in both the factors; and if there be not so many places in the product, supply the defect by prefixing ciphers to the left hand.

5. Multiply 7853 by 3,5.

6. Multiply,007853 by ,035.

7. Multiply 004 by,004.

Ans. $14, 4d. 3c. 8,35m.

8. What cost 6,21 yards of cloth, at 2 dols. 32 cents, 5 mills, per yard? 9. Multiply 7,02 dollars by 5,27 dollars.

100

Ans. 36,9954 dols. or $36 99 cts. 5m. 10. Multiply 41 dols. 25 cts. by 120 dollars. Ans, $4950 11. Multiply 3 dols. 45 cts. by 16 cts.

Ans. $0,5520=55 cts. 2 mills.

12. Multiply 65 cents, by ,09 or 9 cents.

Ans. $0,0585-5 cts. 81⁄2 mills.

13. Multiply 10 dols. by 10 cts. Ans. $1 14. Multiply 341,45 dols. by ,007 or 7 mills. Ans. $2,39 To multiply by 10, 100, 1000, &c. remove the separating point so many places to the right hand, as the multiplier has ciphers.

makes 4,25

by 100, makes 42,5

Multiplied by 10,

So,425

by 1000, is

,425

For ,425 × 10 is 4,250, &c.

DIVISION OF DECIMALS.

RULE.-1. The places of the decimal parts of the divisor and quo Went counted together, must always be equal to those in the divid