12 s. 6 d. fore,

5. What cost 340 yards of cloth, at 12 s. 6 d. per yard? 10 s. (£.) and 2 s. 6 d. (=

£.); there

1. What distinction do you make between simple and compound numbers? (¶ 26.) 2. What is the rule for addition of compound numbers? 3. for subtraction of, &c. ? 4. There are three conditions in the rule given for multiplication of compound numbers; what are they, and the methods of procedure under each? 5. The same questions in respect to the division of compound numbers? 6. When the multiplier or divisor is encumbered with a fraction, how do you proceed? 7. How is the distance of time from one date to another found? 8. How many degrees does the earth revolve from west to east in 1 hour? 9. In what time does it revolve 1°? Where is the time or hour of the day earlier at the place most easterly or most westerly? 10. The difference in longitude between two places being known, how is the difference in time calculated? 11. How may operations, in the multiplication of compound numbers, be facilitated? 12. What are some of the aliquot parts of 1 cwt.? 13. What is this

of 1 s. ?

of 1 £.?
manner of operating usually called?

EXERCISES.

1. A gentleman is possessed of 14 dozen of silver spoons, each weighing 3 oz. 5 pwt.; 2 doz. of tea spoons, each weighing 15 pwt. 14 gr.; 3 silver cans, each 9 oz. 7 pwt.; 2 silver tankards, each 21 oz. 15 pwt.; and 6 silver porringers, each 11 oz. 18 pwt.; what is the weight of the whole? Ans. 18 lb. 4 oz. 3 pwt.

Note. Let the pupil be required to reverse and prove the following examples:

2. An English guinea should weigh 5 pwt. 6 gr.; a piece of gold weighs 3 pwt. 17 gr.; how much is that short of the weight of a guinea?

3. What is the weight of 6 chests of tea, each weighing 3 cwt. 2 qrs. 9 lb. ?

4. In 35 pieces of cloth, each measuring 27 yards, how many yards?

5. How much brandy in 9 casks, each containing 45 gal. 3 qts. 1 pt.?

6. If 31 cwt. 2 qrs. 20 lb. of sugar be distributed equally into 4 casks, how much will each contain?

7. At 4 d. per lb., what costs 1 cwt. of rice?

2 cwt.? 3 cwt.? Note. The pupil will recollect, that 8, 7 and 2 are factors of 112, and may be used in place of that number.

8. If 800 cwt. of cocoa cost 18 £. 13 s. 4 d., what is that per cwt.? what is it per lb.?

9. What will 94 cwt. of copper cost at 5 s. 9 d. per lb. ? 10. If 64 cwt. of chocolate cost 72 £. 16 s., what is that per lb. ?

11. What cost 456 bushels of potatoes, at 2 s. 6 d. per bushel?

Note. 2 s. 6 d. is of 1 £. (See T 42.)

12. What cost 86 yards of broadcloth, at 15 s. per yard? Note. Consult 42, ex. 5.

13. What cost 7846 pounds of tea, at 7 s. 6 d. per Ib. ? at 14 s. per lb. ? at 13 s. 4 d. ?

14. At $94 25 per cwt., what will be the cost of 2 qrs,

of tea?

of 3 qrs.?

of 16 lbs. ?

- of 14 lbs. ?

of 24 lbs. ?

Note. Consult T 42, ex. 4 and 5.

of 21 lbs. ?

15. What will be the cost of 2 pks. and 4 qts. of wheat, at $150 per bushel?

16. Supposing a meteor to appear so high in the heavens as to be visible at Boston, 71° 3', at the city of Washington, 77° 43′, and at the Sandwich Islands, 155° W. longitude, and that its appearance at the city of Washington be at 7 minutes past 9 o'clock in the evening; what will be the hour and minute of its appearance at Boston and at the Sandwich Islands?

FRACTIONS.

43. We have seen, (T 17,) that numbers expressing whole things are called integers, or whole numbers; but that, in division, it is often necessary to divide or break a whole thing into parts, and that these parts are called fractions, or broken numbers.

Thus,

It will be recollected, (T 14, ex. 11,) that when a thing or unit is divided into 3 parts, the parts or fractions are called thirds; when into four parts, fourths; when into six parts, sixths; that is, the fraction takes its name or denomination from the number of parts, into which the unit is divided. if the unit be divided into 16 parts, the parts are called sixteenths, and 5 of these parts would be 5 sixteenths, expressed thus, The number below the short line, (16,) as before taught, ( 17,) is called the denominator, because it gives the name or denomination to the parts; the number above the line is called the numerator, because it numbers the parts. The denominator shows how many parts it takes to make a unit or whole thing; the numerator shows how many of these parts are expressed by the fraction.

1. If an orange be cut into 5 equal parts, by what fraction is 1 part expressed?

- 4 parts?

or a whole orange?

2 parts?

3 parts? 5 parts? how many parts make unity

2. If a pie be cut into 8 equal pieces, and 2 of these pieces be given to Harry, what will be his fraction of the pie? if 5 pieces be given to John, what will be his fraction? what fraction or part of the pie will be left?

It is important to bear in mind, that fractions arise from division, ( 17,) and that the numerator may be considered a

172):

*

dividend, and the denominator a divisor, and the value of the fraction is the quotient; thus, is the quotient of 1 (the numerator) divided by 2, (the denominator;) is the quotient arising from 1 divided by 4, and is 3 times as much, that is, 3 divided by 4; thus, one fourth part of 3 is the same as 3 fourths of 1.

Hence, in all cases, a fraction is always expressed by the sign of division.

3 is the dividend, or numerator.

expresses the quotient, of which is the divisor, or denominator. S

3. If 4 oranges be equally divided among 6 boys, what

part of an orange is each boy's share?

A sixth part of 1 orange is, and a sixth part of 4 oranges is 4 such pieces, = 告。 Ans. of an orange.

4. If 3 apples be equally divided among 5 boys, what part

of an apple is each boy's share? if 4 apples, what? if 2 apples, what? if 5 apples, what?

5. What is the quotient of 1 divided by 3?of 2 by 3?

by 7?

- of 5 of 2 by 14? 6. What part of an orange is a third part of 2 oranges?

of 1 by 4?

of 2 by 4?

of 3 by 4?

- of 6 by 8?

of 4 by 5?

of 3 oranges?

of 2?

of 5?

one fourth of 2 oranges?

of 3 oranges?

4 of 3?

of 4?

of 2?

A Proper Fraction. Since the denominator shows the number of parts necessary to make a whole thing, or 1, it is plain, that, when the numerator is less than the denominator, the fraction is less than a unit, or whole thing; it is then called a proper fraction. Thus, &,, &c. are proper fractions.

An Improper Fraction. When the numerator equals or exceeds the denominator, the fraction equals or exceeds unity, or 1, and is then called an improper fraction. Thus, §, †, †, 2, are improper fractions.

A Mixed Number, as already shown, is one composed of a whole number and a fraction. Thus, 141, 137, &c. are mixed numbers.

7. A father bought 4 oranges, and cut each orange into 6 equal parts; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces; what was each one's fraction?

Was James's fraction proper, or improper? Why?
Was Nancy's fraction proper, or improper? Why?

To change an improper fraction to a whole or mixed number. 44. It is evident, that every improper fraction must contain one or more whole ones, or integers.

1. How many whole apples are there in 4 halves (4) of

an apple? 20?

in ?

in ?

in 12? in 984?

in

in & of a yard?

in &?

in

?

in?

in?

in

in 17?

in 20?

in 48?

3. How many bushels in 8 pecks? that is, in & of a bushel?

This finding how many integers, or whole things, are contained in any improper fraction, is called reducing an improper fraction to a whole or mixed number.

4. If I give 27 children of an orange each, how many oranges will it take? It will take 27; and it is evident, that

OPERATION.

4)27

Ans. 6 oranges.

dividing the numerator, 27, (= the Humber of parts contained in the fraction,) by the denominator, 4, (= the number of parts in 1 orange,) will give the number of whole oranges.

Hence, To reduce an improper fraction to a whole or mixed number,-RULE: Divide the numerator by the denominator; the quotient will be the whole or mixed number.

EXAMPLES FOR PRACTICE.

5. A man, spending & of a dollar a day, in 83 days would spend 83 of a dollar; how many dollars would that be? Ans. $138.

6. In 1417 of an hour, how many whole hours? The 60th part of an hour is 1 minute: therefore the question is evidently the same as if it had been, In 1417 minutes, how many hours?

Ans. 2387 hours.

7. In 8763 of a shilling, how many units or shillings? Ans. 730 shillings.

8. Reduce 14678 to a whole or mixed number.

9. Reduce 38, 106, 175, 1788, 3465, to whole or mix

ed numbers.

4509