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19. If 1 load of hay weigh 1 T. 10 cwt. 2 qrs. 20 lb. 5 oz. 15 dr. what will 33 loads weigh ?

20. Multiply 27 gals. I qt. 1 pt. 3 gi. by 28. 21. Multiply 67 yds. 3 qrs. 3 nls. by 72.

SUBTRACTION.

MENTAL EXERCISES.

$ XXXVII. 1. A goldsmith having 1 lb. of silver, melted

up

7 oz. of it. How much had he left ? 2. A boy having 3 shillings, gave away le. 6d. How much had he left ?

3. A pitcher contains 3 quarts of cider, and a man fills 3 pint tumblers from it. How much cider is left in the pitcher ?

4. Take 2 gi. from 1 pt. From 1 qt. From 1 gal.

5. Take 3 nls. from 1 qr. From 1 yd. From 1 E. E. From 1 E. F.

6. Take 3 oz. from 1 lb. From 1 lb. 1 oz. From 2 lb.

2 oz.

12 gr.

7. Take 19 gr. from 1 3. From 1.0.6 gr. From 3 D.

8. Take 3d. from 1 shilling. From 1s. 1d. From 3s. 2.

9. Take 15 minutes from 1 hour. From lh. 5m. From 7h. 7m.

10. Take 5 drams from 1ib. 13. From 63.33. From 916. 13. 43.

Write the following examples.

11. A man's property amounted to 6,872£..15s., and he lost a ship worth 1,539£. 17s. What was ho then worth?

As we cannot take 17s. from 15s. we must borrow £. a £.=20s. This 20s. added to 15s. makes 358., from 6872; 15 which, 17 being taken, 18 are left. As we borrowed 1539; 17 a £, we must make the pounds 1 less ; that is, 6,871 instead of 6,872. 6871-1539=5332.

5332;

18 12. George had 16s. 8d. and he gave 4s. 9d. for a sled. How inuoh had he left ? Ans. Ils. lld.

It will be better, instead of diminishing the next higher denomination of the minuend, to increase that of the subtrahend by 1; and the result will be the same, for 5 taken from 16, ovidently leaves the same remainder, as 4 from 15, viz. 11. This is the best method in compound numbers, because, sometimes, the figure of the minuend may be a cypher. We should not be able to diminish this.

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8;

16;

13. A man bought a keg of brandy for 3£. Os. 6d., but in carry-
ing it home there leaked out as much as would come to l£. 3s. 8d.
What was the value of the remainder ?
Here we borrowed a shilling. To £.

d.
make up for this, we increased the 3 3; 0; 6;
shillings in the lower number by 1, in. 1; 3;
stead of diminishing the upper number,
which is a cypher.

1;

10; Ans. NOTE. The intelligent pupil will perceive, that, as there are no shillings in the minuend, the borrowed shilling comes from the pounds. If we had taken a pound from the 3 pounds, and after borrowing Is., put down the remaining 19 in place of shillings, there would have been no occasion to increase the lower number. For 19—3=16 and 20–4=16. This however, in compound numbers, would be perplexing.

It will be recollected that this method of allowing, after having borrowed, was noticed in Subtraction of Simple Numbers- ($XXII.)

14. A man commenced business with 1,850£. Os. 10 d. and at the end of the year, found that he was worth 2,570£. 98. 6fd. How much had he gained Ans. 720£. 88. 8d.

15. I borrowed £317; 6, and afterwards paid £178; 18; 54. How much was then due ? Ans. £138; 7; 61.

16. A. man purchased cloth to the amount of £27; 11. In turn ho gave flour to the amount of £19; 17; 6, and the rest in money. How much money did he give ? Ans. £7; 13; 6.

From the above we derive the following rule.

PLACE THE SAME DENOMINATIONS UNDER EACH OTHER. TAKE EACH DENOMINATION IN THE SUBTRAHEND, BEGINNING WITH THE LOWEST, FROM THE SAME IN THE MINUEND, AND TO COMPENSATE FOR BORROWING PROM ONE DENOMINATION TO ANOTHER, INCREASE THE NEXT HIGHER DENOMINATION OF THE SUBTRAHEND BY 1.

EXAMPLES FOR PRACTICE. 18. A merchant had 200 bls. 16 gal. of brandy, of which he sold 82 bls. 15 gal. 1 pt. How much had he left ? Ans. 118 bls. 3 qts. 1 pt.

19. A man borrowed £60; 10, : and paid, at one time £17; Il ; 6; at another, £9; 8; at another, £7; 9; 6; and at another, 19s. 6 d. How much then romained unpaid ? Ans. £25; 1; 54.

20. I pay a debt of £105; 10, as follows; viz. I give an order on another person for £15; 14; 9, and two notes, one for £30; 0; 6, and another for £39 ; 11. The rost I pay down. How much do I pay

down?
21. The war between England and America commenced April
19, 1775, and continued until Jan. 20, 1783. How long did it con-
tinue ? Set it down thus. Yrs. mo. d.

1783;
1775; 3;

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0;

20; 19;

22. From 2 cwt. 2 qrs. 27 1b. 3 oz. 8 dr. take 3 qrs. 0 lb. 9 oz. 7 dr.

23. From 7T. 9 cwt. 3. qrs. 16 lb. 8 oz. 3 dr. take 1T. 11 cwt. 3 qrs. 17 lb. 9 oz. 6 dr.

24. From 15 yrs. 3 mo. 3 w. 2 d. 5 h. 3 m. 3 sec., take 13 yrs. 9 d. 27 sec.

25. From 289 acres, 3 roods 7 rds., take 196 acres 3 roods, 30 rds.

DIVISION.

MENTAL EXERCISES. § XXXVIII. 1. A man divided 4 bushels and 2 pecks of grain, between 2 poor" persons.

How much had each ?

2. A man divided 6 acres, 2 roods, and 8 rods of land into 2 equal fields. How much land was there in each field ?

3. Five boys agreed to share 2 qts. and 1 pt. of nuts equally. How much ought each to have ?

4. What is the 4th part of 5d. ? What is the 6th part of id. 2 qrs. ?

5. What is the 8th part of 1 lb. Avoidupois? 6. What is the 7th part of 1 lb. 2 oz. Troy? 7. What is the 8th part of 93 ? of 11b. 43 ?

8. What is the 4th part of 6d. ? of 9d. ? of 1s. Id.? of 2s. 2d. ? 9. What is the 5th part of 6 gals. I qt. ? of 12 gals.

1 io. What is the 9th part of 2 square yds. ? of 8 square

8 yds.?

11. What is the 11th part 1£ 2s. ? of 3€ 6s. ? of 12£ 2s. ? The following are to be written.

12. A man paid 4 labourers an equal sum each. To the whole he gave £10; 8. What was that apiece ?

In dividing 10£ by 4, 2£ are left. £. $. £, s. 4 will not go in 2, but if 2£ be redu. 4)10 8(2 12 ced to shillings, 4 will divide the 8 number of shillings. Let this reduction be made, by multiplying by 20, 2 and let the 8s in the given sum be 20 added in, at the same time. The whole is 48s., which divided by 48 4=128.

48 Hence, to divide compound numbers,

I. DIVIDE EACH DENOMINATION OF THE DIVIDEND, SEPARATELY BY THE DIVISOR, AND THE SEVERAL QUOTIENTS WILL

2 qts.

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BELONG TO THE DENOMINATIONS OF THEIR RESPECTIVE DIVIDENDS.

II. IF A REMAINDER BE LEFT, IN DIVIDING ANY DENOMINATION, REDUCE IT TO THE NEXT LOWER DENOMINATION, ADD IN ALL OF THAT DENOMINATION, IN THE GIVEN DIVIDEND, AND THEN DIVIDE AS BEFORE.

EXAMPLES FOR PRACTICE. 13. If 11 tons of hay cost £23; 0; 2, what is that per ton ?

A. £2; 1; 10. 14. If 48 lbs. of cheese cost fl ; 16, what is that per Ib. ?

A. 9d. 15. If 13 persons pay equally towards a bill of 5£. 8s. 101d. how in uch must each pay? A. 88.4d. 16. If a nobleman's salary be £150,000 a year, what is that a day?

A. 410£ ; 198. ; 2d. 17. If1 cwt, of raisins cost £3 ; 10, what is that a lb. A. 74d. 18. If 12 quarts of wine cost £4; 15; 6, what is that a qt.

A. 78. 11 d. 19. Divide £115 ; 10, by 90. A. £1; 5; 8. 20. Divide £136; 16; 6, by 108. A. £1; 5 ; 4. 21. Divide 6 T. 11 cwt. 3 qrs. 19 lb. by 4.

A. I T. 12 cwt. 3 qrs. 25 lb. 12 oz. 22. Divide 26 lb. 1 oz. 5 dwt. by 24. A. 1 lb. 1 oz. 1 dwt. 1 qr. 23. Divide 666£ 15s. 9d. 1 qr. by 125. 24. Divide 32 yrs. 5 mo. 2 w. 5 d. 17 h. 27 sec. by 306. 25. Divide 441 16.3 oz. 16 dwt. 17 gr. by 509. 26. Divide 81 T. 16 cwt. 3 qrs. 25 lb. 15 oz.. 13 dr. by 572.

OBSERVATIONS ON COMPOUND NUMBERS, FOR ADVANCED

PUPILS. & XXXIX. Man could not have lived long in the earth, without perceiving the necessity of different measures. In estimating dimensions of length, for example, he would find those minute divis. ions of space,

which, for the measurement of small objects were not only convenient but even necessary, very inadequate to his purpose when applied to the height of a tree, or the breadth of a river ; and much more so, when employed to expross the elevation of a mountain, or the distance traversed on a long journey. Hence, the origin of different denominations. For the purpose of easily comparing the different denominations with one another, and, in some cases, of substituting, one for another without altering the value, it seemed best to make each higher denomination such as to contain an exact number of the next lower.

Space was, doubtless, first measured by man; and for this purpose were employed the dimensions of various members of the human hody; as the breadth of the hand, its extent when spread, called

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the span, the breadth of the nail, and of the thumb, the length of the foot, and of the arm, and also the length of a stop or pace. These things seem to have formed the basis of measures of length, in all nations. From the thumb is derived the inch; from the foot, as being, in longth, about twelve times the thumb's breadth, the meas. ure still used of the same name; from the arm, as being about three times the length of the foot, the yard; and from the hand, the span and the pace, the measures, respectively bearing those names at the present time. It is said that the yard now in use in England, “was adjusted from the arm of Henry I. in 1,101, and that the old French pied du roi, (king's foot) had a similar origin.” From these sprung higher denominations; as the mile, being mille passuum, that is a thousand paces, &c. From lineal measurements, the transition was easy to those of surfaces and solids.

In the ruder ages, men weighed with the natural balance of the arms and hands; a balance, indeed,' quite as rude as the age, in which it was employed. When greater accuracy seemed to become necessary, the artificial balance was constructed, on the hint, thus afforded by nature. From what circumstance weights derive their actual, or their comparative sizes, we do not know.

As the process of coining implies weight, money, properly speak. ing, was not probably employed, until long after the invention of the balance. Its denominations have usually been entirely arbitrary.

The division of time was naturally suggested by the succession of days and nights, by the revolutions of the moon, and the returns of the seasong. The subdivisions into 60s, seem to have had their origin in Ptolemy's sexagesimal Notation. (see $ vi.) It has been supposed by some, on the other hand, however, that, since the moon was observed to make 12 complete revolutions in a year, man naturally made the subdivisions of the day, and likewise of the night the same in number. 12 subdivisions of the day, and 12 of the night made 24 hours. Then, as the month contained 30 days and 30 nights, making 60 parts in the whole, the subdivisions of the hour were made by 608. [See American Almanac for 1830.] This, however seems improbable, and the circumstance that the year contains considerably moro than 12 revolutions of the moon, and . likewise, that one revolution, (which measured the original month,) does not contain 30 days, shows that the supposition has little foundation in fact. The division of time into periods of 7 days, or weeks, has been found to have been very extensively employed, by rude nations.

Any one who will compare the operations on Federal Money, with those on any other set of Compound quantities, which we havo exhibited, will be ready to inquire, or will perhaps rather be able to answer the inquiry, why it is that the former are so much more simple and easy, than the latter. It is evidently because the laro of increase, in Federal Money, is the same as that of simple numbers. The radix of each i: 10. The pupil will now perceive how much reason we have to regret, that weights, measures, &c. should have been originally made to increase by ratios so irregular and so incons venient. But it must be recollected that they had their origin,

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