85;

Same as 10 in 170; that is, 17 times.

The object of these changes is, to give the learner an accurate and complete knowledge of numbers, and of division; and the result is not the only object sought for, as many young learners suppose. How many times is 75 contained in 575? or, divide 575 by 75.

Ans. 73. Quotient, 64.

Quo. 162, or 13.

A person spent 6 dollars for oranges, at 64 cents apiece; how many did he purchase?

Ans. 96.

(ART. 22.) When two or more numbers are to be multiplied together, and one or more of them having a cipher on the right, as 24 by 20, we may take the cipher from one number, and annex it to the other, without affecting the product; thus, 24×20, is the same as 240×2; 286×1300=28600×13; and 350×70×40=35X7X4 X1000, &c.

Every fact of this kind, though extremely simple, will be very useful to those who wish to be skillful in operation.

(ART. 23.) Before we combine division with multiplication, let us take a more systematic view of numbers.

The following are called prime numbers, because no one can be divided by any number less than itself without producing a fraction:

11. 13.
41
. 43

47

67

71.73
101.

73 com

The points represent the composite numbers; and here it can be observed, that there are 27 prime numbers, and, of course, posite numbers in the first hundred, the prime numbers becoming fewer as the numbers rise higher. Observe the following series:

5 10 15 20 25 30 35 40 45 50 55 60,

and so on. Every body knows that our arithmetical scale of numbers is 1, 10, 100, 1000, &c. Now, we wish the student to observe the numbers,

5 20 25 50 75 125 500,

as being not only in the preceding series, but aliquot parts of some number in our arithmetical scale. For example, 25 is of 100; 125 is of 1000, &c.

We now charge the student to make his eye familiar with all the preceding series-the prime numbers as being unmanageable and in

convenient, and the others the very reverse; but the full importance of such a study can only appear in the sequel.

(ART. 24.) When it becomes necessary to multiply two or more numbers together, and divide by a third, or by a product of a third and fourth, it must be literally done, if the numbers are prime.

For example: Multiply 19 by 13, and divide that product by 7. This must be done at full length, because the numbers are prime : and in all such cases there will result a fraction.

But, when two or more of the numbers are composite numbers, the work can always be contracted.

Example: Multiply 375 by 7, and divide that product by 21. To obtain the answer, it is sufficient to divide 375 by 3, which gives 125. The 7 divides the 21, and the factor 3 remains for a divisor. Here it becomes necessary to lay down a plan of operation.

Draw a perpendicular line, and place all numbers that are to be multiplied together under each other, on the right hand side, and all numbers that are divisors under each other, on the left hand side.

EXAMPLES.

1. Multiply 140 by 36, and divide that product by 84. the numbers thus:

We place

We may cast out equal factors from each side of the line without affecting the result. In this case, 12 will divide 84 and 36: then,

the numbers will stand thus:

But 7 divides 140, and gives 20, which, multiplied by 3, gives 60 for the result.

2. Multiply 4783 by 39, and divide that product by 13.

Three times 4783 must be the result.

3. Multiply 80 by 9, that product by 21, and divide the whole by the product of 60×6×14.

In the above, divide 60 and 80 by 20, and 14 and 21 by 7, and those numbers will stand canceled as above, with 3 and 4, 2 and 3 at their sides.

Now, the product 3×6×2, on the divisor side, is equal to 4 times 9 on the other, and the remaining 3 is the result.

Hoping now that the pupil understands our forms, and comprehends the true philosophical principles, we give a few unwrought examples for exercises.

4. Multiply 84 by 56, and divide the product by 14; what is the result?

5. What is the result of 75X21, divided by 7?

6. What is the result of 126×72, divided by 48?

7. What is the result of 5728×49, divided by 56?

8. What is the result of 64X18×48, divided by 16X9X12?
9. What is the result of 125×8×2, divided by 100×24?
10. What is the result of 39x41X360, divided by 82×30?
11. What is the result of 224×13×37), divided by 75×45?
12. What is the result of 71×19×7, divided by 38×21?
13. What is the result of 221X635, divided by 35X5?

APPLICATION.

1. A farmer sold 28 bushels of wheat at 85 cents per bushel, and took his pay in cotton cloth at 14 cents per yard; how many yards I did he receive?

Ans. 170.

2. A person bought 12 yards of cloth at 219 cents per yard, and paid in butter at 9 cents per pound; how many pounds did it require? Ans. 292.

3. How many yards of cloth, at $4.66 à yard, must be given for 18 barrels of flour at $9.32 a barrel?

Ans. 36.

4. The children in a Sunday-school contributed 5 dollars to a charitable object; each giving 64 cents; how many children were there? Ans. 80.

5. A laborer worked 26 days at 87 cents per day, and took his pay in wheat at 65 cents per bushel; how many bushels did he receive? Ans. 35. 6. A person pays 34 dollars a week for board; how many dollars must he pay for 26 weeks? Ans. 13 times 7: $91. 7. A merchant bought 526 barrels of flour at $4 50 per barrel, and paid in cloth at $2.25 per yard; how many yards did it require?

Ans. 1052. 8. How much land, at $2.50 per acre, must be given in exchange for 360 acres at $3.75 per acre? Ans. 540. 9. What will 28 pounds of sugar cost, at 93 cents per pound? Ans. 7 times 39 cents: or, $2.75. 10. An auctioneer sold 55 bags of cotton, each containing 400 pounds, receiving I mill commission on a pound; how many dollars did his commission come to? Ans. $22.

11. How many casks, each containing 1 bushel 1 peck, are required to hold 145 bushels? Ans. 116.

12. How much will 540 yards of cloth cost at 3 shillings 4 pence per yard, in dollars, at 6 shillings each? Ans. $300.

13. If 1 quart cost 10 pence, how many pounds will 12 hogsheads cost? Ans. £126. 14. How many pounds of butter at 94 cents per pound, will pay for 19 yards of muslin at 11 cents per yard?

Ans. 22. 15. How many bushels of oats at 224 cents a bushel, will pay for 75 pounds of sugar at 5 cents per pound? Ans. 164. 16. How many bushels of wheat at 75 cents a bushel, will pay for 6 yards of cloth at 34 dollars a yard?

17. How many bushels of corn, at 45 cents per chase 24 yards of carpeting at 65 cents per yard?

Ans. 28.

bushel, will purAns. 343.

18. How many tons of hay at 5 dollars per ton, will pay for 11 acres of land at 19 dollars per acre?

Ans. 38. 19. If a man travel, on an average, 4 miles an hour and 9 hours a day, how many days will be required to pass over 1260 miles?

Ans. 35.

20. How many bushels of barley at 75 cents per bushel, will be required to pay a debt of $45.75 ? Ans. 61. 21. How many days' work at 125 cents per day, will pay for 80 acres of land at 225 cents per acre?

Ans. 144.

22. What number, multiplied by 23, will give the same product as 75 multiplied by 69?

Ans. 225.

23. What number, multiplied by 163, will give the same product as 300 multiplied by 451 ?

Ans. 8200.

COMPOUND NUMBERS.

(ART. 25.) COMPOUND NUMBERS are such as express quantities consisting of different denominations, but of the same general kind, such as bushels, pecks, quarts, &c.; yards, feet, inches, &c. The most proper appellation for these quantities is Denominate Numbers, because they simply consist of different denominations, and are not compound; but the name compound numbers has been so long attached to them that it would be difficult to change it.

The following Tables of the denominations of compound numbers are to be committed to memory before entering upon reduction.

ENGLISH MONEY.

The denominations of English Money are, guineas, pounds, shillings, pence, and farthings.

TABLE.

4 farthings, marked far., make 1 penny, marked d.

1 shilling,.

1 guinea.

MENTAL EXERCISES.

In £2 38., how many shillings?
In 2s. 3d., how many pence?
In 3s. 2d., how many pence?
In £3 28., how many shillings?

It is evident, from the inspection of the table, that to change pounds to shillings we must multiply the £1 by 20, and to change shillings to pence we must multiply the shillings by 12, &c.; and this changing of a quantity from one denomination to another is called Reduction; for it is reducing.

TROY WEIGHT.

Gold, silver, jewels, and liquors, are weighed by this weight. Its denominations are pounds, ounces, pennyweights, and grains.

In 2 pounds 1 ounce, how many ounces?

In 2 ounces 3 pennyweights, how many pennyweights?
In 2 pennyweights and 2 grains, how many grains?
In 1 pound, how many penny weights?

In 1 ounce, how many grains?

APOTHECARIES' WEIGHT.

10.

This weight is used by apothecaries and physicians in mixing their medicines. Its denominations are pounds, ounces, drams, scruples, and grains. The pound and ounce are the same as the pound and ounce in the Troy weight; the difference between the two weights consists in the different divisions and subdivisions of the ounce.

By this weight are weighed all coarse articles, such as hay, grain, chandlers' wares, and all the metals, excepting gold and silver. Its denominations are tons, hundreds, quarters, pounds, ounces, and drams.

The hundred weight is 112 pounds, as appears from the table; but at the present time the merchants in our principal cities buy and sell by the 100 pounds, and 20 hundreds a ton.