5. Four men bought a piece of land, each paying 12 dollars. What did they all pay?

6. If a horse can trot 11 miles in one hour, how many miles can he travel in 8 hours? how many in 10? 12 ?

7. If a bushel of wheat cost 2 dollars, how many dollars will 8 bushels cost? how many will 9? how many will 10? how many will 11? how many will 12?

8. If a man receive 4 shillings for a day's work, how much will he receive for a week's work?

9. A pound of sugar is worth 8 cents. —9—10—11—12 pounds worth?

What are 6-7-8

10. John bought a writing-book for 6 cents. cost? what will 4? what will 6? what will 8 ?

What will 2

Art. 26.-The scholar should commit to memory the following Table before proceeding any further.

MULTIPLICATION AND DIVISION TABLE

OBS.—The student may be required to write out the table, as an exercise, up to 24 times 24, and commit it to memory.

11. If a man pay 85 dollars for a carriage, what must he pay for 5 carriages?

The answer may be obtained by setting down 85 five times, and adding them up, thus:

It will be seen, by examining this operation, that the product of five times five units is two tens and five units, and five times eight tens is 40 tens. The answer, then, is 40 tens, 2 tens and 5 units, or 425.

85

85

85

85

85

425

This method would be tedious when a number is to be many times repeated, and can be solved much easier by multiplication, thus:

85

5

425

Instead of setting down 85, five times, we write 5, the multiplier, under the unit figure of the number to be multiplied; then say, 5 times 5 are 25, setting down 5, the excess of tens; and reserving in the mind, 2, the number of tens, we say, 5 times 8 are 40; adding the two tens which we reserved from the unit column, we set down 42. The answer, then, is 42 tens and 5 units,

or 425.

Art. 27. From the above we derive the following definitions:

1st. Multiplication is the concise method of performing many additions.

2d. Multiplication consists in repeating a given number a required number of times.

OBS. 1.-It is always true of multiplication, that it can be performed by addition; but it is not always true that addition can be performed by multiplication: it is only the case when a number is to be repeated.

OBS. 2.-The word factor signifies an agent, or doer: it is derived from the Latin word factum, which signifies a deed, or thing done. A person employed to do business for another, is called an agent, or factor. Hence, when two numbers are employed as multipliers, or as the means of obtaining a product, they are called factors. (See def. Art. 54.)

12. If a share in a bridge is worth 142 dollars, how many dollars are 6 shares worth?

13. If one man receive 164 dollars for a year's labor, what ought 32 men to receive, for the same time?

Operation.

164

32 328 492

Since we cannot conveniently multiply by a larger number than 12 collectively, it will be necessary, in this example, to adopt a new mode of operation. The multiplier consists of 2 units and 3 tens. We first multiply

2 times the mult. 30 times the mult. 5248 32 times the mult. each figure of the multiplicand by the units of the multiplier. Units into units give units; units into tens give tens; units into hundreds give hundreds. We next multiply each figure of the multiplicand, beginning with the units, by the tens of the multiplier, observing to set 2, the first figure of the product, in the place of tens, because it is the product of tens. We next multiply the 6 tens of the multiplicand by the 3 tens of the multiplier, carrying one for every ten as in Addition, and set the product in the place of hundreds. The product of tens into tens is hundreds. Lastly, we multiply the hundreds of the multiplicand by the tens of the multiplier, and set the product in the place of thousands. The product of tens into hundreds is thousands. Adding together the several products, we have 5248, the answer.

The above illustration may be better understood by setting the product of each figure of the multiplier into each figure of the multiplicand down by itself; thus,

From the foregoing examples we derive the following

RULE.

I. Place the multiplier directly under the multiplicand, units under units, tens under tens, etc., then draw a line underneath.

II. When the multiplier is 12, or less than 12, begin at the right hand of the multiplicand, and multiply each figure contained in it by the multiplier, setting down the numbers, and carrying as in Addition.

III. When the multiplier is greater than 12, write down the figures, as before directed, and multiply the multiplicand by each figure in the multiplier, commencing with the unit figure; observing to place each figure in the product directly under the figure by which you multiply. In this way proceed, and the sum of the products will be the answer.

There are three methods of proving Multiplication.

First-Make the multiplicand and multiplier change places, and multiply the latter by the former, in the same manner as before; if the latter product be the same as the former, the work is supposed to be right.

Second-Cast the 9's out of the product, or answer, and set down the remainder. Cast the 9's out of the sum of the two factors; multiply the two remainders together, and cast the 9's out of the product. The last remainder, if the work is right, will be equal to the first.

OBS. 3.-The four remainders may be set within the four angular spaces of a cross, as in the following example.

Third.-Multiplication may be proved by Division. The product divided by either of the factors will give the other.

OBS. 4.-The second and third methods can be deferred until the scholar becomes acquainted with Division.

QUESTIONS.-1. What is Multiplication? 2. How many numbers are required to perform the operation? 3. What is the number to be multiplied, called? 4. What is the number by which you multiply. called? 5. Taken together, what are they called? 6. Why called factors? 7. What is the answer called? 8. How many figures are there in the multiplier of the 13th question? 9. By which do we multiply first? 10. What is the product of units multiplied into units? 11. Of tens into tens? 12. Of hundreds into hundreds? 13. How are the numbers placed in Multiplication? How do you proceed when the multiplier is 12, or less than 12? 15. When the multiplier is greater than 12?

14.

Multiply 24 by 2; then double the multiplier; then double the multiplicand; then double the product.

3d.

2×2= 48X2=

24 × 2=
4

48

2

What effect upon the product has multiplying the multiplier? What effect upon the product has multiplying the multiplicand?

23. What will 432 barrels of flour cost, at 14 dollars a barrel ? Ans. 6048 dollars. 24. How many rods in 84 miles, there being 320 rods in a mile? Ans. 26880 rods. 25. What will be the cost of 6328 thousands of boards, at 18 dollars per thousand? Ans. 113904 dollars.

26. How many dollars would a man count in 12 days, if he count 42000 in one day?

Ans. 504000.

How would you solve the above question by Addition? 27. What will 64 cows cost, at 16 dollars apiece?

Ans. 1024 dollars.

The multiplier, 16, in the last example, is a number which can be formed by the multiplication of two numbers—thus: