25. In 1840 the population of New York was 2428921, and in 1830 it was 1913006. What was the increase during this 10 years? Ans. 515915.

QUESTIONS INVOLVING ADDITION AND SUBTRACTION.

1. A lets B have 60 bushels of wheat, worth 70 dollars, a fine horse worth 150 dollars, and 37 dollars' worth of butter. B in turn gives A his note for 110 dollars, and the rest in cash. What is the amount of cash?

Ans. 147 dollars.

2. A borrows of B, at one time, 375 dollars; at a second time he borrows 95 dollars, and at a third time he borrows 413 dollars; he has paid him 319 dollars. How much does he still owe him? Ans. 564 dollars.

3. A person left a fortune of 10573 dollars to be divided between two sons and one daughter; the first son received 4309 dollars, the other son had 4987 dollars. How much did the daughter receive?

Ans. 1277 dollars.

4. Two persons are 375 miles apart; they travel towards each other; at the end of one day, one has travelled 93 miles, and the other 57 miles. How far apart are they? Ans. 225 miles.

5. A farmer sold a span of horses for 150 dollars, a cow for 27 dollars, some cheese for 83 dollars, and 7 tons of hay for 56 dollars. He purchased 10 yards of broadcloth worth 45 dollars, a cook stove for 23 dollars, and a pleasure carriage for 80 dollars. How much money will he have left? Ans. 168 dollars.

6. In the year 1840, the coinage of the United States mint was as follows: 1675302 dollars of gold, 1726703

15. The pupil must also bear in mind that the multiplier and multiplicand may be interchanged without altering the product. Thus :

4x8=8x4=32

9x7=7x9=63

4×6=6×4=24
3x5=5x3=15.

What does multiplication teach? The number to be repeated is called what? The number denoting how many times the multiplicand is to be repeated is called what? What are the multiplicand and multiplier sometimes called? The result obtained is called what? What is the symbol for multiplication? Can the multiplier and multiplicand exchange places without altering the product?

When the multiplicand consists of more than one figure, and the multiplier has but one figure, we proceed as follows:

Multiply 697 by 3.

Place the multiplier under the multiplicand, units under units. First, multiply the 7 units by the 3 units; we obtain 21 units, or 2 tens and 1 unit. Write the 1 under the unit column, and the 2 under the tens' column. Next, multiply the 9 tens by the 3, and we have 27 tens; equal to 2 hundred and 7 tens.

Write the 7 tens under the tens' column, and the 2 under the hundreds' column. Finally, multiply the 6 hundreds by the 3 and we have 18 hundreds, which is the same as 1 thousand and 8 hundreds. Write down the 8 under the hundreds' column and carry the 1 to the thousands' place; that is, write down the whole 18.

We then add these partial products, and obtain 2091 for the total product.

OPERATION.

697 multiplicand. 3 multiplier.

2091 product.

By recalling to mind (ART. 10,) that ten in the place of units are equal to one in the place of tens, ten in the tens' place are equal to one in the hundreds' place, &c., we may perform the above multiplication as follows: First, multiplying 7 of the multiplicand by 3 the multiplier, we obtain 21 units, which are the same as 2 tens and 1 unit. Hence we write down the 1 under the units' column, and reserve the 2 to carry to the tens'. Next, multiplying the 9 by 3, we find 27 tens, to which, adding the 2 tens reserved, we have 29 tens, which are equal to 2 hundreds and 9 tens. Write down the 9 under the tens' column, and reserve the 2 to carry to the hundreds. Finally, multiplying the 6 by 3, we have 18 hundreds; to which add the 2 hundreds reserved, and we have 20 hundreds, the whole of which we write down, obtaining 2093 for the product.

Again, let it be required to multiply 367 by 84. Here the multiplier consists of more than one figure.

Place the multiplier under the multiplicand, units under units, and tens under tens.

Multiplying first by the 4 units, we find 1468 for the product. We are next to multiply by the 8 tens. Now, it is obvious that 1 unit, taken ten times, that is, multiplied by 1

OPERATION.

367 multiplicand. 84 multiplier.

1468 2936

30828 product.

ten, must produce 10 units or 1 ten. So 7 units, (as in the example,) multiplied by 8 tens, must produce 56 tens, or 5 hundreds and 6 tens. Therefore, set the first figure, 6 of this second product under the tens' column and reserve the 5 to carry to the hundreds. The next step is the multiplication of tens by tens, which must produce hundreds, since 1 ten, taken 1 ten times, is equal to 1 hundred. Therefore 8 tens times 6 tens are 48 hundreds; to which add the 5 hundreds reserved, and we obtain 53 hundreds; equal to 5 thousands and 3 hundreds. Place the 3 under the hundreds' column, and carry the 5 to the next column, and so proceed throughout. The sum of these partial products will give the total product, 30828.

If the multiplier consists of three figures, its left-hand or hundreds figure, multiplied into the units of the multiplicand, will give hundreds for the first figure of the product, which must of course be set down under the hundreds' column; hundreds and tens, multiplied together, will give thousands; hundreds and hundreds multiplied together will give ten thousands, &c.

If the multiplier consists of four figures, its left-hand or thousands' figure multiplied into units, will give thousands for the first figure of the product, which must be set down under the thousands' column. Thousands multiplied into tens, gives tens of thousands; into hundreds, gives hundreds of thousands; and so on.

It would be necessary to annex ciphers to the figures in these several products, to show their true places, if these places were not determined by the position of the figures with relation to other figures, whose places are known.

16. If we again take the first example, which is to

inultiply 697 by 3, we remark that since 697 is to be repeated 3 times, it may be done by writing it down 3 times, and then adding, thus:

697

697

697

2091

And it is obvious that all questions of multiplication may be performed by addition.

Hence, multiplication is sometimes defined as being a concise way of performing several additions.

NOTE.-When a zero or 0 occurs in the multiplier, we may observe that its product must remain 0, since nothing repeated any number of times is still nothing.

PROOF OF MULTIPLICATION.

17. If we interchange the multiplier and multiplicand, and then multiply, we shall obtain the same product if the work is right. (See ART. 15.)

As in addition, these two results may be alike, and still the work may be wrong, since mistakes may occur in both operations. As good proof as any, is to carefully repeat the multiplication.

When 0 is multiplied by any number, what is the result? How is multiplication sometimes defined. How may multiplication be proved? Is this method infallible? Why not? What is as good proof as any other ?

CASE I.

18. When the multiplier consists of only one figure.