(1.) Twice 4 are 8, but this is not 8 units but 8 hundreds, because the 2 stands for 2 hundred, hence it is 2 hundred times 4 or 8 hundred; put down the 800 in this case.

(2.) Twice 6 are 12, or it is really 2 hundred times 6 tens are 12 thousands, or 1 ten thousand and 2 thousands; we put down the 2 under the thousands, and carry the 1 ten thousands to the next figure.

(3.) Continue the process.

This same example may be exhibited in the following manner, where it is still more evident that long multiplication consists in finding separate products and taking their sum.

First, 4 times 7564 is taken, then 30 times the same number, next 200 times the 7564, these three products are added, and their sum is the multiplication of 7564 by 234 as before.

In practice this method of multiplication is simple. There is no consideration for hundreds, thousands, etc., simply multiply by the digits in their order, beginning from right to left, taking care that if you are multiplying by the second digit from the right towards the left, the first figure is placed under the second digit; so if you are multiplying by the third digit, the first figure in the operation is placed under the third digit, and so on successively. The cyphers seen, as cancelled, are always omitted during the operation. As the cyphers to the right of the second and third products make nothing in the addition, it saves time and trouble when they are omitted, and the other figures kept in their proper positions without them. When the cyphers are taken away, the second line will be observed to stand one place to the left of the first, and the third line one place to the left of the second. Hence the rule places the first figure found, when multiplying by any figure, under that figure.

Rule for Performing Multiplication.-(1.) Place the multiplier under the multiplicand, units under units, tens under tens, etc. Draw a line under the whole.

(2.) Beginning at the right-hand side, multiply each figure in order in the multiplicand by the figure in the units' place in the multiplier, being careful to carry to the next product the number of tens arising from the multiplication of any of the digits in the multiplicand by the multiplier.

(3.) Be careful to place down the units under the figure multiplied, while the tens are carried.

(4.) Proceed in the same manner with the digit of the multiplier in the tens' place, the first unit (really tens) arising from this produced is placed under the digit by which we are multiplying.

(5.) Continue the operation with the third, fourth, and every succeeding digit in the multiplier till all have been multiplied into the top line.

(6.) Add all the products together, and the sum will be the entire product sought.

Cyphers in Multiplication.-When noughts are in the multiplier, the practice is to take no notice of them, so far as actual multiplication is concerned. They only affect the position of the digits, and the digits produced by multiplication. In multiplying by such a number as 47200, the two cyphers at the commencement do not affect the actual multiplication. They determine the value of the 2 they precede, and make it 2 hundred. To multiply by the 200, we merely put down the two noughts and then proceed as if the 2 were two units. When the 7 is reached it is 7 thousand, so that care must be exercised to place the first figure produced on the multiplication by 7 under the units of thousands, for we take the multiplicand not 7 times, but 7000 times. Again, with such a multiplier as 4009, having multiplied by the 9 units, no notice is taken of the two cyphers so far as they are individually concerned, but we pass on to the 4 and multiply by it; but the two noughts by their position have affected the value of the 4, for it is 4 thousand. In multiplying by it we have to bear this fact in mind, and place the first figure in the second line under the fourth of the line produced by the multiplication with the 9. The simple reason for all this is, that 5, 6, 1000, or any number of nothings, or noughts, amount to nothing. When cyphers are annexed to the multiplier, or form part of it, we have stated that we must proceed, practically, as if hey were absent, thus:

In the first of these examples it will be seen that the noughts are placed after the other figures at a and b, and that they require no consideration; the learner must be careful that he understands that when he multiplies by the 1, and says once 8 is 8, the 8 is placed under the 2, because the 1 by which he is multiplying and the 2 are both the fifth figures from the right towards the left.

In the second example, when we come to multiply by the 2, it being the fourth figure, the 6 is placed under the 8, which is the fourth figure from the right; so also in multiplying by the 7-the 1 is placed under the 6, because the 7 is the seventh figure from the right. Some express the rule for determining the position of the first figure thus: Be careful to place the first figure obtained by multiplication directly under the figure by which you are multiplying.

Another point frequently neglected is this: Scholars are encouraged in the upper forms to multiply by such numbers as 56, 84, etc., when it is far better and more reasonable, to multiply by 7×8 and 12 x 7, etc., thus :

In multiplication this plan is not so good as this.

42195

42195

In the first case 26 figures are used, in the second only 20— a saving of more that 23 per cent.

EXERCISES.-V.

EXAMPLES IN SIMPLE MULTIPLICATION.

Multiply the following numbers by the numbers placed beneath them respectively:

23. In a cathedral there are 179 windows, each window contains 347 pieces of glass; how many pieces of glass did it take to glaze the whole of the windows?

24. Suppose in this country there are consumed 31,423 tons of coal in a day; how many tons are consumed in 213 days? 25. Multiply 306705 by 579002.

26. Multiply three millions and three by one hundred and fifty thousand three hundred and five. Express the result in words. 27. Find the continued product of 87 x 56 x 409.

28. Write in words the number 4,005,672,030, and also its product when multiplied by the number four thousand and thirty. 29. Multiply 631285 by 34009.

30. Multiply 227351 by 429.

31. Multiply 4507 by 3006 and 17119 by 2119.

32. Increase 90658 to 523604 times.

33. Square 37700, that is, multiply it by itself.

34. Cube 37700, that is, multiply 37700 × 37700 × 37700.

35. Multiply 99 by itself four times, or raise it to the fifth power. 36. If a letter-carrier walk 19 miles every day except Sundays, how far does he walk in a year?

37. How many loaves of one pound each can be made out of 14 sacks of flour, each weighing 186 pounds?

38. Five loads of coal, each containing 30 sacks, each sack holding 105 lbs., were delivered at a public school; find the whole quantity in pounds.

SIMPLE DIVISION.

Division.-Division consists in finding how many number is contained in another.

times one

Quotient. The number of times that one number is contained in another is called the quotient.

Divisor. The divisor is the number that we divide by. We have to ascertain how many times the divisor is contained in a second number.

Dividend. The dividend is the number we divide. Division is simply a series of subtractions in the same way that multiplication is a repetition of additions.

Short Division and Long Division.-When we divide by one figure, or up to 12, the method is termed Short Division, when by higher numbers, consisting generally of two or more digits, the process is termed Long Division; but simple division is performed with higher numbers than 12, when they can be separated into factors.

Division consists of a series of subtractions, or it is a short method of performing subtraction.

EXPLANATION of an ordinary sum in simple division. First Method.-Let it be required to divide forty-one thousand seven hundred and eighty-four by six.

Before proceeding to division, the learner must well exercise himself in the multiplication table; for instance, if going to divide by six, repeat the multiplication table thus:

6 times 1 are 6; 6 times 2 are 12; 6 times 3 are 18, etc.

Next vary it thus:

6 times 1 are 6; 6 into 6 goes once.

6 times 2 are 12; 6 into 12 goes twice.

6 times 3 are 18; 6 into 18 goes three times, etc.

When the multiplication table is learned, the division table is known, a very little practice being all that is required. One who has learned the multiplication table already knows that 6 times 7 are 42; nothing is easier than to reverse this and say 6 into 42 goes 7 times, or 7 into 42 goes 6 times.

Suppose it is required to find how many sixes there are in 45, repeat "six times" till you get close to 45. Suppose you say six eights are 48, that is too many: you do not require to go so far, retrace and say six sevens are 42. Now count upon the fingers up to 45, which gives 3; this shows that six goes into 45 seven times, with a remainder 3.