CHAPTER IV

MULTIPLICATION OF INTEGERS

34. Multiplication. The process of taking a number as many times as there are units in another number is called multiplication.

This means multiplication by an integer. Multiplication by a fraction is considered later.

35. Terms. The number that is multiplied is called the multiplicand; the number that shows how many times the multiplicand is taken is called the multiplier; the result of the multiplication is called the product.

4 multiplicand 3 multiplier 12 product

36. Multiple. The product of two or more integers is called a multiple of each.

For example, 15 is a multiple of 3 and 5, and 40 is a multiple of 2, 4, 5 and also of 8, 10, 20.

37. Factors. Integral numbers whose product is a given number are called factors of that number; the multiplicand and the multiplier, for example, are factors of the product.

38. Symbol. The symbol of multiplication is an oblique cross, x. It is read times or multiplied by.

Thus 3 x 4 =

12 is read "3 times 4 equals 12."

The multiplier may be written either before or after the multiplicand, but the tendency in this country is to write the multiplier first, as we naturally read it. For example, 3 × $4 is read "3 times $4," and $4 × 3 is read "$4 multiplied by 3."

39. The Multiplication Table. The following table, already learned, must be reviewed so that you can tell instantly any product when the teacher tells you the factors:

40. Multiplying by a One-Figure Multiplier. Required the product of 587 by 6.

We may multiply each part of 587 by 6 and add the results, as shown in the upper solution. This method is long, so we use it only

587 multiplicand

6 multiplier

42 6 x 7

48 6 x 8

30 6 x 5

(tens)
(hundreds)

3522 6 × 587 product

for explanation, and write the work
as in the lower solution. Beginning
at the right, we have 6 × 7 = 42; we
write the 2 in the units' place and
reserve the 4 tens to add to the prod-
uct of the tens. Then 6 x 8 tens =
= 48
tens, which, with the 4 tens reserved,
makes 52 tens, or 5 hundreds and 2
tens. We therefore write the 2 tens in

the tens' place, and reserve the 5 hundreds to add to the product of the hundreds. Then 6 x 5 hundreds

587

6

3522

30 hundreds, which, with the 5 hundreds reserved, makes 35 hundreds, or 3 thousands and 5 hundreds, which we write in the thousands' and hundreds' places. The product is therefore 3522.

41. Order of the Factors. The product is the same whatever the order of the factors.

That is, 3 x 4 = 4 x 3. If we read these dots by rows, we have 3 x 4 dots; if we read by columns, we have 4 x 3 dots. But the dots do not change, so 3 x 44 x 3. This is true for any number of rows and columns.

For this reason we may always use the smaller factor as the multiplier, thus making the work easier.

Thus, instead of multiplying 3 by 478, we would multiply 478 by 3. In the same way, if we wished to find the product of 478 × $3, we would take the product of 3 x $478, the answer being the same.

42. Nature of the Factors. Since the multiplier shows how many times the multiplicand is taken, we have the following: The multiplier is always an abstract number.

The multiplicand may be either abstract or concrete, and it and the product will always be like numbers.

In 3 x $4 = $12, 3 is abstract and $4 and $12 are like numbers.

61. What will 48 sheep cost at $7 each ?

62. What will 480 dozen pairs of hose cost at $5 a dozen ? 63. A farmer shipped 9 cases, each containing 360 eggs. How many eggs did he ship?

64. A farmer had 7 acres of onions, and the yield was 562 bushels per acre. How many bushels did he get in all?

43. Multiplying United States Money. The process here is the same as with other numbers. Simply keep the decimal point after the dollars.

$23.05

7

$161.35

Thus, to multiply $23.05 by 7, proceed as here shown. The explanation is the same as in § 40.