COR. 1.-When the multiplicand has several figures and the multiplier one that is only units, the first product of units by units will be units, or units and tens; the units must be placed in the right-hand or units place; if there be tens, it must be reserved and placed in or added to the column of tens; in the next product of tens by units, the right-hand figure will be tens, and must be united with the tens reserved, and placed in the column of tens; the left-hand figure, if there be one, must be treated as the previous one, reserved until the next product is obtained, and united with the right-hand figure; the process is the same in every successive order.

COR. 2.-When the multiplier also has several figures, the process of each successive multiplier is the same, except that the right-hand figure of each product must be placed in the order of its multiplier. (Cor. 3, Prob. 2, page 22.)

REM.-A multiplicand may be either an abstract or a concrete number, but a multiplier cannot be concrete, as it cannot refer to things, but merely indicates how many times the multiplicand is to be taken; but the product will be of the same name as the mul tiplicand; for twice $5 are $10; 3 times 20 yards of cloth are 60 yards of cloth; twice 4 are 8; 3 times 4 are 12, etc.

In computation, it is best to regard all numbers as abstract.

REM. The product is not changed by alternating the multipli

cand and multiplier.

EXAMPLES.

346.|

543.

6. Multiply 9876325 by 356. 7. Multiply

879654 by 2175.

697.

8. Multiply

986432 by 8704.

9. Multiply

326875 by 3005.

1. Multiply 54326 by 2. Multiply 23748 by 3. Multiply 46874 by 4. Multiply 36975 by 476. 5. Multiply 236874 by 2134. 10. Multiply 468753 by 2100.

Examples may be added, or the same repeated, as the student will more readily comprehend by repetition than by different examples.

REM. 1.—In multiplication, two factors are given to find their product.

REM. 2.-In division, two numbers also are given to find the third; the one called the dividend corresponds to the product in multiplication, the other given number is called the divisor, and the required number is called the quotient; the two latter correspond to the factors in multiplication.

DIVISION.

PROBLEMS.

When the product of two numbers is 4, and one of the numbers is 2, the other number is also 2; for 2×2 = = 4, and 4 divided by 2, or 4 divided into 2 equal parts, each part is 2, that is, the quotient is 2.

COR. 1.-The product of the divisor and quotient equals the dividend.

COR. 2. The divisor and quotient may be alternated.

COR. 3.-Division is the reverse of multiplication and addition, and is similar to subtraction; for, it is separating a number into equal parts, which is the same as subtracting the same number from a larger one; that is, subtracting the divisor from the dividend and then from the remainder, repeating this process until there is no remainder, or until the remainder is less than the divisor. 6 is subtracted 4 times, hence it is contained four times. 24÷ 64.

COR. 1.-Adding a zero to the right of a number multiplies the number by 10; taking a zero away from the right of a number divides the number by 10.

REM. 1.-The same result is obtained by short division, by putting the first figure of the quotient under the left-hand figure of the dividend (when it is contained in it), as it is of the same order.

REM. 2.-If the unit of the divisor is not contained in the first unit of the dividend, then the first figure of the quotient will be of the same order as the second figure of the dividend and should be placed under it.

COR. 1.-Since the product of any two factors will have as many figures or one less than both factors, so in division the number of figures of the divisor and quotient will either be equal to or one greater than that of the dividend.

COR. 2.-When the divisor is contained in the same number of figures of the dividend as is in the divisor, then the number of figures of the divisor and quotient will be one more than that of the dividend; but when it requires an additional figure of the dividend to contain the divisor, then the number of figures of the divisor and quotient will be equal to that of the dividend.