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plier is hundreds, the right-hand figure must be placed in the column of hundreds; and, in general, whatever the order of the multiplier is, the right-hand figure must be in the column of that order.

COR. 4.-If there be one or more zeros in the multiplier, the product of the next figure will be put back one figure for every zero.

REM.—In the multiplication, each figure may be regarded as the unit of its order.

FROBLEMS.

1. 10 x 10 = 100.

2. 11 x 11 = 121 = 11 × (10 + 1) = 11 × 1 = 11

11 x 10 = 110

121

3. 12 x 12 = 144 = 12 × (10 + 2) = 12 × 2 = 24

12 × 10

4. Multiply 432 by 4 = (400 + 30 + 2) × 4.

120

144

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5. Multiply 432 by 14 = 432 × (10+ 4).

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REM. The problems should be carefully impressed on the

mind before proceeding.

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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 multiplicand; 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.

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REM. The product is not changed by alternating the multipli

cand and multiplier.

EXAMPLES.

346.|

543.

697.

6. Multiply 9876325 by 356. 7. Multiply 879654 by 2175. 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.

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

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

24

6

18

6

12

6

6

610

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 6 = 4.

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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.

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