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

The number to be multiplied is called the multiplicand; the number showing how many times the multiplicand is to be taken is called the multiplier; and the result of multiplication is called the product. The multiplicand and the multiplier are called factors of the product.

When the multiplier is written before the sign ×, as in 4 × 8 ft., the sign is read "times." Thus, 4 x 8 ft. is read 4 times 8 feet. When it is written after the sign, the sign is read "multiplied by." Thus, 4 ft. x 8 is read 4 ft. multiplied by 8.

1. Does the expression "6 apples times 4" make sense? The multiplier cannot name things, as, feet, hours, etc., since it indicates the number of times the multiplicand is to be taken. A number used without reference to a particular thing is called an abstract number; as, 5, 6, 50.

A number used with reference to a particular thing is called a concrete number; as, 5 trees, 6 dollars, 50 inches.

The multiplier is always regarded as an abstract number. The product is always like the multiplicand.

2. Compare the product of 8 x 6 with the product of 6 × 8; the product of 2 × 14 with the product of 14 × 2. Either factor may be regarded as the multiplier.

NOTE. The multiplicand may be either concrete or abstract. When it is concrete, the product has the same name as the multiplicand.

How to multiply and divide by 10, 20, 100, etc.

1. Multiply 42 by 10. Annex a zero to the right of 42. Is there any difference in the products? Which of these two methods of multiplying is the shorter? Multiply

42 by 100.

Annex two zeros to the right of 42.

2. Multiply each of the following numbers by 10; by 100: 43 26 75 96 283 694 786 813 465 710 634

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To test multiplication, use the multiplicand for a multiplier and perform the multiplication again; or, divide the product by the multiplier.

2. How much will 48 chairs cost at $1.25 each?

$ 1.25 48

10 00

500

Multiply as before, and mark off from the right, in the product, two places for cents.

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The process of finding how many times one number contains another, or of separating a number into equal parts, is called division.

Division may be defined also as the process of finding one of two factors when their product and the other factor are given. The number to be divided is called the dividend; the number by which we divide is called the divisor; and the result of division is called the quotient.

The part of the dividend remaining when the division is not exact is called the remainder.

Oral Work

1. Division is indicated in three ways: 153, 15, 3)15. 2. In 153, which number is the dividend? Which is the divisor?

If the dividend and the divisor are concrete, they must have the same The quotient is then abstract. Thus, $7 (divisor) is contained

name.

in $21 (dividend) 3 times (quotient).

When the divisor is abstract and the dividend concrete, the quotient has the same name as the dividend. Thus, $ 21 ÷ 7 = $3, or 4 of $ 21 = $ 3.

When we consider that $7 is contained 3 times in $21, the problem differs from the separation of $21 into 7 equal parts. The latter kind of division is called partition.

3. Find of 120; 160; 200; 400; 480; 960.

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How to divide by 10, 100, 1000, etc.

Oral Work

1. Divide 40 by 10. Cut off a zero from the right of 40. Is there any difference in the quotients? Which of these two methods of dividing is the shorter?

2. Divide 2436 by 100.

100)24 36

24; r. 36

3. Divide each of the following numbers by 10; by 100; by 1000: 3000; 46,000; 273,000; 619,000; 81,400; 2000; 8626; 46,153; 814,256.

Written Work

1. Divide 81,906 by 34.

Notice in the third division that 30 is smaller than the divisor. Place zero in the quotient, write 6 to the right of 30, and divide 306 by 34.

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