Arithmetic

B

Multiplying Decimals

Multiplying decimals is similar to multiplying integers, except that the position of the decimal point must be kept in mind. First, multiply decimal numbers as if they were integers, without considering the decimal points. Then place the decimal point at the appropriate position in the product so that the number of decimal places is the same as the total number of decimal places in the numbers being multiplied. For example, in multiplying 0.3 by 0.5

the 15 in the product is the direct value of 3 times 5. We place the decimal point differently than we do in addition and subtraction. Since the factors, 0.3 and 0.5, each have one decimal place, the product must have two decimal places. Thus, 0.15 is the product.

Another example will serve to clarify this concept. Multiply 0.2 by 0.3. The product of 2 and 3 is 6, but since 0.2 and 0.3 each have one decimal place, the product must have a total of two decimal places. We can fulfill this requirement by placing the decimal point two places to the left of the 6 in the product, then adding a zero to fill the tenths place: 0.06.

More complicated problems are solved similarly:

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Because the top number has three decimal places and the bottom number has two, the product must have a total of five decimal places. Count five places to the left starting with the digit farthest to the right (the 8 of the final product), adding a zero if necessary. The answer is 0.03108.

One final example will also illustrate the importance of counting the proper number of decimal places in determining products of decimal numbers. Multiply .001 and .002. The multiplication of the digits is simple: the answer is 2. However, each number has three decimal places, giving a total of six places that must be preserved in the product. Insert five zeroes to fill the places between the 2 and the decimal point. The answer is 0.000002.

C

Dividing Decimals

Like multiplication, the division of decimal numbers follows the same procedures used to divide integers, except that we must take care to determine the proper placement of decimal points in quotients. Dividing a decimal number by a whole number is straightforward: Place the decimal point in the quotient directly above the decimal point in the dividend and ignore it during the rest of the process of division:

In cases where the divisor is a decimal number, convert the problem to one in which the divisor is an integer; division may then proceed as in the above example. To divide 14 by 0.7, for example, convert the divisor to an integer by multiplying it by 10: (0.7)(10) = 7. Then multiply the dividend by an equal amount. We can understand this procedure more easily by considering the division rewritten as a fraction. Multiplying both numerator and denominator by the same amount will not change the value of the fraction:

Similarly, the division of 2.675 by 0.23 can be considered in the form 2.675/0.23. We can convert this fraction to a division involving an integer divisor, namely 23, if we multiply both numerator and denominator by 100:

We can convert any division problem involving a decimal divisor into a problem with an integer divisor simply by moving the decimal point in the divisor as many places to the right as is necessary to make it an integer. Then move the decimal point in the dividend an equal number of places to the right, and add zeros if necessary. For example, to divide 21.5 by .002, move the decimal point in the divisor three places to the right, giving the integer 2. Move the decimal point in the dividend three places to the right as well:

Carry out the division as usual, placing the decimal point in the quotient directly over the new decimal point in the dividend. The quotient in this case is 10750:

See also Mathematical Symbols; Mathematics, New; Number.

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