K.-Simplification of Fractional Expressions. Rule. Perform the operations indicated by the signs, for which the former rules will be sufficient, taking care to reduce each part of an expression to the simplest form before actually performing the operations. I. TO WRITE A FRACTION WITH TEN, OR ANY POWER OF TEN FOR ITS DENOMINATOR, AS A DECIMAL FRACTION. Rule. Mark off with a point as many figures as there are ciphers in the denominator, beginning from the right hand. If there be not enough figures in the numerator, write ciphers before to make up the number required. If there be too many, those that remain are the whole number which the fraction contains. 11. TO CONVERT A DECIMAL INTO A VULGAR FRACTION, OR THE SUM Rule. Form a fraction having for its numerator all the figures integral and decimal in the given decimal, and for its denominator 1, followed by as many ciphers as there are decimal figures; and reduce to its lowest terms. Or, form a series of fractions having the several decimal figures as numerators, and for denominator 1, followed by a number of ciphers equal to the distance of the figure from the point: the sum of these is the equivalent expression. EXAMPLES. 1. Convert 3.14159 into a vulgar fraction, and the sum of a series of fractions. Rule. Write the decimals under each other, so that the decimal points may be all in one column. Add the different columns together, as in common addition, and place the decimal point under the others. EXAMPLE. Add together 3.14159, 31.4159, 314.159, 314159. 3.14159 31.4159 314.159 314159 314507.71649 Ans. IV. TO SUBTRACT DECIMALS. Rule. Write the smaller number under the greater, keeping the decimal points immediately under each other. Subtract each figure of the lower line from that above it, as in common Subtraction, supplying a cipher, whenever a figure occurs in one line, and not in the other. 3. Subtract .2709 from .4086382. .4086382 .1377382 Ans. V. TO MULTIPLY DECIMALS. Rule-1. To multiply by 10, 100, 1000, &c. remove the decimal point as many places to the right as there are ciphers in the multiplier, supplying ciphers on the right to make up any deficiency. 2. To multiply two decimal fractions, multiply as in whole numbers, and mark off in the product as many figures for decimals as there are decimals in the multiplier and multiplicand together. If there are not sufficient figures in the product, supply ciphers on the left. EXAMPLES. 1. Multiply 57.29577 by 10, 1000, 1000000. 57.29577 X 10=572.9577 57.29577 X 1000 = 57295.77 2. Multiply 21.0735 by .04683. 21.0735 .04683 632205 1685880 1264380 842940 .986869005 Ans. 3. Multiply .008207 by 7.056. .008207 7.056 49242 41035 574490 .057908592 Ans. VI. ABBREVIATED METHOD OF MULTIPLICATION OF DECIMALS, WHEN ONLY A CERTAIN NUMBER OF DECIMAL PLACES ARE REQUIRED IN THE PRODUCT. Rule-1. Reverse the multiplier, and place it under the multiplicand, so that what was the units' figure be under the last place of decimals to be retained, placing ciphers, if necessary, so that every figure in the multiplier may have a figure above it. 2. Multiply as usual, beginning each figure of the multiplier with that |