REDUCTION. 126. To reduce a fraction to a decimal. Ex. Reduce to a decimal. OPERATION. 4) 3.00 .75 ANALYSIS. equals of 3 units. 3 units equal 300 hundredths. of 300 hundredths equal 75 hundredths. 127. RULE.-Annex decimal ciphers to the numerator, and divide by the denominator, pointing off as many decimal places in the quotient as there are ciphers annexed. 128. A fraction in its lowest terms can be reduced to a pure decimal only when its denominator contains no prime factors but 2 and 5. If the denominator or divisor contain any prime factor other than 2 and 5, the division will not end. The decimals thus produced are called Interminate or Repeating Decimals, and the figures repeated, Repetends. When a fraction is in its lowest terms, its numerator and denominator have no common factors (61). Annexing ciphers to the numerator introduces the factors 2 and 5 only; hence, if the denominator is an exact divisor of the numerator with the ciphers annexed, it must contain these prime factors and none others. plying both terms by the denominator 3. (57, 3.) 131. RULE.-Omit the decimal point, supply the proper denominator, and reduce the fraction to its lowest terms. 133. Since decimals, like integers, increase and decrease uniformly according to a scale of ten, with the exception of placing the decimal point in the result (usually called pointing off), they may be added, subtracted, multiplied, and divided in the same manner as integers. Ex. What is the sum of 28.7, 175.28, .037, 25.0045, and 4.08? OPERATION. 28.7 175.28 .037 25.0045 4.08 233.1015 ANALYSIS.-Write the numbers so that units of the same order stand in the same column. If the decimal points are in the same vertical line, tenths will necessarily be under tenths, hundredths under hundredths, etc. Add as in integers, and place the point in the result directly under the points of the numbers. Ex. Add .6, .37, 16.0481, 8.1234, and 24.125. .6 .371 OPERATION. = .6 = = 16.0481 8.12344 = 24.125 .3775 16.04831 8.1234 = 24.125 49.27421 ANALYSIS.-Reduce the complex deci. mals as far as the decimal places extend in the other numbers. Since the fractions now express parts of the same fractional unit, they may be added. In practice, the fractions may be rejected if the decimals are carried one place, at least, farther than accuracy is required. 134. RULE.-Write the numbers so that their decimal points are in the same vertical line. Add as in integers, and place the decimal point in the result directly under the points in the numbers added. EXAMPLES. 135. 1. Add ninety-seven hundredths; three hundred and forty-seven thousandths; sixteen, and seventy-five hundred-thousandths; four hundred and seventy-five, and two thousand and thirty-seven millionths. 2. Add four, and eighty-one thousandths; thirty-seven, and two hundred and one ten-thousandths; seven thousand and eight hundred-thousandths; seven thousand, and eight hundred-thousandths; nineteen hundredths; three hundred and sixty-four, and nine tenths; and fifty-six, and fifty-four thousandths. 3. Add three hundred and seventy-five, and eight hundredths; eighteen thousandths; ninety-six, and eighty-four hundredths; four, and four tenths; and eight hundred and seven ten-millionths. 4. What is the sum of 18 hundredths; 716 hundred-thousandths; 6342 millionths; 11567 ten-millionths; 625 ten-thousandths; 9 tenths; 99 hundredths; and 512 thousandths? 5. Add 81.86; 12.593; 4.004; 18.00129; .443; 400.043 ; .12875; 175.00175; 17.3008; 9000.0016; and .9016. 6. Required, the sum of 99 ten-thousandths; 157 thousandths; 789 millionths; 6 tenths; 18 hundredths; 1728 tenmillionths; and 88 hundredths. 7. Add $1728.64; $0.37; $18.441; $10.18; $6.25; and $0.161. 8. What is the sum of $12.37; $144.18; $6.621; $175.061; $40.174; and $398? 9. Add .1264; 12.875; 187.25; 9.1414; .12; 5.7604; and .0008. 10. Add .261; 4.18; .0017; .008641; .04; 17.387; and .0102075. SUBTRACTION. 136. Ex. From 12.75 subtract 8.125. OPERATION. 12.75 8.125 4.625 ANALYSIS.-Write the subtrahend under the minuend so that units of the same order stand in the same column. Subtract as in integers, and place the point in the result directly under the points of the numbers. If, as in this example, the minuend has not as many decimal places as the subtrahend, suppose decimal ciphers to be annexed until the right-hand figures are of the same order. (120.) Reduce complex decimals as in addition (133). 137. RULE. Write the numbers so that their decimal points are in the same vertical line. Subtract as in integers, and place the point in the remainder directly under the points in the minuend and subtrahend. EXAMPLES. 138. 1. From four, and sixty-five thousandths, subtract eight hundred and forty-seven ten-thousandths. 2. From twenty-seven hundredths take twenty-nine hundredthousandths. 3. From nine thousand, and thirty-four ten-thousandths, subtract nine thousand and thirty-four ten-thousandths. Find the difference between 4. 8.3644 and 7.8996. 5. 17.4586 and .785. 6. 1.010101 and .999999. 7. $173.46 and $87.29. 8. 3 and .873845. 9. 17.24 and 18.9731. 10. $510.60 and $389.451. 11. $1728 and $.06. 12. 17.864 and 16.94. 15. 117.48 and 49.43§. MULTIPLICATION. 139. Ex. Multiply .144 by .12. OPERATION. .144 .12 .01728 ANALYSIS. For convenience, write the right-hand figures of the factors in the same vertical line. 000 1728 10000 .144 x .12: Multiply the numerators of the two factors for the numerator of the product, as in multiplication of fractions. In the above multiplication of fractions, it will be observed that the number of ciphers in the denominator of the product equals the sum of the ciphers in the denominators of the two factors. Since each cipher represents a decimal place, the product should have as many decimal places as both factors. If the number of figures in the product is less than the number of decimal places in the two factors, supply the deficiency by prefixing ciphers. 140. RULE.-Multiply as in integers, and from the right point off as many decimal places in the product as there are decimal places in the two factors. NOTE. To multiply a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the right as there are ciphers in the multiplier, annexing ciphers to the multiplicand, if necessary. EXAMPLES. 141. 1. Multiply three hundred and forty-four ten-thousandths by twelve thousandths. 2. Multiply one hundred and ninety-two thousandths by four, and nineteen hundredths. 3. What is sixteen hundredths of six hundred and thirty-two millionths? 4. What is five hundredths of $864.32? Of 3645.75 francs? 5. What is .0583 of 784.65? Of 943.25 ? |