4. Reduce .75 to a common fraction in its lowest Art. 93. RULE.-To reduce a decimal to a common fraction in its lowest terms, Omit the decimal point and supply the denominator, and then reduce the fraction to its lowest terms. CASE III.-COMMON FRACTIONS REDUCED TO DECIMALS. ? 1. How many tenths in ? In ? ? 17. Reduce .621 to thousandths. 18. Reduce .0123 to ten-thousandths. 19. Reduce 12.061 to millionths. Art. 94. RULE.-To reduce a common fraction to a decimal, Annex decimal ciphers to the numerator and divide by the denominator, and point off as many decimal places in the quotient as there are annexed ciphers. LESSON III. Addition of Decimals. 1. What is the sum of 5 tenths and 4 tenths? 6 tenths and 9 tenths? SOLUTION.-6 tenths and 9 tenths are 15 tenths, which is equal to 1 unit and 5 tenths. 2. What is the sum of 8 hundredths and 7 hundredths? 18 hundredths and 7 hundredths? 3. What is the sum of 28 thousandths and 9 thousandths? 56 thousandths and 22 thousandths? WRITTEN EXERCISES. 4. What is the sum of 24.6, 307.08, 93.609, .456, 400.06, 37.027. 5. What is the sum of .4506, .709, and 27.0508? 6. Add 15.34, 6.078, 60.804, and 99.875. 7. Add $21.94, $87.075, $9.858, and $807.621. 8. Add thirty-nine hundredths, six hundred and eight ten-thousandths, and eighty-seven thousandths. 9. What is the sum of 47.6 miles, 19.48 miles, 34.75 miles, and 76.625 miles? Art. 95. RULE.-To add decimals, 1. Write the numbers so that figures of the same order shall be in the same column. 2. Beginning at the right, add as in the addition of integers, and place the decimal point at the left of the tenths' order in the amount. LESSON IV. Subtraction of Decimals. 1. From 8 tenths take 5 tenths. 2. From 5 tenths take 5 hundredths. SOLUTION.-5 tenths are equal to 50 hundredths, and 50 hundredths less 5 hundredths are 45 hundredths. 3. From 7 hundredths take 4 thousandths. WRITTEN EXERCISES. 4. From 56.403 take 18.6. PROCESS: 56.403 37.803, Ans. Since only like units can be subtracted, write the numbers so that figures of the same order shall be in the same column. Since ten units of any decimal order 5. From 56.6 take 18.403. make one unit of the next 56.600 PROCESS: 18.403 38.197, Ans. 6. From 45.3 take 28.756. 7. From .0407 take .008075. left-hand order, subtract as in integers, and place the decimal point at the left of the tenths' order in the remainder. 8. From twelve thousandths take eight millionths. 9. From eight tenths take eight ten-thousandths. 10. From 47.065 + 36.87 take 9.08 + 43.375. 11. From the sum of twenty-five thousandths and forty-six ten-thousandths take their difference. Art. 96. RULE. To subtract decimals, 1. Write the numbers so that figures of the same order shall be in the same column. 2. Subtract as in the subtraction of integers, and place the decimal point at the left of the tenths' order in the remainder. LESSON V. Multiplication of Decimals. 1. How much is 3 times 2 tenths? 3 times ? 2. How much is 4 times 3 tenths? 4 times .4? 3. How much is 5 times 18? 5 times .05? 8. What is the denominator of the product when tenths are multiplied by units? Tenths by tenths? 9. What decimal order is produced when tenths are multiplied by hundredths? Hundredths by hundredths? Hundredths by thousandths? 10. What is the number of decimal orders in the product of any two decimals? PROCESS. = 4.35 X 10 Since the value of decimal orders increases from right to left tenfold (Art. 90, Pr. 2), the removal of the decimal point one place to the right removes each figure one order to the left, and hence multiplies 4.35 by 10. The removal of the decimal point two places to the right multiplies 4.35 by 100. 27. Multiply 4.085 by 100. By 1000. PRINCIPLES AND RULES. Art. 97. PRINCIPLES.-1. The number of decimal places in the product equals the number of decimal places in both factors. 2. Each removal of the decimal point one place to the right multiplies a decimal by 10. Art. 98. RULES.--1. To multiply one decimal by another, Multiply as in the multiplication of integers, and point off as many decimal places in the product as there are decimal places in both multiplicand and multiplier. NOTE.-If there be not enough decimal figures in the product, supply the deficiency by prefixing decimal ciphers. 2. To multiply a decimal by 10, 100, 1000, etc., |