of the same name or denomination. The amount, both in decimals and whole numbers, takes its name from the lowest, or right-hand place of the numbers added: thus, 1 hundred, 2 tens and 5 units, when added, are read 125 units; and one tenth, 2 hundredths and five thousandths, when added, are read, 125 thousandths. Decimal fractions may also be added and illustrated in the same manner as vulgar fractions. 2. Add two and five tenths; four and six hundredths; seven and three thousandths. Then 2.5=25, and 4.06=106, and 7.003-7003 1000 25×100-2500, and 106 × 10-1068 OBS. Multiplying the terms of a fraction by the same quantity does not alter its value. (See Art. 61.) The fractions added: 1588+1868+7003-13563-13.563 Ans. The same, added by decimal fractions: From the foregoing it is evident, that decimal fractions are reduced to a common denominator by writing tenths in the place of tenths, and hundredths in the place of hundredths, and supposing those decimal places, which are deficient, to be supplied by ciphers. Applying the decimal point to the amount, is equivalent to dividing it by its own denominator, which we have seen is the denominator of the lowest of the given decimals, or that decimal whose denominator is the largest. But the decimal places in the numerator of a decimal fraction, are equal to the number of ciphers in its denominator, the denominator being understood; therefore, addition of decimals may be performed by the following QUESTIONS.-16. How is the first decimal place produced? 17. The second, third, &c.? 18. How are decimals to be added, written? 19. From what does the amount take its name? 20. Applying the decimal point is equal to what? 21. How are deci mal fractions reduced to a common denominator? RULE. Place the numbers, tenths under tenths, hundredths under hundredths, etc.; or, so that the decimal points may stand directly under each other. Add as in whole numbers; observing to point off as many places for decimals in the amount, as will be equal to the greatest number of decimals in any of the given numbers. 4. Add thirty-five and four tenths; five hundred twentynine and seven millionths; sixty-nine, four hundred and sixtythree thousandths; two hundred, sixteen and two hundredths; seventy-seven, nine hundred and two tenths. Ans. 1827.083007. 5. Add forty-nine and sixty-seven hundredths; six hundred seventy-nine, two hundred seventy-five thousandths; one thousand four hundred, fifty-five thousandths, nine hundred and ninety-nine millionths. 6. Add 249.39; 6712.9123; 6.3219; 2739.235; 5.671; 723.2674; 926.679; 72.601. 7. Add .7+9.2+.321+279.+4.67+349.2+3.956. 8. Purchased of one man 325.5 lbs. of beef; of another, 175.75; of another, 178.028; what was the amount? 9. I receive of A. $183.25; of B. $138.89; of C. $372.218; of D. $88.99; of E. $137.29; what is the amount of the whole? 10. Add $59.67; $158.355; $375.752; $167.375; $567.756. SUBTRACTION OF DECIMALS. Art. 90.—1. From three and two tenths, take one and five tenths. Operation. 3.2 1.5 1.7 Ans. Lastly, Because five tenths cannot be taken from two tenths, we borrow 1 from the unit's place, which, reduced to tenths, equals 10 tenths; 10+2=12, then 12-%==.7. 1 from 2, and 1 remains. Again, three and two tenths is the quotient of 32 divided by 10, (see definition of a mixed number, Art. 54;) therefore, 3.2=32, and 1.5=15; then· 32-15=17=1.7 Ans., as before. Pointing off the remainder is dividing it by its own denominator. Hence the RULE. Write the numbers and point the result, as in Addition of Decimals, and subtract as in whole numbers. 6. From two hundred and sixty-nine and three tenths, take fifty-seven and thirty-nine hundredths. Ans. 211.91. 7. Take twenty-four thousandths from nine hundredths. Ans. .066. 8. Take sixty-five millionths from five tenths. 9. From three hundred seventy-five thousand and three tenths, take two hundred forty-nine and thirty-nine one hundred thousandths. Ans. 374751.29961. 10. From 361.2 take 276.75. 11. From 456.35 take 27.356. 12. From 5678.0002 take 3980.96715. QUESTIONS.-22. How are decimals to be subtracted, written? 23. How can five tenths be taken from two tenths? 24. What is done with the unit borrowed? MULTIPLICATION OF DECIMALS. Art, 91.-1. Multiply three and five tenths by five tenths. Operation. 3.5 .5 1.75 Answer. 10 The product of tenths into tenths is hundredths: o×10= 25.25. The product of tenths into units is tenths: 3 X ==1.5. The sum of the product, .25+1.5=1.75 Ans. Again, 3.5-35 and 35×5=175=1.75 Ans., as before. The value of the product is the quotient of its numerator divided by the denominator. Hence the figures cut off from the right of the numerator are equal to the ciphers in the denominator; but the ciphers in the denominator of the product, it will be perceived, are equal to the decimal places in both factors; therefore the multiplication of decimals may be performed by the following RULE. Multiply as in whole numbers, and point off as many places for decimals in the product as there are decimal places in both factors. If there are not so many places, supply the defect by prefixing eiphers. EXAMPLES. 2. Multiply five hundredths by five tenths. Operation. .05 .5 .025 Answer. The product of tenths into hundredths is thousandths. In this example, the tenth's place in the product is wanting; we must, therefore, supply it by prefixing a cipher. QUESTIONS.-25. What is the product of tenths into units? 26. Of tenths into tenths? 27. What is the rule for the multiplication of fractions? 28. What is the value of the product? 1485. 158.40 4. Multiply 569.39 by 27.05. 7. Multiply six hundred and seventy-five by twenty-seven and three tenths. 8. Multiply sixty-seven thousand by three hundredths. 9. Multiply 34.56 by 1.3. 10. Multiply 674.49 by 37.16. 11. Multiply 5648 by 6.78. 12. Multiply 7864 by 467. 13. Multiply fifty-seven and three tenths by twenty-nine. 14. Multiply thirty-seven thousand by three hundredths. 15. Multiply fifty thousand and seven tenths by four hun. dredths. DIVISION OF DECIMALS. Art. 92.-1. Divide twenty-five hundredths by five tenths. Divide as in whole numbers, and point off so many places for decimals in the quotient, that the decimal places in the quotient |