inator 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 ciphers. EXAMPLES. 2. Multiply five hundredths by five tenths. Operation. .05 .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. 3. Multiply 49.5 by 3.2. 3.2 4. Multiply 569.39 by 27.05. 99.0 1485. 158.40 5. Multiply 6.791 by 2.67. 6. Multiply 549.05 by 35.257. 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 hundredths. 16. Multiply sixty-nine and five tenths by three thou sandths. QUESTIONS. 26. What is the rule for the multiplication of fractions? 27. What is the value of the product? DIVISION OF DECIMALS. 1. Divide twenty-five hundredths by five tenths. By Vulgar Fractions. Operation. 10)1%(1%=.5, Ans. When no remainder will arise from the division, the terms of the dividend may be We have seen that the deci divided by the corresponding mal places in the product of terms of the divisor. (See page 78.) It will be seen that any two factors are equal to the decimal point implies a those factors. the decimal places in both division of the numerator of the quotient by its own denominator. The divisor and quotient are factors of the dividend; therefore, the decimal places in the quotient and divisor, taken together, must be equal to the decimal places in the dividend. Hence the RULE. Divide as in whole numbers, and point off so many places for decimals in the quotient, that the decimal places in the quotient and divisor, taken together, shall equal the decimal places in the dividend; or, so many as the decimal places in the dividend exceed those of the divisor. If there are not so many, supply the deficiency by prefixing ciphers. NOTE. The above rule may be illustrated by reference to the operation of the preceding question by Vulgar Fractions, thus; the ciphers in the denominator of the divisor and quotient are equal to the ciphers in the denominator of the dividend; but the decimal places in the numerator of a decimal fraction are equal to the ciphers in its denominator; therefore the decimal places in the numerator of the quotient and divisor, taken together, must be equal to the decimal places in the numerator of the dividend. 2. Divide five tenths by twenty-five hundredths. Operation. .5===.50, then .25).50(2, Answer. .50 NOTE. Annexing a cipher to a decimal fraction multiplies the terms of the fraction by 10, and, therefore, does not alter the value. (See page 65.) Whenever the decimal places in the divisor exceed those of the dividend, annex a cipher or ciphers to the dividend; this reduces it to the denomination of the divisor. QUESTIONS. 28. What is the rule for the division of decimals? 29. How is the quotient pointed? 30. Illustrate the rule. 3. Divide three hundred and sixty-nine thousandths by nine. Operation. .041, Answer. 369 1000 The necessity of prefixing a cipher to the quotient will be more readily seen by the following: ÷9=1800. If we remove the denominator of the quotient, and prefix the decimal point to the numerator, it will then be 41 hundredths, which is not its true value; but, by placing a cipher between the decimal point and the left hand figure, the right hand figure of the quotient will be made to occupy the thousandths' place, which will denominate the parts into which the unit is divided, or show their true value. Prefixing a cipher, therefore, divides the fraction, by multiplying its denominator. 9. Divide one hundred and seventeen and nine tenths by nine tenths. Answer, 131. 10. Divide four hundred fifty-six and three hundred thirtythree thousandths by three hundredths. 11. If three hundred fifty pounds of beef cost twelve dollars twenty-five hundredths, what cost one pound? Answer, .035. 12. If 565.05 pounds cost 25.42725 dollars, what will one pound cost? Answer, .045. FEDERAL MONEY. FEDERAL MONEY is the coin of the United States. Its denominations are eagles, dollars, dimes, cents and mills. From the above examples and illustrations in Decimal Fractions, we have seen that a decimal is the division of the unit into tens, and that from the unit's place towards the right hand it decreases in a tenfold proportion. If we examine the denominations of Federal Money, we shall find that all bear a decimal relation to the dollar, which is considered the unit. This will be seen by the following TABLE. NOTE. The eagle is a gold coin, the dollar and dime are silver coins, the cent is a copper coin. The mill is only imaginary, there being no coin of that denomination. The dime being 1 tenth of a dollar, it occupies the first, or right hand place from the dollar-thus, 0.1. The cent, being 1 tenth of a dime, and consequently 1 hundredth of a dollar, occupies the second place, or place of hundredths -thus, 0.01. The mill, being 1 tenth of a cent, and consequently I thousandth of a dollar, occupies the third place, or place of thousandths-thus, 0.001. Placing them tod. d. c. m. gether, 1 1 1 1. This may be read, one dollar, one dime, one cent and one mill; or, one dollar, eleven cents and one mill-as eleven cents is equal to one dime and one cent. The same may be said of eagles and dollars-thus, 25 dollars may be read 2 eagles and 5 dollars, since 20 dollars are equal to 2 eagles. Write 4 eagles, 5 dollars, 8 E. D. D. C. M. dimes, 3 cents, 5 mills-4 5 8 3 5. This may be read, 4 eagles, 5 dollars, 8 dimes, 3 cents and 5 mills; or, 45 dollars, 83 cents and 5 mills. Hence, it is evident that the denominations in Federal Money are dollars and decimals of a dollar, and may be treated as Decimal Fractions. Federal Money is denoted by this character ($) placed before the figure. ADDITION OF FEDERAL MONEY. RULE. Write the denominations, add and point the result as in Addition of Decimals. EXAMPLES. 1. If I buy a bushel of wheat for $2.25-a bushel of corn QUESTIONS. 1. What is Federal Monday? 2. What are its denominations? for $1.32-four yards of cloth for $14.285; how much do for the whole? pay 2.25 NOTE. The scholar will do well to turn now to the rule for reducing a vulgar fraction to a decimal. 1.32 14.285 $17.855, Answer. 2. Bought 8 yards of cloth for $16.25-a pair of shoes for 87 cents; a hat for $4.33; a whip for 42 cents —a knife for 371⁄2 cents. How much did I pay for the whole? Ans. $22.25.5. 3. Bought a cart for $17.62; a wagon, $62; a plough, $7.48; 4 rakes, $1.26; 3 hoes, $2.15; a pitch-fork, 87 How much did the whole cost? cents. Ans. $91.88. 4. Purchased a barrel of flour for $9.25; 4 pounds of tea, $2.08; 2 gallons of molasses, 64 cents; 3 pounds of raisins, 37 cents; 9 pounds of sugar, $1.21; 8 yards of calico $2.234. What is the amount of the whole? Ans. $15.805. 5. Add forty dollars, sixty-seven cents and three mills; six hundred seventy-nine dollars, twenty-five cents and seven mills; one thousand and four dollars, five cents and five mills; nine hundred, ninety-nine dollars, thirty-nine cents and nine mills. Ans. $2723.384. SUBTRACTION OF FEDERAL MONEY. RULE. Write the numbers, subtract and point the result as in Subtraction of Decimals. EXAMPLES. 1. A man bought 50 bushels of wheat for $125.50-sold it for $145.75. How much did he gain? Ans. $20.25. 2. Bought 26 bushels of oats for $8.49-sold the same for $8.94. How much did I gain? Ans. $0.45. 3. Purchased a horse for $92-lost on the sale of him $15.25. For how much did I sell him? Ans. 76.75. 4. Bought 2 barrels of flour for $22.50-but, it being damaged, I am willing to sell it at $4.25 less. What must I receive for it? Ans. $18.25. 5. Bought 8 yards of cloth for $36-gave a $50 bill. What must I receive in change? Ans. $14. |