Ans. 36 Examples. 1. Divide 92 by 1 of 7. 3. Divide [ by 4. 91= 55 dividend. 4. Divide of by of 4. of 1=1 divisor. Then, Ans. Q: 55 x==239=2U Quot. 5. Divide 53 by 73. Ans. 43 2. Divide by .. Ans. = 6. Divide } by 6. Ans. QUESTIONS. 1. What are Fractions ? 20. What is the least common mula 2. Of how many kinds are Frac tiple? tions ? 21. What is a prime number? 3. Wherein do they differ? 22. What is a perfect number? 4. How is a Vulgar Fraction exa 23. What is Reduction of Vulgar pressed? Fractions ? 5. What is the number below the 24. How do you find the greatest line called ? common measure of two or more 6. What does it express ? numbers ? 7. What is the number above the | 25. How do you find the least comline called ? mon multiple ? 8. What does that show? 26. How are fractions redaced to 9. How is a Decimal Fraction ex their lowest terms ? pressed? 27. How are mixed numbers redu. 10. How is it distinguished from a ced to improper fractions ? whole number? 28. How is an improper fraction re11. How may a Decimal be changed duced to whole or mixed numinto a Vulgar Fraction ? bers? 12. What is a Vulgar Fraction? 29. How is a compound fraction re13. What is a proper fraction ? duced to a single one ? 14. What is an improper fraction ? 30, How are fractions reduced to a 15. What is a compound fraction ? common denominator ? 16. What is a mixed number? 31. How to the least common deno17. What is the common measure of minator? two or more numbers ? 32. What is the rule for the Addi18. What is the greatest common tion of Vulgar Fractions ? measure of two or more num. 33. for the Subtraction ? bers ? 34. for the Multiplication? 19. What is the common multiple of 35. for the Division ? two or more numbers: 2. Decimal Fractions. A Decimal Fraction is one whose numerator only is expressed. Were the denominator to be written, it would, in all cases, be 10, 100, or 1, with as many ciphers annexed as there are figures in the decimal. When there are whole numbers and decimals in the same sum, it is called a mixed number, as, 16.44, which is read sixteen and forty-four hundredths. In decimals, unity is considered a fixed point, each way from which the value of the numbers varies in a ten-fold proportion, increasing towards the left hand, and decreasing towards the right, as in the following TABLE. 2 Ciphers in the right hand of decimals do not alter their value, but on the left hand, decrease their value in a ten-fold proportion. Thus, .5.50 and .500 express the same value, viz. ì, and are read 5 tenths, 50 hundredths, and 500 thousandths ; but .5 .05 and .005 decrease in value in a ten fold proportion, and are read 5 tenths, 5 hundredths, and 5 thousandths. Tie order to make the scholar familiar with the notation of deci. mals, he is requested to write out the following Examples. 1. Express the decimal .36 in. 1. Write forty-three thouwords. sandths in characters. 2. Express .03 in words. 2. Write 60 and nine hundred 3. Express 1002 in words. thousandths in characters. 4. Express 27.27 in words. 3. Write one hundred and four 5. Express 34.14 in words. thousandths. 1. ADDITION OF DECIMALS. Rule. Place the numbers under each other according to the value of their places, and add as in whole numbers. Point off as many decimal places from the sum as are equal to the greatest number of decimal places in either of the gived numbers.* *When the numbers are all written properly, and the amount properly point. ed, the separatrices, or decimal poiuts, will all stand in a coluon, or directly Examples. 1. What is the sum of 25.4 rods, 16.05 rods, 8.342 rods and 46.004, when added together? 25.4 Here it will be seen that the decimal points all fall in 16.05 the same column, that the decimals are arranged towards 8.842 the right hand from this column, and the whole numbers 46 004 towards the left, and that the number of decimals in the sum is equal to the greatest given number of decimals. 96.296 Ans. 2. What is the sum of 312.984, 4. What is the sum of .014, 21.3919, 2700.42, 3.153, 27.2, anu | .9816, .32, .15914, .72913 and 58 L06 : Ans. 3646.2088. .0047 ? Ans. 2.20857 3. What is the sum of thirtyseven and eight hundred twenty 5. What is the sum of six one thousandihs; five hundred thousand and six thousandths; and forty-six and thirty-five hun five hundred and five hundredths, dredths ; eight and four tenths, and forty and four tenths ? and thirty seven and three hun Ans. 6540.456 dred twenty-five thousandths ? Ans. 6.9.896. 2. SUBTRACTION OF DECIMALS. RULE.---Place the less number under the greater, with one of the decimal points directly under the other ; then subtract as in whole numbers, and point off as ja addition. Examples. . 1. From 468.742 take 76.4815 | 4. From .9173 subtract .2138 468.742 Ans. .7035 76.4815 5. From 742 subtract 195.127 Rem. 392.2605 Ans. 546.873 2. From 273 take 1.9183 Rem. 0.8115 6. From 9.005 subtract 8.728 Ans. 0.277 3. From 428, subtract 14.76 Ans. 413.21 over one another. All the difficulty in Decimal Fractions is in placing the numbers, and pointing off the decimals. In other respects, they are managed precisely as whole numbers. The scholar should endeavor to become familiar with the management of Decimals, as to us they form one of the most useful parts of Arithmetick. Our lawful mode of reckoning money is purely decimals 3. JIULTIPLICATION OF DECIMALS. Rule.- Write the multiplier under the inultiplicand; then multiply as in whule numbers, and from the procluct point off as many places for deciinals, as there are decimal places in both the factors. Il' there be not so many figures in the product as there ought to be decimals, supply the deficiency by prelixing ciphers. Examples. 1. Multiply 25.63 by 2.4 3. Muitiply .026 by .003 25.63 Here because there are .026 Here because there 24 three decimal places in both .003 are six decimal places factors, 1 point off threc in both factors, I make 10252 places in the product. Pro. .000078 up the deficiency of 5126 the product by pla cing four ciphers at 61.512 the left hand of 78. 2. Multiply 25.238 by 12.17 Ans. 307.14646 4. Multiply 17.6 by .75 Ans. 13.2 4. DIVISION OF DECIMALS. Rule.t-Divide as in whole numbers, and point off so many places for decimals in the quotient as the decimal places in the dividend exceed those in the divisor. If there are not so many figures in the quotient as the number of decimals required, sapply the defect by prefixing ciphers. If the decimal places in the divisor exceed those in the dividend, make them equal by annexing ciphers to the latter. When there is a remainder, by annexing ciphers, more decinal places may be obtained in the quotient. The truth of this rule will appear by considering that .026 and .003 are equivalent to 1367 and 10; whence 1360x100=1078000 = .000078 by the nature of notation ; that is, the dechual consists of as many places as there are ciphers in the denominator, and when the product falls short of this number, the deficiency must be made up by ciphers on the left hand. There are usually given several methods of contraction under this rule ; but they are of no essential service, and might perplex the young scholar. It may not be amiss, however, to observe that in dividing by 10, 100, or 1 with any number of ciphers, we have only to remove the separatrix as many places towards the right hand as there are ciphers in the multiplier ; thus 2.71 multiplied by 10 is 27.1 ; by 100, it is 271, &c. † The reason of the rule for pointing off the decimal places in the quotient will appear obvious by considering that the divisor and quotient are two factors whose product is the dividend, and that the decimal places in both the factors are equal to the decimal places in their product, as was shown in Multiplication of decimals, Examples. 1. Divide 487.653 by 24.21 24.21)487 653(20.144 Here are four decimals in the dividend, 4842 (counting the cipher added to the remainder after bringing down all the figures in the divi3453 dend) and only two in the divisor, therefore 2421 there must be two decimal places pointed off in the quotient, that the decimal places to the 10320 quotient and divisor counted together may 9684 equal those in the dividend. The sign + plus. after the quotient, shows that more decimals 636 may be procured by annexing ciphers to the: remainder. 2. Divide 7.02 by .18 Ans. 39. 3. Divide .0081892 by .547 Ans. 0236 The Scholar is requested to point the two following examples. 4. Divide 4263 by 2.5 5. Divide 4.2 by 36. Ans. 17052 Ans. 116+ RECIPROCALS. The Reciprocal of a given number is one, which, multiplied by the given number, gives a unit for the product; thus.2 is reciprocal of 5, becausc 5 X.l=1. If the given number be a multiplier, its reciprocal may be em ployed as a divisor of the multiplicand, and the quotient will be equal to the product of the muliiplicand by the multiplier ; but if the given number be a divisor, its reciprocal may be employed as a multiplier of the dividend, and the product will be equal to the quotient of the dividend by the divisor. Examples.-1. Multiply 7 by 5. 7 X5=35, and 7--2--35. 2. Divide - by 5.7.-3=1.4, and 77.21.4. Problem. To find the reciprocal of any number. RULE.--Divide a unit by the given number, and the quotient will be its reciprocal. Examples. 1. What is the reciprocal of 125? 3. By what number shall I 125)1.000.008 Ans. multiply 240, that the product 1000 may be equal to the quotient of 240 divided by 25 ? 125 X.008=1. proof. 25)1.000.04 recip. Ans. 100 2. What is the reciprocal of .4? Ans. 2.5 240-225–9:6} proof. |