number; 2376 to a whole number; 613 to 15ths; of ofofto a simple fraction. A fraction may be reduced to a decimal, by performing the division which the fraction expresses, annexing decimal zeroes to the numerator. Thus, 33.0-5.6. A decimal may be reduced to a fraction, by writing the decimal for a numerator, and the denomination tenth, hundredth, &c. for a denominator. Thus .06; .193= A fraction may be reduced to its lowest terms, by dividing the numerator and denominator by their greatest common divisor. Thus 65=5, because 135, and 13 1. This reduction should always be made in giving the result of any operation. If the denominator is 10, 100, 1000, &c., Divide both terms of the fraction by 10, 5, and 2, as often as it can be done without a remainder. This process will evidently effect the reductions desired, because 10 contains no prime factors other than 2 and 5. In the reduction of decimals, this rule will be found the most convenient. 3 7 6. Reduce to a decimal, ;; 1; 8; 3; 9; 9;;; 275 94 2 3; 1; 5. 28 7. Reduce to their lowest terms, 75 38 1 256 90 76 5 8. Reduce each of the following decimals to a fraction, and reduce the fraction to its lowest terms: .8; .014; .08; .008; .045; .1768; .0375; .25. 9. Reduce to their lowest terms, 364; 2500 765 392 10000 972 13 181. 45 10. Reduce to a decimal, ; ; ; 4; 12; 193; 783 810. 463 25 1277 1804 ; 3096. Two or more fractions may be reduced to a common denominator, by dividing the least common multiple of all the denominators, by each given denominator, and multiplying both terms of the fraction by the quotient. EXAMPLE FOR THE BOARD. If we wish to reduce the fractions, 5, 3, 7, 13 and, to a common denominator, we first find that the least common multiple is 36. Then 1 In the first mode of performing the operation, the multipliers are each equal to 1, and the product by 1 is of course equal to the multiplicand. 11. Reduce to a common denominator, 3, §, 12, and 1. 5, and 1. 12. Reduce to a common denominator, 4, 8, 14 13. Reduce to a common denominator,ofof, 3, of and of of 3. The compound fractions must first be reduced to simple ones. 14. Reduce to a common denominator, of, 7 1. 11 and 1 of 3. 1463 793 131 of 15. Reduce 29163 to an improper fraction; 9847 to & mixed number; 287 to a decinal; to its lowest terms; 19 7 and to a common denominator; .004625 to a fraction; 295 to 19ths. 36' 54 ADDITION OF FRACTIONS. The addition of Fractions having the same denominator, is as easy as the addition of any other numbers. Thus, as 5 houses+3 houses+2 houses 10 houses, so ++ 5 2=40, and 13+13+=19. Therefore, to add fractions, Reduce all the given fractions to a common denominator, and add their numerators. 1. Add 21, 1, 151, and 285. 2. Add of,of, and 198. 11, 18. 3. Add ,,, and . §,, 4. Add 19, 281, 1611, and 34. 5. Add 1⁄2 of 1 of 1 of 4, 270, 11, and 1518. 6. Add 123, § of §, 115, of of 1, and 191. 7. Add of of 4 of 9, SUBTRACTION OF FRACTIONS. 8 As 8 houses-3 houses=5 houses, so 8-3=5, 5. and 14 44. The subtraction of fractions having a comInon denominator, is, therefore, devoid of difficulty, except when the numerator of the subtrahend is greater than that of the minuend. 162 161 EXAMPLE FOR THE BOARD. If we wish to subtract 3 from 163, we must, as in simple subtraction, increase the fraction in the minuend by 1, or, and carry 1 to the units of the 12 subtrahend. Therefore, to subtract fractions, Reduce both fractions to a common denominator, and subtract the numerator of the subtrahend from the numerator of the minuend. If the numerator of the minuend is less than that of the subtrahend, increase it by the denominator, and carry 1 to the units figure of the subtrahend. 1. What is the difference between 5 and? 2. Subtract 31 from 117. 6. What is the difference between 419 and 23? 7. What is the difference between of 10 and 2 of ? 8. Subtract of 16 from of of 953. MULTIPLICATION OF FRACTIONS. If any number be multiplied by 2, it is evident that the product will be as large as if multiplied by 1. Then × 11 of; × = of §. Multiplication of fractions is therefore performed in the same way as reduction of compound fractions to simple ones. Then, when either the multiplier or multiplicand is a fraction, Change whole or mixed numbers (if any) to improper fractions, and multiply the numerators together for a new numerator, and the denominators for a new denominator. To MULTIPLY A FRACTION BY A WHOLE NUMBER.-It is evident from the nature of fractions, that =2×}; } =3×1;=4×, &c. Hence, we may multiply a frac tion by a whole number, either by multiplying the nume 169 ། rator, or dividing the denominator. When one factor is a fraction, and the other an integer, either of these rules may be employed. When the multiplier is a mixed number, the product may often be obtained most conveniently, by multiplying first by the whole number, and afterwards by the fraction, and adding the two products. 1. Multiply by 31; 27 by 51. 2. Multiply of 8 by 16. 3. Multiply 157 by 91; 2811 by 1. 4. Multiply 341 by 4; by 8; 12; 16; 24; 96. 5. Multiply 223 by of 4 of 1611. 6. 14,3×15? 89×418=? 7. Multiply 10 by 2; 4; 31; 8; 248. 248 8. Multiply of by of of 95. 9. What is the product of 331 by 41? by 1? by 15? DIVISION OF FRACTIONS. The smaller the divisor, the more times will it be contained in the dividend, and the larger will be the quotient. Thus, if be divided by 1, the quotient will be will be contained 4 times as often as 1, therefore 3+1=1×3. will be contained only as often as 1, therefore 3-3of 4×3×3. Then, whenever either the divisor or dividend is a fraction, Change whole or mixed numbers (if any) to improper fractions, and if the numerators and denominators cannot be directly divided into each other, invert the divisor, and proceed as in multiplication. COMPLEX FRACTIONS, or such as contain fractions in their numerator or denominator, or both, may be resolved into simple frac35 tions by this rule. Thus, 35×5=4• TO DIVIDE A FRACTION BY A WHOLE NUMBER.-It is evident that 1÷2; }=}÷3; 1÷4, &c. Therefore, We may divide a fraction by a whole number, either by dividing the numerator, or multiplying the denominator. When the divisor is an integer, either of these rules may be employed. 1. Divide 1 by ; by ; by . 5. What is the quotient of 78 by 1011? By 15? 6. of of of 4-1 of 5 of 131=? 9. Divide 41 by 813; by 161; by 98. CIRCULATING OR INFINITE DECIMALS. 444 3 In changing fractions to decimals, if the divisor contains any prime factors, other than 2 or 5, we shall always find that our work would continue without end, the same figures being repeated again and again. The decimal is then called a circulating, or infinite decimal. Thus, =.333+; =.22222+; .090909+; 317.71396396+. The figures that are thus repeated, are called the repetend. The figures preceding the repetend (if any) are called the finite part of the decimal. The repetend is usually distinguished by placing a point over the first and last figThus .3 is the same as .3333+; 09 is the same as 090909+;.71396 is the same as .71396396+. In the latter example .71 is the finite part, and 396 is the repetend. We find, by reducing the following fractions to decimals, that =.i, therefore =.2, §=.5, &c. =.01, therefore,=.02, 14.14, 88.95, &c. .001, therefore, .002, 4= ure. .045, 163.163, &c. 45 999 Hence, every repetend is equivalent to a fraction, having the repetend for its numerator, and an equal number of 9's for its denominator. FRACTIONS. TO REDUCE INFINITE DECIMALS ΤΟ We have seen that 71. 342-71343 71271 = 999 • Therefore, .071342 (which is 1000 of 71.342),=99906. In this example, the numerator is equivalent to 71342-71; that is, to the whole decimal, minus its finite part. The denominator contains as many 9's as there are figures in the repetend, with as many O's as there are finite figures. |