number; 2376 to a whole number; 613 to 15ths; of 1 of 1% of to a simple fraction. A fraction may be reduced to a decimal, by performing the division which the fraction expresses, annexing deci. mal zeroes to the numerator. Thus, =3.0-5=.6. A decimal may be reduced to a fraction, by writing the decimal for a numerator, and the denomination tenth, hundredth, foc. for a denominator. Thus .06=ico; .193= 193 is 1000. A fraction may be reduced to its lowest terms, by dividing the numerator and denominator by their greatest common divisor. Thus 44=$, because 11:13=\, and is =). 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. 6. Reduce to a decimal, si ; S;J; ; 5; 5; ; 7. Reduce to their lowest terms, 75; 28 28. 256 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, ; 392) 2 500 10000 765 972 345. 3990 495) 5890* 810 : 4 63 25 10. Reduce to a decimal, ; ; ; ; 18; 28; 13; 1277; 1304 ; 3076. 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, ā, , ji and , to a common denominator, we first find that the least common multiple is 36. Then 26 36 30 36 36-9= 4 and Xf=;8 or as there are ji in 1, ý= and = 38 = and=1 = and 1= 36-18= 2 and 13 x i T'E=1 and 1=3 36+ 6 = 6 and š xg š=3% and =38 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, s, 5, 12, 1 and 1. 12. Reduce to a common denominator, q, , 14 13. Reduce to a common denominator, of of 5, s, 1 of and 3 of of 3. The compound fractions must first be reduced to simple 5 and 1 ones. 14. Reduce to a common denominator, of ģ, 15, šof di and į of . 15. Reduce 2916i to an improper fraction ; 947 to a mixed number; 28.7 to a deciinal; 1993 to its lowest terms; 51 and to a common denominator ; .004625 to a fraction; 295 to 19ths. 19 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 +&+ Ž= 49, and it is + i=18. Therefore, to add fractions, Reduce all the given fractions to a common denominator, and add their numerators. 1. Add 23, 1, 151, and 28. 3 * SUBTRACTION OF FRACTIONS. As 8 houses—33 houses=5 houses, so :- =, and -= . 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. EXAMPLE FOR THE BOARD. ? 16 16 If we wish to subtract 3 from 16ặ, we must, as 33 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 and a 4. Subtract 49,13 from 56213. 6. What is the difference between 414 and 23 ? 7. What is the difference between į of 19 and į of ? 8. Subtract of 164 from g of 1 of 953. MULTIPLICATION OF FRACTIONS. If any number be multiplied by :, it is evident that the product will be as large as if multiplied by 1. Then 4x11=of 1 ;=off. 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 =2x; } =3x ; f=4*, &c. Hence, we may multiply a fraction by a whole number, either by multiplying the nume 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 14 by 33; 27 by 53. DIVISION OF FRACTIONS. The smaller the divisor, the more times will it be con. tained in the dividend, and the larger will be the quotient. Thus, if be divided by 1, the quotient will be . I will be contained 4 times as often as 1, therefore 3:1=1x will be contained only j as often as 4, therefore = = of 4 x ž=1x ž. 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 frac 3 tions by this rule. Thus, 1-35 -=X=414. TO DIVIDE A FRACTION BY A WHOLE NUMBER.— It is evident that £=*=2; þ=}=3; 1=1;4, &c. There. fore, 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 by ; by 3 ; by 36. 2. Divide 21} by 5; by 14; by 8%. ure. CIRCULATING OR INFINITE DECIMALS. 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, į=.333+; z=.22222 +; i=.090909+; &l=.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 fig. Thus.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, 5.5, &c. n=.oi, therefore, v=.02, 1:=.14, 95=.95, &c. ooo=.001, therefore, gio=.002, .045, 1635.163, &c. Hence, every repetend is equivalent to a fraction, having the repetend for its numerator, and an equal number of 9's for its denominator. TO REDUCE We have seen that 71. 342=71343=1371. Therefore, .071342 (which is doo of 71.342),=50300 In this example, the numerator is equivalent to 71342—71 ; that is, to the whole decimal, minus its finite part. The de. nominator contains as many 9's as there are figures in the repetend, with as many O's as there are finite figures. INFINITE DECIMALS ΓΟ FRACTIONS. |