To Reduce Fractions to their Least Common Denominator. First. Find the L.C.M. of all the Denominators. This is the L.C.D. Second. Divide this L.C.D. by each Denominator, and Multiply its Numerator by Quotient. This gives the New Numerators. Third. Write the L.C.D. under these. 5 EXAMPLE WORKED OUT. Reduce to their L.C.D. 3, 8, = L.C.M. of 9', 6', 4′, 3', 12, 1, 3, 12 and 13. 18', = 36.. 369. = L.C.D. 4.; = 3.; 12 ៖ This is, in fact, Multiplying Numerator and Denominator of each Fraction by the same Number; namely, by that Factor of L.C.D. which corresponds with Denominator of that Fraction. A Decimal Fraction has for its Denominator some Power of 10.. Certain Vulgar Fractions are equal to Decimal Fractions. = = 125, &c. Thus, %, 20%, Now, seeing that the Denominator of the Decimal must be a Multiple of the Denominator of the Vulgar Fraction, it is evident that, if a Vulgar Fraction be equal to a Decimal, its Denominator will measure some Power of 10.. And, since the only prime Factors of any Power of 10∙ are twos and fives, and so on, it is plain that a Vulgar Fraction is not equal to a Decimal, unless that Vulgar Fraction, when in its lowest terms, has a Denominator composed of the prime Factors 2 and 5 ̊, or of their Powers. Hence, any Vulgar Fraction is equal to a Decimal, if its Denominator be any of the following; namely, 2′, 4′, 5′, 8′, 16′, 20′, 25′, 32•, 40, 50, 64, &c. &c. And no Vulgar Fraction is equal to a Decimal, if, when in its lowest terms, its Denominator be any of the following; namely, 3', 6., 7., 9., 11', 12′, 13′, 14°, 15°, 17., 18', 19′, &c. &c. Nevertheless, each of the following is equal to a Decimal: &,, 131⁄2, 15, 15, 13, 12, 12, 12, 1%, because, when reduced to their lowest terms, these Fractions become 1, 1, 1, 2, 1, 1, 1, 1, 1, 1. It is most useful to be able to decide, at a glance, whether a Vulgar Fraction has, or has not, any equivalent Decimal. EXERCISE LXXI. Select from the following Vulgar Fractions those which have equivalent Decimals. 3′ ÷ 8 ̊, Prin. XV., and 3' ÷ 8· ⚫375 = .375. = And we say, that the Vulgar Fraction is here Reduced to its equivalent Decimal, which process consists merely in expressing the Quotient of Numerator and Denominator as a Decimal instead of as a Vulgar Fraction. (See last twenty-seven Examples of Exercise LVIII., page 111.) To Reduce a Vulgar Fraction to its Equivalent Decimal. Divide Numerator by Denominator, and carry out Quotient-figures in Decimal Places. EXERCISE LXXII. Reduce to Equivalent Decimals, 1. 4, 12, 4, 7, 74, 89 29 6. 6 48, & 452 7. 1431300', & 11÷÷÷ 26,3% INTERMINATE OR ENDLESS DECIMALS. In attempting to Reduce to an Equivalent Decimal a Vulgar Fraction which has no such equivalent, we obtain an Interminate Decimal; that is, one in which the Division can never be completed, for one or more of the Quotientfigures constantly recur. In a Repeating Decimal a single Digit is constantly repeated. Thus, = ∙1111 &c., a Repeating Decimal. In a Circulating Decimal a set of two or more Digits is constantly repeated. Thus, =142857 142857 &c., a Circulating Decimal. The repeating figure or figures constitute a Period, Repetend, or Circle. A Repetend is distinguished by a dot over its first, and another over its last figure. A Pure Repetend commences at the Decimal Point. A Mixed Decimal is partly terminate and partly interminate. = Thus, 17 125142857, a Mixed Decimal, for the last six REDUCTION OF INTERMINATE DECIMALS TO THEIR EQUIVALENT VULGAR FRACTIONS. A Repetend of two Figures has for its Denominator 99. And, similarly, any Pure Repetend is the Numerator of a Vulgar Fraction whose Denominator is as many nines as there are Places of Figures in the Repetend. If there be one or more Significant Figures between the Decimal Point and the Repetend, the Denominator will be the same as if these were ciphers, as in above examples. For 81 = 73 468 Therefore, a Repetend separated from the Decimal Point by any number of Figures must have the same number of ciphers annexed to the nines in its Denominator. III. NUMERATOR OF A MIXED DECIMAL. A Mixed Decimal is equal to two Fractions. First, a Terminate Decimal, which has a Power of 10 for its Denominator; and, second, an Interminate Decimal which has its Denominator expressed by nines, or by nines and ciphers. The sum of these two is the Vulgar Fraction equivalent to the whole Mixed Decimal. Their Common Denominator may always be the Denominator of the Repetend Decimal. And to Reduce the first Fraction (the Terminate Decimal) to this Common Denominator, its Numerator and Denominator must merely be multiplied by the nines in Denominator of second Fraction (the Interminate Decimal). .. If Repetend be a Single Digit, the Multiplier will be 9. If Repetend contain two Digits, the Multiplier will be 99. and so on. Now, Subtracting any Number from its 10th Multiple gives its 9th Multiple, And, Subtracting any Number from its 100th Multiple gives its 99th Multiple. Which Examples shew, that the Numerator of the Vulgar Fraction, equivalent to a Mixed Decimal, is found by subtracting the Terminate part from the whole Mixed Decimal. To Reduce any Interminate Decimal to its Equivalent Vulgar Fraction. First. Take for Numerator all the Figures from Decimal Point to the end of the First Period, Subtracting the Number expressed by the non-repeaters, if any. Second. Write for Denominator as many nines as there are Repeating Figures, followed by as many ciphers as there are non-repeaters. Third. Reduce Vulgar Fraction to its lowest terms. EXERCISE LXXIII. Reduce to Equivalent Vulgar Fractions, 13, 45, 613, 4578; 6. 84316, 085403; 2. ·064, 00513, 00014; 3. 018, 06, 0640; 4. 00043, 0007649, 005; 5. ·7432, ·3412, ·05189; 7. ·26428571, ·75983258; 8. 0001684, 09005; 9. 17, 17, 017, 017; 10 017, 0017, 0017, ·0017. ADDITION, MULTIPLICATION, SUBTRACTION, AND First. Reduce the Interminate Decimals to their equivalent Vulgar Fractions. Second. Add, Subtract, Multiply, or Divide, as may be required, and Reduce the Result to its Equivalent Decimal. |