uNEGATIVE FRACTIONSA fraction preceded by a minus sign is

NEGATIVE FRACTIONSA fraction preceded by a minus sign is nega-tive. Any negative fraction is equivalent to apositive fraction multiplied by -l. For example,

The number.T is read"minus two-fiftha "We know that the quotient of two numbewith unlike signs is negative. Therefore

This indicates that a negative fraction is equivalent to a fraction with either a negative numerator or a negative denominatorThe fractionis read two over minusfIve. The fractionis read "minus twoover five 'A minus sign in a fraction can be moveabout at will. It can be placed before thmerator. before the denominator or before thefraction itself, Thus

Moving the minus sigm from numerator todenominator, or vice versa, is equivalent tomultiplying the terms of the fraction by-1This ia shown in the following examples:

A fraction may be regarded as having threeBigma associated with it-the sign of the numerator, the aign of the denominator, and the signpreceding the fraction. Any two of these signsmay be changed without changing the value ofthe fraction. Thus

OPERATIONS WITH FRACTIONSIt will be recalled from the discussion ofdenominate numbers that numbers must be ofthe same denomination to be added. We can addpounds to pounds, pints to pints, but not ouncesto pints. If we think of fractions loosely as de-nominate number, it will be seen that the ruleof likeness applies also to fractions, We canadd eighths to eighths, fourths to fourths, butnot eighths to fourths, To adds inch to T inchwe simply add the numerators and retain thedenominator unchanged. The denomination isfifths: as with denominate numbers, we add 1fifth to 2 fifths to get 3 fifths, orLIKE AND UNLIKE FRACTIONSWe have shown that like fractiona are addedby simply adding the numerators and keeping thedenominator. Thus

Similarly we can subtract Ilke fractions bysubtracting the numerators

The following examples will show that likefractions may be divideddividing the nu-merator of the dividendthe numeratorthe divisor

SOLUTION: We may state the problem asquestion:"How many times does appear in aor how many times may f be taken from 2?

We see thatcan be subtracted from 3/8three times. Therefore3/8+1/8·3

When the denominators of fractions are unual. the fractions are said to be unlikeition, subtraction, or division cannot be pcanB0that theybecome like fractions: then all the rules forlike fractions applyLOWEST COMMON DENOMINATORo change unlike fractions to like fractionsit is necessary to find a COMMON DENOMINALOWEST COMMON DENOMINATOR (LC DIThis is nothing more than the least commonmultiple af the denominatoLeast Common MultipleH a number is a mullH出E,mhH,2116 and 2. There are manythege numbers. The numbeme a few, are also ctiples af 6The smallest of the common multiples of acalled the LEl[ULTIPLE. It is abbreviated Lcommon multiple of 6 and 2 is 6. To find thleast common multiple of a set of numbersfirst separate each of the numbers into primSuppose that we wish to find the LCM of 14,prime factors we have142·35The LCM will contain each of the various primefactors shown. Each prime factor isne of the numbers. Notice that 3, 5, and 7 eaoccur only once in any one number. On theoccurs three times in one number

Thus, 840 is the least common multiple of 14,24,and30

Greatest Common DivisorThe largest number that can be divided intoeach of two or more given numbers without aremainder is called the GREATEST COMMONDIVISOR of the given numbers. It is abbreviatedGCD. It is also sometimes called the HIGHESTCOMMON F ACTORIn finding the GCD of a set of numbers, sparate the numbers into prime factors just asfor LCM. The GCD is the product of only thosefactors that appear in all of the numbers. Noticein the example af the previous section that 2 isthe greatest common divisor of 14, 24, and 30Find the GCD of 650, 900, and 700. The pro

Notice that 2 and 5 are factora of each num-ber, The greatest common divisor is 2 x 25- 50USING THE LCDConsider the example

The numbers 2 and 3 are both prime; so theLCD is 6

Practice problems, Change the fractions ineach of the following groups to like fractionswith least common denominators

It has been shown that in adding like frac-tions we add the numerators, In adding unlikefractions, the fractions must first be changed sothat they have common denominators, We applythese same rules in adding mixed numbers. Itwill be remembered that a mixed number is anIndicated sum. Thus, 2 is really 2+3. Add-ing can be done in any order. The followingexamples will show the application of theserules:

We first change the fractions so that they arelike and have the least common denominatorand then proceed as before.

Practice problems. Add, and reduce thesums to simplest terms:

The following example demonstrates a prtical application of addition of fractionsEXAMPLE: FInd the total length of the pleceof metal shown in figure 4-5 (A)SOLUTION: First indicate the sum as follows:

Changing to llke fractions and adding numerators,

The total length is 3k inchesPractice problem.the distance fromthe center of the firstto the center of thelast hole in the metal plate shown in figure4-5(B)neheRSUBTRACTIONThe rule of likeness appliesthe subtraction of fractions as well a如additionSome examples will show that cases lkely toarise may be solved by use of ideas previouslydeveloped