Fraction Rules

Pronunciation: /ˈfræk.ʃən rulz/ Explain

numerator/denominator is defined as numerator divided by denominator.
Figure 1: Fraction

Fraction rules are a set of algebraic rules for working with fractions. A fraction has a numerator and a denominator. A fraction represents a division operation. The numerator is the dividend. The denominator is the divisor.

Fraction Rules
OperationEquationsExamplesDescription
Adding two fractions[2](a/b)+(c/d)=(ad+bc)/(bd)(1/2)+(2/3)=(1*3+2*2)/(2*3)=(3+4)/6=7/6To add fractions, transform each fraction so they have a common denominator. Add the numerators and use the common denominator as the denominator. Reduce the fraction. See Operations on Fractions: Addition and Subtraction.
Subtracting two fractions(a/b)-(c/d)=(ad-bc)/(bd)(2/3)-(3/4)=(2*4-3*3)/(3*4)=(8-9)/12=(-1)/12=-1/12To subtract fractions, transform each fraction so they have a common denominator. Subtract the numerators and use the common denominator as the denominator. Reduce the fraction. See Operations on Fractions: Addition and Subtraction.
Multiplying two fractions[2](a/b)*(c/d)=(ac)/(bd)(5/12)*4=(5*4)/12=(5*4)/(3*4)=5/4To multiply fractions, multiply the numerators and multiply the denominators. Reduce the fraction. See Operations on Fractions: Multiplication.
Multiplying a fraction and a whole number.(a/b)*C=(aC)/b(1/2)*(2/3)=(1*2)/(2*3)=1/3To multiply a fraction and a whole number, multiply the numerator by the whole number. The denominator remains unchanged. Reduce the fraction if possible.
Dividing two fractions[2](a/b)/(c/d)=(ad)/(bc)(2/3)*(3/4)=(2/3)*(4/3)=(2*4)/(3*3)=8/9To divide fractions, flip the divisor upside down then multiply by the dividend. Reduce the fraction. See Operations on Fractions: Division.
Dividing a fraction by an integer.(a/b)/C=a/(b*C)(14/3)/7=(14/3)/(7/1)=(14/3)*(1/7)=(14*1)/(3*7)=(2*7)/(3*7)=2/3To divide a fraction by a whole number, convert the whole number to a fraction, the divide the fractions.
(a/b)/C=(a/b)/(C/1)=(a/b)*(1/C)=a/(b*C)
Raising a fraction to a power.(a/b)^m=(a^m)/(b^m)(2/3)^3=(2^3/3^3)=8/27See Operations on Fractions: Exponentiation.
Converting a mixed number to an improper fraction.2+(3/8)=(2*8+3)/8=(16+3)/8=19/8To convert a mixed number to an improper fraction, multiply the whole part by the denominator and add the product to the numerator. The denominator remains unchanged. See How to Convert a Mixed Number to a Fraction.
Converting an improper fraction to a mixed number.21/5, 21=4*5+1, 21/5=4+(1/5)2+(3/8)=(2*8+3)/8=(16+3)/8=19/8To convert an improper fraction to a mixed number, divide the numerator by the denominator using a remainder. The mixed number is the quotient plus the remainder divided by the denominator. See How to Convert a Fraction to a Mixed Number..
Zero numerator.0/a=0, a!=00/5=0Applying the property of multiplying by zero, a zero numerator with a zero denominator is zero. See Property of Multiplying by 0.
Zero denominator.a/0=undefined8/0=undefinedSince division by zero is undefined, a zero denominator makes the fraction undefined.
One minus sign.(-a)/b=-(a/b), a/(-b)=-(a/b)(-3)/2=-(3/2), 3/(-2)=-(3/2)Since -a=(-1)*a, apply the associative property of multiplication to get (-a)/b=(-1)*a/b=(-1)*a/b=-(a/b)
Two minus signs.(-a)/(-b)=a/b(-3)/(-5)=3/5Since -a=(-1)*a, apply the associative property of multiplication to get (-a)/(-b)=((-1)*a)/((-1)*b)=(-1)/(-1)*a/b=1*a/b=a/b
If a fraction has the same nonzero numerator and denominator, the fraction's value is 1.a/a=1, a!=0(-3)/(-2)=1Anything except 0 divided by itself is 1.
Any integer can be made into a fraction.a=a/15=5/1Since 1/1=1, apply the property of multiplying by 1: a=a*1=a*(1/1)=(a*1)/1=a/1. See Property of Multiplying by 1.
Reducing fractions.c*d=a,c*e=b,a/b=(c*d)/(c*e)=d/ea/b=(c*d)/(c*e)=d/eGiven two arbitrary values a and b, and values c, d, and e such that a = c · d and b = c · e, a/b=(c*d)/(c*e)=d/e. See Reducing Fractions.
Building fractions.\array{Given a/b, and d; find c such that a/b=c/d. There exists e such that b*e = d. Then (a/b)*(e/e)=(a*e)/(b*e) = c/d5/4=?/12. 4*?=12. 4*3=12. (5/4)*(3/3)=(5*3)/(4*3)=15/12Given a fraction a / b and a number d that is a multiple of d, find e such that b · e = d, then a / b = (a · e) / (b · e).
Operations on complex fractions.Simplify the complex fractions, then use the rules for simple fractions.4/(3/5)+1/2=4*(5/3)+1/2=20/3+1/2=40/6+3/6To manipulate a complex fraction, convert it to a simple fraction, then follow the rules for simple fractions. See Complex Fraction.
Converting a decimal number to a fraction.A.bcd=Abcd/100042.895=42895/1000=8579/200To convert a decimal to a fraction, change the decimal to a whole number and divide it by 10n where n is the number of digits after the decimal point.
Converting a percentage to a fraction.a%=a/10032%=32/100=(8*4)/(25*4)=8/25To convert a percentage to a fraction, use the percentage as the numerator, 100 as the denominator, then simplify.
Comparing fractions with like denominators.(a/b)<(c/b), b>0 iff a<c; (a/b)=(c/b), b>0 iff a=c; (a/b)>(c/b), b>0 iff a>c23/7<=>?27/8->(23/7)<27/7To compare fractions with like denominators, compare the numerators. The relationship between the fractions is the same as the relationship between the denominators.
Comparing fractions with unlike denominators.(a/b)=?(c/d)37/7<=>?24/5->(37/7)*(5/5)<=>?(24/5)*(7/7)->185/35<=>?168/35->185/35>168/35To compare fractions with unlike denominators, either convert them to a decimal or transform them to a common denominator, then compare them.
37/7<=>?24/5->5.28571<=>?4.8->5.28571<4.8
Table 1

References

  1. McAdams, David E.. All Math Words Dictionary, fraction rules. 2nd Classroom edition 20150108-4799968. pg 82. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pp 12-15. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 7/11/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n21/mode/1up/search/fraction. Buy the book
  3. Oberg, Erik. Arithmetic Simplified. pp 21-31. www.archive.org. Industrial Press. 1914. Last Accessed 7/11/2018. http://www.archive.org/stream/arithmeticsimpli00oberrich#page/21/mode/1up/search/fraction. Buy the book
  4. Oberg, Erik. Elementary Algebra. pg 23. www.archive.org. Industrial Press. 1914. Last Accessed 7/11/2018. http://www.archive.org/stream/elementaryalgebr00oberrich#page/n26/mode/1up/search/fraction. Buy the book
  5. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 9-11,36-40. www.archive.org. International Textbook Company. January 1963. Last Accessed 7/11/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book

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Cite this article as:

 McAdams, David E. Fraction Rules. 4/21/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/fractionrules.html.

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Revision History

4/21/2019: Modified equations and expression to match the new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/28/2018: Corrected spelling. (McAdams, David E.)
7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/5/2010: Added "References". (McAdams, David E.)
1/14/2009: Initial version. (McAdams, David E.)

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