Fraction Rules
Operation | Equations | Examples | Description |
Adding two fractions[2] | ![(a/b)+(c/d)=(ad+bc)/(bd)](../../equations/f/fractionruleseqn01.png) | ![(1/2)+(2/3)=(1*3+2*2)/(2*3)=(3+4)/6=7/6](../../equations/f/fractionruleseqn02.png) | To 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)](../../equations/f/fractionruleseqn03.png) | ![(2/3)-(3/4)=(2*4-3*3)/(3*4)=(8-9)/12=(-1)/12=-1/12](../../equations/f/fractionruleseqn04.png) | To 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)](../../equations/f/fractionruleseqn05.png) | ![(5/12)*4=(5*4)/12=(5*4)/(3*4)=5/4](../../equations/f/fractionruleseqn06.png) | To 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](../../equations/f/fractionruleseqn35.png) | ![(1/2)*(2/3)=(1*2)/(2*3)=1/3](../../equations/f/fractionruleseqn36.png) | To 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)](../../equations/f/fractionruleseqn07.png) | ![(2/3)*(3/4)=(2/3)*(4/3)=(2*4)/(3*3)=8/9](../../equations/f/fractionruleseqn08.png) | To 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)](../../equations/f/fractionruleseqn39.png) | ![(14/3)/7=(14/3)/(7/1)=(14/3)*(1/7)=(14*1)/(3*7)=(2*7)/(3*7)=2/3](../../equations/f/fractionruleseqn38.png) | To 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)](../../equations/f/fractionruleseqn37.png) |
Raising a fraction to a power. | ![(a/b)^m=(a^m)/(b^m)](../../equations/f/fractionruleseqn09.png) | ![(2/3)^3=(2^3/3^3)=8/27](../../equations/f/fractionruleseqn10.png) | See Operations on Fractions: Exponentiation. |
Converting a mixed number to an improper fraction. | ![](../../equations/f/fractionruleseqn11.png) | ![2+(3/8)=(2*8+3)/8=(16+3)/8=19/8](../../equations/f/fractionruleseqn12.png) | To 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)](../../equations/f/fractionruleseqn13.png) | ![2+(3/8)=(2*8+3)/8=(16+3)/8=19/8](../../equations/f/fractionruleseqn14.png) | To 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!=0](../../equations/f/fractionruleseqn16.png) | ![0/5=0](../../equations/f/fractionruleseqn17.png) | Applying 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=undefined](../../equations/f/fractionruleseqn18.png) | ![8/0=undefined](../../equations/f/fractionruleseqn19.png) | Since division by zero is undefined, a zero denominator makes the fraction undefined. |
One minus sign. | ![(-a)/b=-(a/b), a/(-b)=-(a/b)](../../equations/f/fractionruleseqn20.png) | ![(-3)/2=-(3/2), 3/(-2)=-(3/2)](../../equations/f/fractionruleseqn21.png) | Since , apply the associative property of multiplication to get ![(-a)/b=(-1)*a/b=(-1)*a/b=-(a/b)](../../equations/f/fractionruleseqn23.png) |
Two minus signs. | ![(-a)/(-b)=a/b](../../equations/f/fractionruleseqn24.png) | ![(-3)/(-5)=3/5](../../equations/f/fractionruleseqn25.png) | Since , apply the associative property of multiplication to get ![(-a)/(-b)=((-1)*a)/((-1)*b)=(-1)/(-1)*a/b=1*a/b=a/b](../../equations/f/fractionruleseqn26.png) |
If a fraction has the same nonzero numerator and denominator, the fraction's value is 1. | ![a/a=1, a!=0](../../equations/f/fractionruleseqn27.png) | ![(-3)/(-2)=1](../../equations/f/fractionruleseqn28.png) | Anything except 0 divided by itself is 1. |
Any integer can be made into a fraction. | ![a=a/1](../../equations/f/fractionruleseqn29.png) | ![5=5/1](../../equations/f/fractionruleseqn30.png) | Since , apply the property of multiplying by 1: . See Property of Multiplying by 1. |
Reducing fractions. | ![c*d=a,c*e=b,a/b=(c*d)/(c*e)=d/e](../../equations/f/fractionruleseqn33.png) | ![a/b=(c*d)/(c*e)=d/e](../../equations/f/fractionruleseqn34.png) | Given two arbitrary values a and b, and values c, d, and e such that a = c · d and b = c · 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/d](../../equations/f/fractionruleseqn40.png) | ![5/4=?/12. 4*?=12. 4*3=12. (5/4)*(3/3)=(5*3)/(4*3)=15/12](../../equations/f/fractionruleseqn41.png) | Given 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+3/6](../../equations/f/fractionruleseqn43.png) | To 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/1000](../../equations/f/fractionruleseqn44.png) | ![42.895=42895/1000=8579/200](../../equations/f/fractionruleseqn45.png) | To 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/100](../../equations/f/fractionruleseqn46.png) | ![32%=32/100=(8*4)/(25*4)=8/25](../../equations/f/fractionruleseqn47.png) | To 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>c](../../equations/f/fractionruleseqn48.png) | ![23/7<=>?27/8->(23/7)<27/7](../../equations/f/fractionruleseqn49.png) | To 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)](../../equations/f/fractionruleseqn50.png) | ![37/7<=>?24/5->(37/7)*(5/5)<=>?(24/5)*(7/7)->185/35<=>?168/35->185/35>168/35](../../equations/f/fractionruleseqn51.png) | To 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](../../equations/f/fractionruleseqn52.png) |
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Table 1 |
4/21/2019: Modified equations and expression to match the new format. (
12/21/2018: Reviewed and corrected IPA pronunication. (
8/28/2018: Corrected spelling. (
7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
2/5/2010: Added "References". (
1/14/2009: Initial version. (