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Fractions arithmetic

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2 Fractions arithmetic in symbolic form is shown below:

a/b + c/d = (a*d + b*c) / (b*d)
a/b - c/d = (a*d - b*c) / (b*d)
a/b * c/d = (a*b) / (b*d)
a/b / c/d = (a*d) / (b*c)

This computational technique is rather straightforward, but it does not guarantee the result to be in the lowest terms. Thus, it may require additional step aimed at reducing the result fraction to lowest terms by prime factoring of numerator and denominator, finding and applying the greatest common factor (GCF).

3 Fractions calculation, in particular addition and subtraction can be performed in two different ways:

Method A: two-step procedure based on the associative property of arithmetic operations, i.e., applying the corresponding arithmetic operator to the first and second fractions, then applying the second arithmetic operator to to the result and the third fraction. Associative property of 3 fractions arithmetic as it applies to the addition and subtraction as shown below (note: in the absence of parenthesis the standard order of mathematical calculation is from left to right).

a/b ± c/d ± e/f = (e/f ± c/d) ± e/f

Corresponding numeric examples:

1/2 + 2/9 - 1/6 = (1/2 + 2/9) - 1/6 = 13/18 -1/6

13/18 -1/6 = 5/9

Method B: a single step computation formula

a/b ± c/d ± e/f = (a*d*f ±c*b*f ± e*b*d) / (b*d*f)

This computational technique is rather straightforward and presented in a compact form. It's relevant to mention, that neither methods guarantee the result to be in the lowest terms, thus, additional step is needed to reduce the result fraction to the lowest terms. In general, fraction calculation can be a rather labor-intensive and error-prone operation, especially in case of 3 fraction arithmetic that involves both regular fractions and mixed number. Such calculations can be easily performed using popular free online 3 fractions calculator available free of charge at [1] in several rather simple steps:

Step 1. Open the web page [1] (see sample screenshot attached to the article)

Step 2. Select computation mode, either 2 or 3 fractions

Step 3. Enter the fractions and/or mixed numbers in the text boxes and select the arithmetic operators using drop-downs

Step 4. Click on 'Calculate' button to get the result displayed as fraction and equivalent decimal number (rounded upon necessity)


  1. Online Fraction Calculator iFC-3
  2. Mobile Online Fraction Calculator


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