Properties of Addition of Fractions

The associative and commutative properties of natural numbers hold good in the case of fractions also.


I: Commutative Property of Addition of Fractions: 

While adding two fractions we may add them in either order. The sum of the fractions remains the same.

Add:

\(\frac{2}{3}\) + \(\frac{1}{4}\)

L.C.M. of 3, 4 is 12

\(\frac{2}{3}\) + \(\frac{1}{4}\)

= \(\frac{8}{12}\) + \(\frac{3}{12}\)

= \(\frac{11}{12}\)

Add:

\(\frac{1}{4}\) + \(\frac{2}{3}\)

L.C.M. of 4, 3 is 12

\(\frac{1}{4}\) + \(\frac{2}{3}\)

= \(\frac{3}{12}\) + \(\frac{8}{12}\)

= \(\frac{11}{12}\)

                       i.e., \(\frac{2}{3}\) + \(\frac{1}{4}\) = \(\frac{1}{4}\) + \(\frac{2}{3}\)


II: Associative Property of Addition of Fractions:

While adding more than two fractions, we may add them in any order; the sum of the fractions remains the same.

Add:

[\(\frac{2}{5}\) + \(\frac{3}{4}\)] + \(\frac{1}{3}\)    

L.C.M. of 5 and 4 is 20.

[\(\frac{2}{5}\) + \(\frac{3}{4}\)] + \(\frac{1}{3}\) = [\(\frac{8}{20}\) + \(\frac{15}{20}\)] + \(\frac{1}{3}\)

[\(\frac{2}{5}\) + \(\frac{3}{4}\)] + \(\frac{1}{3}\) = \(\frac{23}{20}\) + \(\frac{1}{3}\)

L.C.M. of 20 and 3 is 60.

= \(\frac{69}{60}\) + \(\frac{20}{60}\) = \(\frac{89}{60}\) = 1\(\frac{29}{60}\)

Add:

\(\frac{2}{5}\) + [\(\frac{3}{4}\) + \(\frac{1}{3}\)]

L.C.M. of 4 and 3 is 12.

= \(\frac{2}{5}\) + [\(\frac{9}{12}\) + \(\frac{4}{12}\)]

= \(\frac{2}{5}\) + \(\frac{13}{12}\)

L.C.M. of 5 and 12 is 60. 

= \(\frac{24}{60}\) + \(\frac{65}{60}\)

\(\frac{89}{60}\) = 1\(\frac{29}{60}\)

              i.e., [\(\frac{2}{5}\) + \(\frac{3}{4}\)] + \(\frac{1}{3}\) = \(\frac{2}{5}\) + [\(\frac{3}{4}\) + \(\frac{1}{3}\)]


III: Zero Property of Addition of Fractions:

If zero is added to any fraction we get back the same fraction.

Add:

\(\frac{1}{2}\) + 0 = \(\frac{1}{2}\) + \(\frac{0}{2}\) = \(\frac{1 + 0}{2}\) = \(\frac{1}{2}\)

Therefore, \(\frac{1}{2}\) + 0 = 0 + \(\frac{1}{2}\) = \(\frac{1}{2}\)

Add:

\(\frac{5}{6}\) + 0 = \(\frac{5}{6}\) + \(\frac{0}{6}\) = \(\frac{5 + 0}{6}\) = \(\frac{5}{6}\)

Therefore, \(\frac{5}{6}\) + 0 = 0 + \(\frac{5}{6}\) = \(\frac{5}{6}\)

Properties of Addition of Fractions


Questions and Answers on Properties of Addition of Fractions:

I. Fill in the Blanks:

(i) \(\frac{3}{5}\) + \(\frac{1}{4}\) = \(\frac{1}{4}\) + ______

(ii) \(\frac{1}{6}\) + ______ = \(\frac{1}{8}\) + ______

(iii) \(\frac{7}{15}\) + \(\frac{3}{5}\) + \(\frac{2}{9}\) = \(\frac{2}{9}\) + ______ + \(\frac{3}{5}\)

(iv) \(\frac{9}{20}\) + \(\frac{4}{15}\) + ______ = \(\frac{3}{5}\) + \(\frac{9}{20}\) + \(\frac{4}{15}\)


Answer:

I. (i) \(\frac{3}{5}\)

(ii) \(\frac{1}{8}\); \(\frac{1}{6}\)

(iii) \(\frac{7}{15}\)

(iv) \(\frac{3}{5}\)



II. Verify the following (show that left hand side = right hand side)

(i) \(\frac{3}{5}\) + \(\frac{2}{8}\) = \(\frac{2}{8}\) + \(\frac{3}{5}\)

(ii) [\(\frac{1}{6}\) + \(\frac{2}{3}\)] + \(\frac{1}{4}\) = \(\frac{1}{6}\) + [\(\frac{2}{3}\) + \(\frac{1}{4}\)]



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