# 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 + $$\frac{5}{6}$$ = $$\frac{5}{6}$$

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

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