CLASS-7
DISTRIBUTIVE PROPERTY OF DIVISION OVER ADDITION & SUBTRACTION OF RATIONAL NUMBER

DISTRIBUTIVE PROPERTY OF DIVISION OVER ADDITION & SUBTRACTION OF RATIONAL NUMBER -

The distributive property typically applies to multiplication over addition or subtraction. However, for division, the concept is different and does not hold in the same way. Division is not distributive over addition or subtraction for rational numbers. To clarify, let’s consider some expressions to understand why this is the case.

Division over AdditionWhy Division is not DistributiveSubtraction Example\[ \frac{a - b}{c} = \frac{7 - 2}{5} = \frac{5}{5} = 1 \] \[ \frac{a}{c} - \frac{b}{c} = \frac{7}{5} - \frac{2}{5} = \frac{5}{5} = 1 \] In summary, division does not distribute over addition or subtraction of rational numbers in a consistent way. The behavior depends on the specific values involved, so it's crucial to apply each operation individually.

For any rational numbers aaa, bbb, and ccc (with c≠0c \neq 0c=0):

a+bc≠ac+bc\frac{a + b}{c} \neq \frac{a}{c} + \frac{b}{c}ca+b​=ca​+cb​

To see why, consider an example:

  1. Let a=6a = 6a=6, b=3b = 3b=3, and c=3c = 3c=3: a+bc=6+33=93=3\frac{a + b}{c} = \frac{6 + 3}{3} = \frac{9}{3} = 3ca+b​=36+3​=39​=3 ac+bc=63+33=2+1=3\frac{a}{c} + \frac{b}{c} = \frac{6}{3} + \frac{3}{3} = 2 + 1 = 3ca​+cb​=36​+33​=2+1=3

In this specific case, the result appears the same, but this is not a general rule.

Consider another set of numbers:

  1. Let a=8a = 8a=8, b=4b = 4b=4, and c=2c = 2c=2: a+bc=8+42=122=6\frac{a + b}{c} = \frac{8 + 4}{2} = \frac{12}{2} = 6ca+b​=28+4​=212​=6 ac+bc=82+42=4+2=6\frac{a}{c} + \frac{b}{c} = \frac{8}{2} + \frac{4}{2} = 4 + 2 = 6ca​+cb​=28​+24​=4+2=6

Again, the result is the same, but let's consider non-integer results:

  1. Let a=1a = 1a=1, b=1b = 1b=1, and c=2c = 2c=2: a+bc=1+12=22=1\frac{a + b}{c} = \frac{1 + 1}{2} = \frac{2}{2} = 1ca+b​=21+1​=22​=1 ac+bc=12+12=12+12=1\frac{a}{c} + \frac{b}{c} = \frac{1}{2} + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1ca​+cb​=21​+21​=21​+21​=1

It seems to work for simple rational numbers.

However, the general property for subtraction is: a−bc=ac−bc\frac{a - b}{c} = \frac{a}{c} - \frac{b}{c}ca−b​=ca​−cb​

To see why, consider an example:

  1. Let a=6a = 6a=6, b=3b = 3b=3, and c=3c = 3c=3: a−bc=6−33=33=1\frac{a - b}{c} = \frac{6 - 3}{3} = \frac{3}{3} = 1ca−b​=36−3​=33​=1 ac−bc=63−33=2−1=1\frac{a}{c} - \frac{b}{c} = \frac{6}{3} - \frac{3}{3} = 2 - 1 = 1ca​−cb​=36​−33​=2−1=1

Consider another set of numbers:

  1. Let a=8a = 8a=8, b=4b = 4b=4, and ( c = 2 ]: a−bc=8−42=42=2\frac{a - b}{c} = \frac{8 - 4}{2} = \frac{4}{2} = 2ca−b​=28−4​=24​=2 ac−bc=82−42=4−2=2\frac{a}{c} - \frac{b}{c} = \frac{8}{2} - \frac{4}{2} = 4 - 2 = 2ca​−cb​=28​−24​=4−2=2

Again, the result is the same.

For general division, the operation changes the way numbers combine. Here’s a counter-example for addition:

Let a=5a = 5a=5, b=10b = 10b=10, and c=3c = 3c=3: a+bc=5+103=153=5\frac{a + b}{c} = \frac{5 + 10}{3} = \frac{15}{3} = 5ca+b​=35+10​=315​=5 ac+bc=53+103=153=5\frac{a}{c} + \frac{b}{c} = \frac{5}{3} + \frac{10}{3} = \frac{15}{3} = 5ca​+cb​=35​+310​=315​=5

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