# Show that in the Denominator Addition/Subtraction Property of proportions: If #a/ b = c/ d#, then #( a + b)/ b = ( c + d)/ d# or #( a − b)/ b = ( c − d)/ d#?

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Show that in the Denominator Addition/Subtraction Property of proportions: If #a/ b = c/ d# , then #( a + b)/ b = ( c + d)/ d# or #( a − b)/ b = ( c − d)/ d# ?

Show that in the Denominator Addition/Subtraction Property of proportions: If

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To show the Denominator Addition/Subtraction Property of proportions, we start with the given proportion:

[ \frac{a}{b} = \frac{c}{d} ]

We want to prove that:

[ \frac{a + b}{b} = \frac{c + d}{d} ]

First, let's consider ( \frac{a + b}{b} ):

[ \frac{a + b}{b} = \frac{a}{b} + \frac{b}{b} ]

[ = \frac{a}{b} + 1 ]

Now, let's substitute ( \frac{a}{b} = \frac{c}{d} ) into this expression:

[ \frac{a + b}{b} = \frac{c}{d} + 1 ]

[ = \frac{c}{d} + \frac{d}{d} ]

[ = \frac{c + d}{d} ]

Thus, we have shown that ( \frac{a + b}{b} = \frac{c + d}{d} ).

Similarly, we can prove that ( \frac{a - b}{b} = \frac{c - d}{d} ) using the same logic:

[ \frac{a - b}{b} = \frac{a}{b} - \frac{b}{b} ]

[ = \frac{a}{b} - 1 ]

Now, substitute ( \frac{a}{b} = \frac{c}{d} ):

[ \frac{a - b}{b} = \frac{c}{d} - 1 ]

[ = \frac{c}{d} - \frac{d}{d} ]

[ = \frac{c - d}{d} ]

Therefore, both ( \frac{a + b}{b} = \frac{c + d}{d} ) and ( \frac{a - b}{b} = \frac{c - d}{d} ) hold true, demonstrating the Denominator Addition/Subtraction Property of proportions.

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