Duplicating the operators?

Algebra Level 3

Given that a b × c d = a × c b × d \dfrac ab \times \dfrac cd = \dfrac {a\times c}{b\times d} . Is it also true that a b ÷ c d = a ÷ c b ÷ d \dfrac ab \div \dfrac cd = \dfrac {a\div c}{b\div d} ?

Clarification: None of a , b , c , d a,b,c,d are equal to 0.

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Zach Abueg
Apr 18, 2017

a b ÷ c d \displaystyle \frac {\color{#20A900}{a}}{\color{#CEBB00}{b}} ÷ \frac {\color{#D61F06}{c}}{\color{#3D99F6}{d}}

= a b × d c \displaystyle = \frac {\color{#20A900}{a}}{\color{#CEBB00}{b}} \times \frac {\color{#3D99F6}{d}}{\color{#D61F06}{c}} , where b , c ≠ 0 {\color{#CEBB00}{b}}, {\color{#D61F06}{c}} \neq 0

= a c × d b \displaystyle = \frac {\color{#20A900}{a}}{\color{#D61F06}{c}} \times \frac {\color{#3D99F6}{d}}{\color{#CEBB00}{b}}

= a c ÷ b d \displaystyle = \frac {\color{#20A900}{a}}{\color{#D61F06}{c}} ÷ \frac {\color{#CEBB00}{b}}{\color{#3D99F6}{d}}

= a ÷ c b ÷ d \displaystyle = \frac {{\color{#20A900}{a}} ÷ {\color{#D61F06}{c}}}{{\color{#CEBB00}{b}} ÷ {\color{#3D99F6}{d}}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Pi Han Goh what if c=0?

Harsh Shrivastava - 4 years, 1 month ago

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In that same vein what if b=0 or d=0?

Anthony Holm - 4 years, 1 month ago

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Whoops. Fixed!

Thanks for the color solution Zach

Pi Han Goh - 4 years, 1 month ago

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