Testwiki:Reference desk/Archives/Mathematics/2022 October 3

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October 3

True or not?

Is it true that if (a/b) = (c/d) where a>c and b>d that ab/cd = a²? I have tried with lot of examples and it came true each time. ExclusiveEditor _{Notify Me!} 09:01, 3 October 2022 (UTC)

Try a=b=3, c=d=2. Then ab/(cd) = 9/4, which is not a². Staecker (talk) 10:33, 3 October 2022 (UTC)

More generally: $a=\frac{bc}{d}⟹a^2=\frac{abc}{d}$, so c is in the numerator, not the denominator. The inequalities do not enter at all. --Wrongfilter (talk) 11:57, 3 October 2022 (UTC)

So equality holds if and only if $|c|=1.$ For example, when $a=3,b=21,c=1,d=7,$ we have

$\frac{a}{b}=\frac{3}{21}=\frac{3\times 1}{3\times 7}=\frac{1}{7}=\frac{c}{d}.$

Then

$\frac{ab}{cd}=\frac{3\times 21}{1\times 7}=\frac{63}{7}=\frac{7\times 9}{7\times 1}=\frac{9}{1}=9=3^2=a^2.$

 --Lambiam 13:39, 3 October 2022 (UTC)