Can ratios really be manipulated as fractions?In high-school Maths, we were taught that it was possible to manipulate ratios as fractions. For example, 1 : 7 = 3 : x 1 7 = 3 x x 7 = 3 x = 3 × 7 x = 21 My question relates to the second line of working, where we represent 1 : 7 as 1 7 . As I understand it, 1 : 7 s better represented as one part of 8 total parts, i.e. 1 8 My question, then, is this:Is there a simple way of showing (or better still, a rigorous proof!) that a ratio-equation of the form a : n = b : m can in all cases be rewritten as a n = b m ?

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Can ratios really be manipulated as fractions?In high-school Maths, we were taught that it was possible to manipulate ratios as fractions. For example,  1  :  7  =  3  :  x        1    7    =      3    x          x    7    =  3    x  =  3  ×  7    x  =  21  My question relates to the second line of working, where we represent   1  :  7 as       1    7  . As I understand it,   1  :  7 s better represented as one part of 8 total parts, i.e.       1    8   My question, then, is this:Is there a simple way of showing (or better still, a rigorous proof!) that a ratio-equation of the form   a  :  n  =  b  :  m  can in all cases be rewritten as       a    n    =      b    m  ?

rynosluv101swv2s · Accepted Answer

It does not actually matter; if you want to do this algebraically, consider writing  a  :  n  =  b  :  mas a fraction in the terms that you&#039;re thinking in, i.e.:      a          a      +      n        =      b          b      +      m      then, so long as neither a nor b is 0, we can apply the function   x  ↦      1    x   to both sides of the equation without affecting its truth, yielding            a      +      n        a    =            b      +      m        b  which reduces to  1  +      n    a    =  1  +      m    b        n    a    =      m    b  and applying   x  ↦      1    x   again to both sides, assuming neither n nor m is zero, gives:      a    n    =      b    m    .So, assuming none of the parts of the ratios are 0, the fraction representation will work just the same.You would expect this to be true since exactly what a ratio means is ambiguous; the ratio a:n certainly is best interpreted as &quot;a parts of one thing per b parts of another&quot; - and you can uniquely associate a few rational numbers to such an idea. For instance,        a    b   is the &quot;conversion&quot; ratio from b to a - i.e. if you multiply how much of the second thing you have by        a    b    you retrieve how much of the first thing you have. Your fraction,       a          a      +      b        represents how much of the total stuff between a and b is of the first thing. So, of course, you&#039;d expect either representation to be valid where well-defined (since we&#039;re only concerned that the two fractions are equal if and only if the ratios were).

Can ratios really be manipulated as fractions? In high-school Maths, we were taught that it was pos

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