Question

Is 24/40=4/7 a true proportion? Justify your answer

Answer 1

A proportion

$a\div b = c\div d$

is nothing but a comparison between two fractions: we can rewrite it as

$\dfrac{a}{b}=\dfrac{c}{d}$

So, we can multiply both sides by the two denominators b and d to get

$\dfrac{a}{b}=\dfrac{c}{d} \iff ad = bc$

In other words, a proportion is true if the product of the inner terms is the same as the product of the outer terms.

In your case, we have the check is the following:

$24 \div 40 = 4\div 7 \iff 24\cdot 7 = 40\cdot 7 \iff 168 = 280$

which is clearly false. So, 24:40 = 4:7 is not a true proportion. In fact, if we convert fractions into numbers, we have

$\dfrac{24}{40} = 0.6,\quad \dfrac{4}{7} = 0.\overline{571428}$

which makes even more clear that the proportion doesn't hold.