Guess the next number, \[42:20::64: ? \]?

Hint: The ratio concept is used to compare two amounts, but the proportion concept is used to establish equivalency between specified ratios. The ratio of two numbers a and b, which is written as \[\dfrac{a}{b}\] or \[a:b\] . While \[a:b::c:d\] or \[\dfrac{a}{b} = \dfrac{c}{d}\] is the relationship between two ratios of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] , the proportion is the relationship between two ratios of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] .

Complete step-by-step solution:
 We can write ratio and proportion in the form of \[\dfrac{a}{b} = \dfrac{c}{d}\].
We can also write it as $a \times d = c \times b$ .
Now, we have given \[42:20::64:\]
we have to find the missing number.
We assumed the missing number would be x.
 Above ratio and proportion can be written as
$\dfrac{{42}}{{20}} = \dfrac{{64}}{x}$
We cross multiply it
$42 \times x = 64 \times 20$
$x = \dfrac{{64 \times 20}}{{42}}$
On further solving
$x = \dfrac{{640}}{{21}}$
So, the value of the missing number is $\dfrac{{640}}{{21}}$ .

Note: Whatever way a ratio is written, we must be able to be simplified down to the smallest whole numbers possible, just like any fraction. This can be accomplished by calculating the greatest common factor between the two numbers and then dividing them accordingly.