Question

Find the mean proportion between  $64$  and  $144$.

Answer 1

Hint: The question asks about the mean proportion between two numbers. If we have the mean proportion of two numbers  $ a,b $  as  $ x $  then it has a property that the ratio between  $ a $  and  $ x $  is the same as  $ x $  and $ b $  So taking this in mind we will use the formula for generating the mean proportion of the given numbers which will serve as the answer of this question.

Formula used:

If we have the mean proportion of two numbers  $ a,b $  as  $ x $  then,

 $ \dfrac{a}{x} = \dfrac{x}{b} $  where,  $ a \leqslant x \leqslant b $  

Complete answer:

The given numbers are  $ 64 $  and  $ 144 $ 

If we put  $ a = 64,b = 144 $  in the mean proportion formula we have,

 $ \dfrac{{64}}{x} = \dfrac{x}{{144}} $ 

 $  \Rightarrow {x^2} = 64 \times 144 $ 

Now the squares of  $ 8 $  and  $ 12 $  are  $ 64 $  and  $ 144 $  respectively so,

 $  \Rightarrow {x^2} = {8^2} \times {12^2} = {(8 \times 12)^2} $ 

The fact that  $ {\alpha ^2} = {\beta ^2} \Rightarrow \alpha  = \beta  $  gives us 

 $ x = 8 \times 12 $ 

The product of  $ 8 $  and  $ 12 $  gives us  $ 96 $  so,

 $ x = 96 $ 

Therefore, the mean proportion of the numbers given to us is  $ 96 $  

So, the correct answer is “ $ 96 $ ”.

Additional information:

The ratio between two numbers  $ a,b $  is the fractional number  $ \dfrac{a}{b} $  which is denoted as  $ a:b $  and pronounced as  $ a $  is to  $ b $ . Two ratios  $ a:b $  and  $ c:d $  are said to be in proportion if  $ \dfrac{a}{b} = \dfrac{c}{d} $  or in words the ratios are the same. This is also written as  $ a:b::c:d $  and is pronounced as  $ a $  is to  $ b $  proportion to  $ c $  is to  $ d $   .

Note:  While calculating the mean proportion  $ x $  always try to simplify the numbers to small factors so that you can easily find the root of that number. Multiplying the numbers will result in a larger number whose root calculation then will be a messy work.