Question

The mean proportional between two numbers is 28 and their third proportional is 224. Find the two numbers.

Hint: In algebra, a mean proportional is a number that comes between two numbers . We used a mean proportional formula which is $\sqrt {ab} =$mean proportional. And the formula of third proportional $ac = {b^2}$.

Mean proportional of two number is given in the question but numbers are not
So first we have to let a, b are the required numbers.
Formula of mean proportional
$\sqrt {ab} =$mean proportional
$\sqrt {ab} = 28$
Now take the square both side
${(\sqrt {ab} )^2} = {28^2}$
$ab = 28.28$
$ab = 784$
So we can find the value of number a
$a = \dfrac{{784}}{b}$ ……… equation (1)
We have the third proportional given in the question that is 224
The formula of third proportional
$ac = {b^2}$
$c = \dfrac{{{b^2}}}{a}$
Put the values
Here c is the third proportional
$224 = \dfrac{{{b^2}}}{a}$
Now put the value of a
$224 = \dfrac{{{b^2}}}{{\dfrac{{784}}{b}}}$
Simplifying the equation
$224 = {b^2}.\dfrac{b}{{784}}$
Multiply the R.H.S
$224 = \dfrac{{{b^3}}}{{784}}$
Apply the cross-multiplication method
${b^3} = 224.784$
${b^3} =$175616
$b{ = ^3}\sqrt {175616}$
$b = 56$
So here we the second number
We can find the first number a with the help of equation (1)
$a = \dfrac{{784}}{b}$
$a = \dfrac{{784}}{{56}}$
$a = 14$
Hence, we have both the numbers
First is 14 and the second number is 56.

Note: In this type of question the most important point is calculation, always do the calculation carefully. We can check our answer by using a mean proportional method
Mean proportional =$\sqrt {a.b}$
We have a = 14
And b = 56
After putting the values, we get
= $\sqrt {14.56}$
$= \sqrt {784}$
$= 28$
So here we get mean proportional that is already given in the question
Our answer is correct by alternative checking method.