Find two numbers such that the mean proportional between them is 28 and proportional to them is 224.

Consider a and b as the two numbers

It is given that \[28\] is the mean proportional

\[a:\text{ }28\text{ }::\text{ }28:\text{ }b\]

We get

\[ab\text{ }=\text{ }{{28}^{2}}~=\text{ }784\]

Here \[a\text{ }=\text{ }784/b\]…… (1)

We know that \[224\] is the third proportional

\[a:\text{ }b\text{ }::\text{ }b:\text{ }224\]

So we get

\[{{b}^{2}}~=\text{ }224a\]….. (2)

Now by substituting the value of a in equation (2)

\[{{b}^{2}}~=\text{ }224\text{ }\times \text{ }784/b\]

So we get

\[\begin{array}{*{35}{l}}

{{b}^{3}}~=\text{ }224\text{ }\times \text{ }784 \\

{{b}^{3}}~=\text{ }175616\text{ }=\text{ }{{56}^{3}} \\

b\text{ }=\text{ }56 \\

\end{array}\]

By substituting the value of b in equation (1)

\[a\text{ }=\text{ }784/56\text{ }=\text{ }14\]

Therefore, \[14\] and \[56\] are the two numbers.

Answer

Verified

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 $ac = {b^2}$.

Complete step-by-step answer:

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 224The 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} $$b = 56$So here we the second numberWe can find the first number a with the help of equation (1)$a = \dfrac{{784}}{b}$$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 = 14And b = 56After 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.

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The mean proportional between two numbers is 28 and their third proportional to them is 224 . The two numbers are

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