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
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
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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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Question 11 Ratio and Proportion Exercise 7.2
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Answer:
Consider a and b as the two numbers
It is given that 28 is the mean proportional
a: 28 :: 28: b
We get
ab = 28^{2} = 784
Here a = 784/b …… (1)
We know that 224 is the third proportional
a: b :: b: 224
So we get
\mathrm{b}^{2} = 224a ….. (2)
Now by substituting the value of a in equation (2)
\mathrm{b}^{2} = 224 × 784/b
So we get
\begin{array}{l} b^{3}=224 \times 784 \\ b^{3}=175616=56^{3} \end{array}
b = 56
By substituting the value of b in equation (1)
a = 784/56 = 14
Therefore, 14 and 56 are the two numbers.
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Find two numbers such that the mean proportional between them is 28 and the third proportional to them is 224.
Let the two numbers are a and b.∵ 28 is the mean proportional∵ a : 28 : : 28 : b
∴ ab = (28)2 = 784
⇒ a = `(784)/b` ...(i)∵ 224 is the third proportional∴ a : b : : b : 224⇒ b2 = 224a ...(ii)
Substituting the value of a in (ii)b2 = `24 xx (784)/b`
⇒ b3 = 224 x 784
⇒ b2 = 175616 = (56)3∴ b = 56Now substituting the value of b in (i)a = `(784)/(56)` = 14
Hence numbers are 14, 56.
Concept: Concept of Proportion
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