Question

Find the two numbers whose A.M is \[25\] and GM is \[20\].

Answer 1

Hint: We are asked to find two numbers, their arithmetic mean and geometric mean are given. First recall the formulas for arithmetic and geometric mean then use these formulas to form two equations. There are two variables and two equations so, using these, find the two numbers.

Complete step by step solution:
Given, two numbers have \[{\text{A}}{\text{.M}} = 25\] and \[{\text{G}}{\text{.M}} = 20\].Let the two numbers be \[x\] and \[y\].
By A.M we mean arithmetic mean. Arithmetic mean can simply be called as average, it can be written as,
\[{\text{A}}{\text{.M}} = \dfrac{{\sum {{x_i}} }}{n}\], \[i = 1,2,3....n\]
where \[{x_i}\]represents the numbers of a set and \[n\]is the count of numbers.
Here, count of numbers is \[2\]. So the arithmetic mean will be,
\[{\text{A}}{\text{.M}} = \dfrac{{x + y}}{2}\]
Putting the value of \[{\text{A}}{\text{.M}}\] we get,
\[25 = \dfrac{{x + y}}{2}\]
\[ \Rightarrow x + y = 50\]
\[ \Rightarrow y = 50 - x\] (i)

By G.M we mean geometric mean. Geometric mean of a set of \[n\] numbers is given by,
\[{\text{G}}{\text{.M}} = {\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)^{\dfrac{1}{n}}}\]
where \[{x_i}\]represents the numbers of the set.
Here, there are two numbers so, geometric mean will be,
\[{\text{G}}{\text{.M}} = {\left( {xy} \right)^{\dfrac{1}{2}}}\]
Putting the value of \[{\text{G}}{\text{.M}}\] we get,
\[{\text{20}} = {\left( {xy} \right)^{\dfrac{1}{2}}}\]
\[ \Rightarrow {\text{2}}{{\text{0}}^2} = {\left( {xy} \right)^{\dfrac{2}{2}}}\]
\[ \Rightarrow 400 = xy\]
\[ \Rightarrow xy = 400\] (ii)
Substituting the value of \[y\] from equation (i) in (ii), we get
\[x\left( {50 - x} \right) = 400\]
\[ \Rightarrow 50x - {x^2} = 400\]
\[ \Rightarrow {x^2} - 50x + 400 = 0\]
\[ \Rightarrow {x^2} - 10x - 40x + 400 = 0\]
\[ \Rightarrow x\left( {x - 10} \right) - 40\left( {x - 10} \right) = 0\]
\[ \Rightarrow \left( {x - 10} \right)\left( {x - 40} \right) = 0\]
\[ \Rightarrow x = 10\,{\text{or}}\,x - 40\]
We get two values for \[x\], now we find the value of \[y\] for both values of \[x\].
Substituting \[x = 10\] in equation (i), we get
\[y = 50 - 10\]
\[ \Rightarrow y = 40\]
And substituting \[x - 40\] in equation (i), we get
\[y = 50 - 40\]
\[ \therefore y = 10\]
Therefore the two sets we get for \[(x,y)\] are \[(10,40)\] and \[(40,10)\].

Hence, the two numbers are \[10\] and \[40\].

Note: Most of the time students get confused between arithmetic mean and geometric mean, so carefully remember their formulas. For arithmetic mean we use the formula \[{\text{A}}{\text{.M}} = \dfrac{{\sum {{x_i}} }}{n}\] and for geometric mean we use the formula \[{\text{G}}{\text{.M}} = {\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)^{\dfrac{1}{n}}}\]. Also, in such types of questions where we need to find variables by forming equations, try to form as many equations as the number of variables only then you can find the values of the variables.