Question

The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller.

Answer 1

Hint: We will first let the two numbers be \[a\] and \[b\]. Use the arithmetic mean and geometric mean to find the relations between two numbers. Then according to the question, we will form the expression and simplify it until we get the reduced form of the expression. Then we will let one term in the expression as a variable \[x\]and simplify it to find the value of \[x\] which will give us the ratio of greater to smaller.

Complete step by step answer:

We will start by letting the two distinct numbers as \[a\] and \[b\].
Now, as we know the general form of arithmetic mean and geometric mean so, we will apply it here and find the relation between two distinct numbers.
Thus, we get,
\[ \Rightarrow A.P. = \dfrac{{a + b}}{2}\]
And \[G.P. = \sqrt {ab} \]
Now, as given in the question that arithmetic mean is twice of geometric mean so, we will form an expression and determine the relationship between the two.
Thus, we have,
\[
   \Rightarrow \dfrac{{a + b}}{2} = 2\sqrt {ab} \\
   \Rightarrow a + b = 4\sqrt {ab} \\
 \]
Now, we will square on both sides of the equation,
\[
   \Rightarrow {\left( {a + b} \right)^2} = 16ab \\
   \Rightarrow {a^2} + {b^2} + 2ab = 16ab \\
   \Rightarrow {a^2} + {b^2} = 14ab \\
   \Rightarrow \dfrac{a}{b} + \dfrac{b}{a} = 14 \\
 \]
Next, we will let the term \[\dfrac{b}{a} = x\] then the term \[\dfrac{a}{b} = \dfrac{1}{x}\], we get,
\[
   \Rightarrow \dfrac{1}{x} + x = 14 \\
   \Rightarrow {x^2} + 1 = 14x \\
   \Rightarrow {x^2} - 14x + 1 = 0 \\
 \]
Now, we will use the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] to determine the roots of the equation where \[a = 1,b = - 14\] and \[c = 1\].
Thus, we get,
\[
   \Rightarrow x = \dfrac{{ - \left( { - 14} \right) \pm \sqrt {{{\left( {14} \right)}^2} - 4\left( 1 \right)\left( 1 \right)} }}{{2\left( 1 \right)}} \\
   \Rightarrow x = \dfrac{{14 \pm \sqrt {192} }}{2} \\
   \Rightarrow x = \dfrac{{14 \pm 2\sqrt {48} }}{2} \\
   \Rightarrow x = 7 \pm \sqrt {48} \\
 \]
The negative value of \[x\] can be ignored thus, we will consider the positive value only.
Thus, the ratio of greater to smaller is given by \[\dfrac{b}{a} = x\].
Hence, we get the ratio as \[7 + 4\sqrt 3 \].

Note: We have ignored the negative value of \[x\] and have considered the positive value of \[x\] only. The arithmetic mean is given by \[\dfrac{{{x_1} + {x_2}}}{2}\] and geometric mean is given by \[\sqrt {{x_1}{x_2}} \] so, we need to remember this. Letting the term \[\dfrac{b}{a}\] as \[x\] makes the calculation easier and we have easily determined the ratio of greater to smaller by just finding the value of \[x\]. Substitute \[\dfrac{b}{a} = x\] and \[\dfrac{a}{b} = \dfrac{1}{x}\] or vice versa to find the values of \[x\].