Respuesta :
The ratio of two numbers a:b is 9:1.
What is arithmetic mean?
The arithmetic mean (AM) for evenly distributed numbers is equal to the middlemost number. Arithmetic Mean is the mean or average of the set of numbers which is computed by adding all the terms in the set of numbers and dividing the sum by a total number of terms.
What is geometric mean?
is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. when numbers in the series are not independent of each other or if numbers tend to make large fluctuations.
For the given situation,
The ratio of the arithmetic mean to the geometric mean is 5:3.
Let the numbers be a, b.
Then arithmetic mean = [tex]\frac{a+b}{2}[/tex]
The geometric mean = √ab
⇒ [tex]\frac{\frac{a+b}{2} }{\sqrt{ab} } = \frac{5}{3}[/tex]
Divide both sides by b,
⇒ [tex]\frac{\frac{a+b}{2b} }{\frac{\sqrt{ab} }{b} }= \frac{5}{3}[/tex]
⇒ [tex]\frac{\frac{a}{b}+1 }{\frac{\sqrt{a} }{\sqrt{b} } } =\frac{10}{3}[/tex]
⇒ [tex]\frac{a}{b} +1=\frac{10}{3}\sqrt{\frac{a}{b} }[/tex]
Let us consider [tex]\frac{a}{b}=x^{2}[/tex] , [tex]a > b[/tex]
⇒ [tex]x^{2} +1=\frac{10}{3} \sqrt{x^{2} }[/tex]
⇒ [tex]3x^{2} +3=10x[/tex]
⇒ [tex]3x^{2} -10x+3=0[/tex]
⇒ [tex]3x^{2} -9x-x+3=0[/tex]
⇒ [tex](3x-1)(x-3)=0[/tex]
∴ [tex]x=3[/tex] or [tex]x=\frac{1}{3}[/tex]
⇒ [tex]x^{2} =9[/tex] or [tex]x^{2} =\frac{1}{9}[/tex]
For [tex]x^{2} ,a > b[/tex] so [tex]x^{2} =9[/tex]
⇒ [tex]\frac{a}{b}=9[/tex]
⇒ [tex]\frac{a}{b} = \frac{9}{1}[/tex]
Hence we can conclude that the ratio of two numbers a:b is 9:1.
Learn more about arithmetic mean and geometric mean here
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