Question

If the AM of two numbers is greater than GM by 2 and the ratio of number is 4:1 then numbers are

A) 4,1

B) 12, 3 

C) 16,4 

D) None of the above 

Answer 1

Hint: To solve this question, we would first define AM and GM. If two numbers are a and b then \[\text{AM=}\dfrac{a+b}{2}\text{ and GM=}\sqrt{ab}.\] Arithmetic mean is the average of two numbers and GM or geometric mean is the ${{n}^{th}}$ root of product of all terms in the geometric sequence.

Complete step-by-step solution:

AM- Arithmetic mean is the mean or average of the set of numbers which is completed by adding all the terms in the set of numbers and dividing the sum by the total number of terms.

Let 'a' and 'b' be two numbers then,

\[\text{AM=}\dfrac{a+b}{2}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)}\]

GM- Geometric mean is the mean value or the central term in the set of numbers in geometric progression. Geometric mean of a geometric sequence with n terms is computed as ${{n}^{th}}$ root of the product of all the terms in sequence taken together.

Let 'a' and 'b' be two numbers then,

\[\text{GM=}\sqrt{ab}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)}\]

Given that, the AM of two numbers is greater than the GM by 2.

\[\Rightarrow \text{AM=GM+2}\]

Substituting equation (i) and (ii), we get:

\[\Rightarrow \dfrac{a+b}{2}\text{=}\sqrt{ab}+2\]

Taking LCM and cross multiplying, we get:

\[\Rightarrow a+b\text{=2}\sqrt{ab}+4\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (iii)}\]

We are given that the ratio is 4:1, then consider a number as $x$ as multiple for the ratio, then the first number is $4x$ and the other number is $x$.

Substituting $a = 4x$ and $b = x$ in equation (iii), we get:

$ \Rightarrow \left( 4x+x \right)=2\sqrt{\left( 4x \right)x}+4 $

$ \Rightarrow 5x=2\times 2x+4 $

$ \Rightarrow 5x-4x=4 $

$ \Rightarrow x=4 $

So, we have a value of $x = 4$.

Then, the first number is $4x$.

Substituting the value of $x = 4$ we have \[4x=4\times 4=16\] and the other number is \[\text{x }=\text{ 4}\]

So, two numbers are $14, 4$ which is option C.

Note: The possibility of error in this question can be at the point where the student has to assume a number as a multiple of x to get the two numbers. It is given in the question that the ratio is 4:1. So, we can easily assume 4x and x as the two numbers. Here, assuming 'x' is important, do not go for directly taking 4 and 1 as the number that would be wrong.