Question

# The geometric mean of two numbers exceeds by $$12$$ the smaller of the numbers, and the arithmetic mean of the same numbers is smaller by $$24$$ than the larger of the numbers. Find the numbers.

Solution

## Let the two numbers be $$a$$ and $$b$$$$GM=\sqrt { ab }$$ and $$AM=\cfrac { a+b }{ 2 }$$$$\sqrt { ab } =12+a\quad -(i)\\ \cfrac { a+b }{ 2 } =b-24\quad -(ii)\\ a+b=2b-48\\ b=a+48$$Putting value of $$b$$ in $$(i)$$$$\sqrt { a(a+48) } =12+a$$Squaring both sides$${ a }^{ 2 }+48a={ a }^{ 2 }+24a+144\\ \Rightarrow 24a=144\\ a=\cfrac { 144 }{ 24 } =6\\ So,b=48+6\\ =54$$Numbers are $$54\quad \&\quad 6$$Mathematics

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