Question

Find two positive numbers whose difference is 12 an whose A.M. exceeds the G.M. by 2.

Answer 1

Let the required positive numbers be a and b, where `a gt b`.
The, `a-b=12` ...(i)
Also, `AM-GM=2`
`rArr (a+b)/2-sqrt(ab)=2`
`rArr a+b-2sqrt(ab)=4`
`rArr (sqrt(a)-sqrt(b))^(2)=4`
`rArr sqrt(a)-sqrt(b)=2`. ...(ii)
Now, `a-b=12` [from (i)]
`rArr (sqrt(a)+sqrt(b))(sqrt(a)-sqrt(b))=12`
`rArr (sqrt(a)+sqrt(b))xx2=12 rArr sqrt(a)+sqrt(b)=6` ...(iii) [using (ii)]
On solving (ii) and (iii), we get `sqrt(a)=4` and `sqrt(b)=2`.
`:. a=16` and `b=4`
Hence, the required numbers are 16 and 4.