Question

The sum of two number is 55 and hcf anf lcm is 5 and 120 respectively. Find the sum of reciprocals of number

Answer 1

Let the numbers be A and B
H.C.F = 5
L.C.M = 120
120 = 2* 60 = 2* 2*30 = 2*2*2*15 = 2*2*2*3*5
Since the H.C.F is '5', both the numbers contain '5' as a common factor.
Hence, from the L.C.M: 2*2*2*3 is to be distributed among the two numbers.
A = 5*x = 5*2*2*2 = 40
B = 5*y = 5*3 = 15
A+B = 55 = 40+15
Hence the numbers are 40,15.
Sum of reciprocals = 1/40 + 1/15 = (3+8)/120 = 11/120 or 0.092

Alternatively:
A+B = 55.... B = 55-A
L.C.M * H.C.F = A*B
120* 5 = A*(55-A)
600 = 55A - A^2
A^2 - 55A +600 = 0
A^2 - 15A - 40A + 600 = 0
A(A-15) -40(A - 15) = 0
(A-40)*(A-15) = 0
A= 40 or 15
Then B = 15 or 40
Hence numbers are  40 ,15
Sum of reciprocals: 1/40 + 1/15 = 3+8/120 = 11/120 = 0.092

Hope it helps.