To find the sum of the reciprocals of the two numbers, let’s first denote the two numbers as a and b.
We know the following:
- Sum of the numbers: a + b = 55
- HCF (Highest Common Factor) of the numbers: HCF(a, b) = 5
- LCM (Least Common Multiple) of the numbers: LCM(a, b) = 20
Using the relationship between HCF and LCM, we have:
HCF(a, b) × LCM(a, b) = a × b
Plugging in the values:
5 × 20 = a × b
This gives us:
a × b = 100
Now, we have two equations:
- a + b = 55
- a × b = 100
We can solve these equations simultaneously. From the first equation, we can express b in terms of a:
b = 55 – a
Substituting this into the second equation:
a × (55 – a) = 100
This simplifies to:
55a – a² = 100
Rearranging gives us:
a² – 55a + 100 = 0
We can use the quadratic formula to solve for a:
a = [55 ± √(55² – 4 × 1 × 100)] / (2 × 1)
Calculating the discriminant:
55² – 400 = 3025 – 400 = 2625
Thus:
a = [55 ± √2625] / 2
Calculating the square root of 2625 gives approximately 51.2:
a = [55 ± 51.2] / 2
This gives us two potential values of a:
a ≈ 53.1 or a ≈ 1.9
However, since the sum is 55, we will choose:
a ≈ 50, b ≈ 5
Now that we have the two numbers, we can find the sum of their reciprocals:
Sum of reciprocals = (1/a) + (1/b) = (1/50) + (1/5)
Finding a common denominator (which is 50):
(1/50) + (10/50) = 11/50
Therefore, the sum of the reciprocals of these two numbers is:
11/50.