If the sum of two numbers is 27 and their HCF and LCM are 3 and 60 respectively, then the sum of the reciprocal of the two numbers is:

Let's denote the two numbers as a and b.

Step 1: We know that the sum of the two numbers is 27, so we can write this as an equation: a + b = 27.

Step 2: We also know that the highest common factor (HCF) is 3. This means that both a and b are divisible by 3. Let's denote a = 3p and b = 3q, where p and q are integers.

Step 3: Substituting a and b from step 2 into the equation from step 1, we get: 3p + 3q = 27. Simplifying this, we get: p + q = 9.

Step 4: We also know that the least common multiple (LCM) is 60. The product of two numbers is equal to the product of their HCF and LCM. So, we can write this as an equation: a * b = 3 * 60.

Step 5: Substituting a and b from step 2 into the equation from step 4, we get: 3p * 3q = 180. Simplifying this, we get: p * q = 20.

Step 6: Now, we need to find the sum of the reciprocals of the two numbers. The reciprocal of a number is 1 divided by that number. So, the sum of the reciprocals of a and b is 1/a + 1/b.

Step 7: Substituting a and b from step 2 into the equation from step 6, we get: 1/(3p) + 1/(3q) = 1/p + 1/q.

Step 8: We know from step 3 that p + q = 9 and from step 5 that p * q = 20. So, the sum of the reciprocals of a and b is 1/p + 1/q = (p + q) / (p * q) = 9 / 20.

So, the sum of the reciprocals of the two numbers is 9/20.