Question

The sum of two numbers is 55 and the HCF and LCM of these two numbers are 5 and 20 respectively. Find the sum of reciprocals of these numbers.

Answer 1

Hint: Here in the given question we need to find two numbers, whose sum is given and LCM and HCF is also given. Here we know that lowest common factor and highest common factor is the property of factors, a number and we need to solve accordingly.

Complete step by step answer:
To solve the given question, here we need to find two numbers, whose sum is given along with LCM and HCF is also given, let’s assume the two numbers and thus solve further with the given conditions, on solving we get:
\[ \Rightarrow numbers = a,b\]
Here we know the sum of to numbers as 55, we have:
\[ \Rightarrow a + b = 55...............eq1\]
Now we know the HCF as 5, we have:
\[ \Rightarrow HCF[a,b] = 5\]
Here the numbers can be written into the multiple of HCF and co-prime numbers, say “k” and “l”, we have:
\[ \Rightarrow a = 5k,\,b = 5l\]
Putting this value into equation one we get:
\[
   \Rightarrow 5k + 5l = 55 \\
   \Rightarrow k + l = \dfrac{{55}}{5} = 11.........eq2 \\
 \]
Again using LCM, as 120 we have:
\[ \Rightarrow LCM[a,b] = 120\]
We know that “k” and “l” are also multiple of the factors hence we can write:
\[
   \Rightarrow 5 \times k \times l = 120 \\
   \Rightarrow k \times l = \dfrac{{120}}{5} = 24........eq3 \\
 \]
From equation two and three we have:
\[
   \Rightarrow k \times (11 - k) = 24\,(\sin ce\,k + l = 11 \to l = 11 - k) \\
   \Rightarrow 11k - {k^2} - 24 = 0 \\
   \Rightarrow {k^2} - 11k + 24 = 0 \\
 \]
Here we need to solve quadratic equation we have:
\[
   \Rightarrow {k^2} - 11k + 24 = 0 \\
   \Rightarrow {k^2} - (8 + 3)k + 24 = 0 \\
   \Rightarrow {k^2} - 8k - 3k + 24 = 0 \\
   \Rightarrow k(k - 8) - 3(k - 8) = 0 \\
   \Rightarrow (k - 8)(k - 3) = 0 \\
 \]
Now for value of “l” we have:
\[
   \Rightarrow k + l = 11 \\
   \Rightarrow 8 + l = 11 \\
   \Rightarrow l = 3 \\
 \]
If we go for the other value of “k” then also the final answer will be the same.
Now we have:
\[
   \Rightarrow a = 5k,\,b = 5l \\
   \Rightarrow a = 5 \times 8 = 40,\,b = 5 \times 3 = 15 \\
 \]
Hence we got the two respective numbers here.

Note: Here in the given question, we need to find the two numbers, hence it is sure that either two equations will form or one quadratic equation will be formed, which will give us two values, and hence this needs to be solved accordingly.