Finding Sum of Inverses Using HCF and LCM

This solution explains how to calculate the sum of the inverses (reciprocals) of two numbers when their sum, Highest Common Factor (HCF), and Least Common Multiple (LCM) are known.

Given Information

• Let the two numbers be $a$ and $b$.
• Sum of the numbers: $a + b = 36$
• Highest Common Factor (HCF): $HCF(a, b) = 3$
• Least Common Multiple (LCM): $LCM(a, b) = 105$

Solution Steps

1. Recall the formula for the sum of inverses: $1/a + 1/b$. Find a common denominator to combine the fractions.

   $1/a + 1/b = (b + a) / (a * b)$

2. Use the property that the product of two numbers is equal to the product of their HCF and LCM.

   $a * b = HCF(a, b) * LCM(a, b)$

3. Substitute the given HCF and LCM values to find the product of the numbers ($a * b$).

   $a * b = 3 * 105$

   $a * b = 315$

4. Now substitute the values of ($a + b$) and ($a * b$) into the formula for the sum of inverses derived in Step 1.

   $Sum of inverses = (a + b) / (a * b)$

   $Sum of inverses = 36 / 315$

5. Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 9.

   $Sum of inverses = 36 \div 9 / 315 \div 9$

   $Sum of inverses = 4 / 35$

Therefore, the sum of the inverses of the two numbers is $4/35$.