The ratio between the ages of Ram and Mohan is $4 : 5$ and that between the ages of Ram and Anil is $5 : 6$. If the sum of the ages of the three is 69 years, what is Mohan's age?

### Concept & Logic The core concept is unifying multiple independent ratios into a single continuous ratio by finding the Least Common Multiple (LCM) of the common linking entity. Once a combined ratio is established, you can divide the total sum proportionally to find individual values. ### Step-by-Step Solution * **Given:** * Ram : Mohan = $4 : 5$ * Ram : Anil = $5 : 6$ * Sum of ages (Ram + Mohan + Anil) = $69$ years. * **Calculation / Deduction:** * Ram is the common entity in both ratios. Let's express both ratios with Ram as the connecting link. * Mohan : Ram = $5 : 4$ * Ram : Anil = $5 : 6$ * To combine them into Mohan : Ram : Anil, equalize Ram's value in both ratios. The LCM of $4$ and $5$ is $20$. * Multiply the first ratio by $5$: Mohan : Ram = $(5 \times 5) : (4 \times 5) = 25 : 20$ * Multiply the second ratio by $4$: Ram : Anil = $(5 \times 4) : (6 \times 4) = 20 : 24$ * Combined Ratio = Mohan : Ram : Anil = $25 : 20 : 24$ * Let their ages be $25x$, $20x$, and $24x$. * Sum of the ages: * $$25x + 20x + 24x = 69$$ * $$69x = 69$$ * $$x = 1$$ * We need to find Mohan's age, which is $25x$. * Mohan's age = $25 \times 1 = 25$ years. ### Exam Strategy & Shortcut Skip writing $x$. Simply sum the parts of the combined ratio: $25 + 20 + 24 = 69$ parts. Since $69$ parts perfectly equals $69$ years, $1$ part = $1$ year. Mohan represents $25$ parts, so he is directly $25$ years old. This observation allows you to solve the question mentally in seconds once the combined ratio is formed. ### Common Pitfall A frequent mistake is directly combining the numbers without equalizing the common linking variable (Ram). Students might incorrectly assume the ratio is $4 : 5 : 6$, leading to a sum of $15$ parts, which completely breaks the calculation and leads to an incorrect age. Always ensure the common link has the exact same value before combining. ### Final Answer **Therefore, the correct answer is 25 years.**