The ages of three friends A, B and C satisfy the following: A : B = 3 : 5 and B : C = 3 : 5. If the sum of their ages is 147 years, what is the present age of B?

Introduction / Context:
This is a typical age and ratio problem where we are given pairwise ratios between three people and the total of their ages. The task is to determine the age of one person, here friend B. To do this, we must convert the pairwise ratios into a single three-term ratio that involves A, B and C together, and then use the total age to find the exact values.

Given Data / Assumptions:
• Ratio A : B = 3 : 5. • Ratio B : C = 3 : 5. • A + B + C = 147 years. • Ages are positive real numbers.

Concept / Approach:
We first express the ratios in terms of a common variable so that B is consistent in both ratios. From A : B = 3 : 5, we set A = 3x and B = 5x. From B : C = 3 : 5, we set B = 3y and C = 5y. By equating the two expressions for B, we find a relationship between x and y. We then derive a combined ratio A : B : C and use the sum 147 to determine the individual ages.

Step-by-Step Solution:
Step 1: From A : B = 3 : 5, write A = 3x and B = 5x. Step 2: From B : C = 3 : 5, write B = 3y and C = 5y. Step 3: Since both expressions equal B, equate them: 5x = 3y. Step 4: Choose x and y so that this holds; for example, x = 3k and y = 5k. Step 5: Then A = 3x = 3 * 3k = 9k, B = 5x = 15k, C = 5y = 5 * 5k = 25k. Step 6: So the combined ratio A : B : C = 9 : 15 : 25. Step 7: Sum of ratio parts = 9 + 15 + 25 = 49. Step 8: Let the common multiplying factor be t, so total age = 49t = 147. Step 9: Solve for t: t = 147 / 49 = 3. Step 10: Actual ages are A = 9 * 3 = 27, B = 15 * 3 = 45, C = 25 * 3 = 75.

Verification / Alternative check:
Check the given conditions: A + B + C = 27 + 45 + 75 = 147, which matches the total. Also, A : B = 27 : 45 = 3 : 5 and B : C = 45 : 75 = 3 : 5 after dividing both terms in each pair by 9 and 15 respectively. Therefore, all the ratios and the sum are satisfied by these values.

Why Other Options Are Wrong:
• 27 and 75 are the ages of A and C, not B. • 49 does not appear as any of the derived ages when the ratios and total are correctly applied.

Common Pitfalls:
A common mistake is to add or average the given ratios directly instead of forming a consistent three-term ratio. Another error is to miscalculate the sum of the ratio parts or misapply the total sum, leading to incorrect scaling factors. Always verify that all original ratio conditions are satisfied by the final numbers.

Final Answer:
The present age of friend B is 45 Years.