The sum of the present ages of Rajesh, Suresh, Mahesh and Dinesh is 76 years. Seven years from now, the ratio of their ages will be 7 : 6 : 5 : 8 respectively. What is the present age of Mahesh?

Introduction / Context:
This problem deals with four people whose combined present ages are known. We are given the ratio of their ages seven years into the future and asked to determine the present age of one of them. The question is a good example of using ratios along with total sum information to solve for individual ages.

Given Data / Assumptions:

• Present total of the ages of Rajesh, Suresh, Mahesh and Dinesh is 76 years.
• After 7 years, their ages will be in the ratio 7 : 6 : 5 : 8 respectively.
• We must find the present age of Mahesh.
• All ages are in years and positive.

Concept / Approach:
The ratio provided applies to their ages in the future, not now. We start by representing their ages after 7 years as multiples of a common factor. The sum of those future ages can be related to the sum of their present ages. Once we find the common factor, we can determine each person future age and then subtract 7 to obtain the present ages. From there, Mahesh present age is read off directly.

Step-by-Step Solution:
Let the ages of Rajesh, Suresh, Mahesh and Dinesh after 7 years be 7 * k, 6 * k, 5 * k and 8 * k respectively.Then the sum of their ages after 7 years is (7 + 6 + 5 + 8) * k = 26 * k.The sum of their present ages is 76 years.In 7 years, each person will be 7 years older, so the total increase in sum of ages will be 4 * 7 = 28 years.Therefore, sum of their ages after 7 years is 76 + 28 = 104 years.So we have 26 * k = 104.Solve for k: k = 104 / 26 = 4.Thus, ages after 7 years are:Rajesh: 7 * 4 = 28 years.Suresh: 6 * 4 = 24 years.Mahesh: 5 * 4 = 20 years.Dinesh: 8 * 4 = 32 years.Present ages are each of those minus 7:Mahesh present age = 20 - 7 = 13 years.

Verification / Alternative check:
Present ages: Rajesh = 28 - 7 = 21, Suresh = 24 - 7 = 17, Mahesh = 13, Dinesh = 32 - 7 = 25.Check sum: 21 + 17 + 13 + 25 = 76 years, which matches the given total.After 7 years, their ages are 28, 24, 20 and 32, and the ratio 28 : 24 : 20 : 32 simplifies to 7 : 6 : 5 : 8 by dividing each term by 4.

Why Other Options Are Wrong:
17 years: This would correspond to a different value of k and would not maintain both the future ratio and the total sum.14 years or 16 years: These also lead to inconsistencies in either the total sum or the required future ratio.

Common Pitfalls:
A common error is to apply the ratio directly to the present ages rather than to the future ages as specified.Some students forget that with four people, the total increase in sum over 7 years is 28 years, not just 7 years.There can also be mistakes in simplifying or computing the sum of the ratio parts.

Final Answer:
The present age of Mahesh is 13 years.